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Controllability Aspects of The

Lindblad-Kossakowski Master

Equation

A Lie-Theoretical Approach

Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades derBayerischen Julius-Maximilians-Universitat Wurzburg

vorgelegt von

Indra Kurniawan

aus

Surabaya, Indonesien

Wurzburg, Dezember 2009

Eingereicht am: 07.01.2010

bei der Fakultat fur Mathematik und Informatikder Bayerischen Julius-Maximilians-Universitat Wurzburg

1. Gutachter: Prof. Dr. Uwe Helmke

2. Gutachter: Prof. Dr. Lars Grune

Tag der mundlichen Prufung: 17.06.2010

revised version: June 22, 2010

Dedicated to my parents

Ayahanda Sudjito and Ibunda Tuti Mudajati

my children

Ihsan, Ilham and Hilya

and my beloved wife

Emil

In the name of God,

the most Merciful and Compassionate.

Declaration

This thesis is an account of research carried out between April 2006 and August2009 at the Faculty of Mathematics and Computer Science, University of Wurzburg,Germany, under the framework of the International Doctorate Program in Engineeringand Computer Sciences : “Identification, Optimization and Control with Applicationsin Modern Technologies” funded by the Elite Network of Bavaria (ENB), an initiativeof the Bavarian State Ministry of Sciences, Research and Arts. This research studywas conducted under the supervision of Prof. Dr. Uwe Helmke.

The work contained in this thesis, except when acknowledged in the customarymanner, to the best of my knowledge, is original research performed by the authorunder the guidance of Prof. Dr. Uwe Helmke. The material presented in this thesishas not been submitted for a degree at any other university or institution. Some partsof the research contained in this thesis have been published or submitted to journaland conference as listed below.

1. I. Kurniawan, G. Dirr and U. Helmke, A unified approach to control-lability of closed and open quantum systems, to appear in Proceedings of the19th International Symposium on Mathematical Theory of Networks and Sys-tems (MTNS), Budapest, Hungary, July 2010.

2. G. Dirr, U. Helmke, I. Kurniawan and T. Schulte-Herbruggen, Lie-semigroup structures for reachability and control of open quantum systems:Kossakowski-Lindblad generators form Lie-wedge to Markovian channels, Re-ports on Mathematical Physics, Vol.64, pp.93–121, 2009.

3. I. Kurniawan, G. Dirr and U. Helmke, The dynamics of open quantumsystems : accessibility results, Proceedings in Applied Mathematics and Mechan-ics (PAMM), Vol.(7)1, pp.4130045-4130046, 2007.

i

Acknowledgement

It has been another fruitful and meaningful stage of my lifetime being at the Universityof Wurzburg and living in Germany, thousands of miles away from home, with differentculture, life, experiences and many more. I have had many lessons to learn, spanningfrom educational life to family life. Personally I believe, this thesis marks anothermilestone of my career and moreover, through accomplishing it, I gained importantmathematical knowledge in control theory, which is significant and complementary tomy original background as a practical-oriented control engineer.

Therefore with this respect, I would like to first of all thank Prof. Uwe Helmkefor inviting me to Wurzburg and giving me opportunity to do research in his group,even though he knew that I did not have a strong mathematical background. I thankfor his advice, suggestion and pressure during my research work.

I would like also to express my gratitude to all staff members of the Departmentfor their hospitalities during my stay and I wish to thank all colleagues in the Dy-namics and Control research group : Jens, Gunther, Oana, Sven, Martin K, Martin S,Christian, Knut, Michael, Rudolf, Frau Bohm and Prof. Wirth. Special mention shallbe addressed to Gunther for his patience, assistance and sharpness in helping me toget in hand with mathematics; I know it is not that easy and sometimes perhaps an-noying to convince an engineer about abstract mathematics (smile); this thesis wouldnot have been possible without his supports. My thanks also to the former “interna-tional student group” who helped me to settle and move around for the first time inWurzburg : Ebru, Jose, Gregorius, Francesco, Radja, Lucia, Laura and others.

I wish to acknowledge the full financial support of the Elite Network of Bavaria(ENB) for providing this research position under the framework of the InternationalDoctorate Program (IDP) in Engineering and Computer Sciences : “Identification,Optimization and Control with Applications in Modern Technologies”. I send manythanks also to all professors, students and members within our IDP.

Finally, I thank my beloved family; Emil, Ihsan, Ilham and Hilya; for beingthe soul of my life, for coloring and enriching my life, and our parents and relativesin Indonesia for their encouragement, moral and spiritual supports, to whom I shalldedicate this thesis.

ii

A theory is something nobody believes,except the person who made it.

An experiment is something everybody believes,except the person who made it.

– Albert Einstein –

Contents

Introduction 1

1 Open Quantum Systems and Completely Positive Operators 7

1.1 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Completely Positive Operators . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Properties of Completely Positive Operators . . . . . . . . . . . . . . . 16

2 The Lindblad-Kossakowski Master Equations 19

2.1 Linear Operators on Matrices . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Quantum Dynamical Semigroup . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Various Forms of the Infinitesimal Generators . . . . . . . . . . . . . . 31

2.3.1 Diagonal Lindblad Forms . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Decompositions of the Dissipative Terms . . . . . . . . . . . . . 36

2.4 Properties of the Lindblad-Kossakowski Generator . . . . . . . . . . . . 37

2.4.1 Unitality and Purity . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.2 Trace and Decomposition . . . . . . . . . . . . . . . . . . . . . 41

2.4.3 Dimensional Aspect . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Matrix Representation of the Master Equation . . . . . . . . . . . . . . 47

2.5.1 Vector of Coherence Representation . . . . . . . . . . . . . . . . 47

2.5.2 Positivity and Confinement in the Bloch Ball . . . . . . . . . . . 48

2.5.3 Master Equation for the Vector of Coherence . . . . . . . . . . . 51

3 Controllability of the Lindblad- Kossakowski Master Equation 56

3.1 Bilinear Control Systems on Lie Groups . . . . . . . . . . . . . . . . . 56

3.1.1 Controllability and Accessibility on Lie Groups . . . . . . . . . 56

3.1.2 Induced Control Systems and Lie Group Actions . . . . . . . . . 60

3.2 Controllability of Quantum Systems . . . . . . . . . . . . . . . . . . . . 62

iii

iv Contents

3.2.1 Controllability of Closed Quantum Systems . . . . . . . . . . . 62

3.2.2 Controllability Issues of Open Quantum Systems . . . . . . . . 68

3.2.3 Reachable Sets of Two-Level Systems . . . . . . . . . . . . . . . 74

3.3 Accessibility of the Lindblad-Kossakowski Master Equations . . . . . . 83

3.3.1 The Unital Case . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.2 The Non-Unital Case . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Genericity of Accessibility and an Algorithm for Determining Acces-sibility 107

4.1 Genericity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Genericity Results for the Lindblad-Kossakowski Master Equation . . . 109

4.3 Algorithm for Determining Accessibility . . . . . . . . . . . . . . . . . 118

4.3.1 Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3.2 Type B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.3.3 Type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A Lie Algebras and Lie Groups 132

B List of Transitive Lie Algebras 137

C Some Calculations for the Vector of Coherence Representation 139

C.1 Gell-Mann Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.2 Calculations for LH , Ljk and pjk . . . . . . . . . . . . . . . . . . . . . . 140

C.3 Calculations for Tr(LD) . . . . . . . . . . . . . . . . . . . . . . . . . . 142

D Some Related Algorithms for Transitivity 145

Index of Notations 152

Introduction

In the last decade there has been a significantly growing interest in the frontier re-search area of quantum control. This is somehow motivated by the belief of theso-called second quantum revolution [22]; quantum mechanics, which originally gavenew rules to merely explain physical nature in its beginning, is now being exploitedto develop new technologies. Control of classical (i.e. non-quantum) systems has al-ways played a central role in current technologies. In a similar way, many futurequantum technologies will depend on control of quantum systems. Controlling andmanipulating quantum systems therefore is an important challenge and will serve asa fundamental key resource for future technologies. Applications of quantum controlfor example can be found in quantum computing, nuclear magnetic resonance (NMR)spectroscopy and spintronics.

In order to effectively control quantum systems, a precise dynamical model ofquantum systems is useful. For closed quantum systems which have no interactionwith the environment, the time evolution is described by the Schrodinger equation ona Hilbert space, or more generally, by the Liouville master equation on the set P ofdensity operators,

ρ = LH(ρ) = −i[H, ρ]. (1)

In contrast with this ideal situation, for real-world open quantum systems interactingwith the environment, dissipation or relaxation have to be taken into account. Thetime evolution of open quantum systems can be modelled by the so-called Lindblad-Kossakowski master equation [5, 24, 50]

ρ = L(ρ) = LH(ρ) + LD(ρ), (2)

where LD accounts for the interaction and is of the form

LD(ρ) =

NA∑

j=1

VjρV†j − 1

2V †

j Vjρ−1

2ρV †

j Vj .

Moreover, we call Eq.(2) unital and non-unital when LD(I) = 0 and LD(I) 6= 0,respectively. Unlike the Liouville master equation which preserves the spectrum ofdensity operators, the Lindblad-Kossakowski master equation destroys the isospectralproperty of the flow. This situation makes the dynamics of the Lindblad-Kossakowski

1

2 Introduction

master equation more complicated, with the dissipation effects not being sufficientlywell understood so far. Therefore, it is both of theoretical and practical importanceto study the dynamical behaviour of the Lindblad-Kossakowski master equation.

The Lindblad-Kossakowski master equation is one of the most important modelsof open quantum systems. It is widely used in quantum physics and covers sufficientlylarge class of quantum physical processes [4, 5, 16, 30, 52]. One particularly importantand remarkable mathematical property of the Lindblad-Kossakowski master equation,which is also physically relevant, is that of complete positivity [15, 24, 50]. Thus, theflow exp(tL), t ≥ 0, of Eq.(2) generates a one-parameter semigroup of completelypositive linear maps on P, or a quantum dynamical semigroup. To the best of ourknowledge, Kossakowski [43] was the first who derived sufficient and necessary condi-tions for a (bounded) linear map to be the infinitesimal generator of a one-parametersemigroup of positive linear maps (not necessarily completely positive) acting on someinfinite dimensional vector spaces. Later on, Kossakowski with his colleagues [24], de-rived the explicit form of the infinitesimal generator L of the completely positiveone-parameter semigroup for the finite dimensional case, based on his previous resulton the infinitesimal generator of positive semigroups [43]. At about the same time,Lindblad [50] independently derived the same form for the infinite dimensional case.

In this thesis, we focus on the finite dimensional (N-level) Lindblad-Kossakowskimaster equation. Especially, we investigate additional features arising when controlproperties are taken into account. Thus, we suppose that the Hamiltonian H in Eq.(1)and (2) can be manipulated by some external input. We assume that H is of the form

H = H0 +

m∑

k=1

uk(t)Hk, (3)

with inputs or control functions t 7→ uk(t) ∈ R. Then, the corresponding Liouville orLindblad-Kossakowski master equation is within the class of bilinear control systemson the set of (complex) Hermitian matrices. In quantum control, a basic task is toexplore the possibilities of steering the state of a quantum system from an initial stateρ0 = ρ(0) to a target state ρF = ρ(T ) in finite time T . Such control tasks are consider-ably important in applications e.g. nuclear magnetic resonance (NMR) spectroscopy[37, 38, 39, 40] and quantum computing e.g. [18, 19, 20, 52, 55]. Related to the abovecontrol problem, this thesis focuses on fundamental control-theoretical issues whicharise when viewing the Lindblad-Kossakowski master equation as a bilinear controlsystem. A crucial mathematical question here is that of the structure of the reachablesets, i.e. the set R(ρ0) of quantum states that can be reached from the initial stateρ0 using arbitrary inputs uk(·).

For the Liouville master equation, the reachable set R(ρ0) is a subset of theunitary orbit of ρ0. Therefore, it is interesting to investigate whether the reachableset R(ρ0) equals to the full unitary orbit of ρ0, for all ρ0 ∈ P. This defines thenotion of (density operator) controllability. In the literature, different notions of

Introduction 3

controllability exist for closed quantum systems, i.e. operator controllability, purestate and projective state controllability, and their complete characterizations havebeen already achieved in [1, 53]. In this thesis, we unify these approaches by exploitingthe classification of Lie groups which act transitively on Grassmann manifolds andGrassmannians, see [18, 60].

While controllability is a meaningful concept for closed quantum systems, thesituation is different for open quantum systems. Here, the reachable set R(ρ0) of theLindblad-Kossakowski master equation is no longer confined to the unitary orbit ofρ0. Assuming the control uk(t) enters bilinearly via H as in Eq.(3), it is easily seenthat the Lindblad-Kossakowski master equation is never controllable. Therefore, it isreasonable to ask when the reachable set R(ρ0) has non-empty interior in P. Thisdefines the accessibility property of the Lindblad-Kossakowski master equation. Usingthe known classification of Lie groups which act transitively on Rd\0 [11, 12, 18, 44],we derive accessibility results for open quantum systems.

The main contributions achieved in this thesis are :

1. A new proof of Kossakowski’s result [43] using the Lie-semigroup techniques.Specifically, we characterize the infinitesimal generator of one-parameter semi-group of positive linear maps via the linear invariance of closed convex cones, cf.Theorem I.5.27 in [29], for the convex cone of positive semidefinite Hermitianmatrices.

2. A rigorous analysis of the structure of the Lindblad-Kossakowski generator L.Particularly, we reveal self- and skew-adjoint decompositions of L and showthat the trace of L is always negative definite. We also prove that the setof admissible Lindblad-Kossakowski generators forms a closed convex cone ingl(d,R) with non-empty interior. This holds for both the unital and non-unitalcase.

3. A unified Lie-theoretical analysis for controllability aspects of finite dimensional(N-level) quantum control systems. This constitutes the main results of thisthesis. Precisely,

• We characterize controllability of the Liouville master equation (closedquantum systems) using the classification of (matrix) Lie groups whichact transitively on Grassmann manifold [18, 60]. It is then shown how dif-ferent notions of controllability (pure state, projective state, pure state likeand density operator controllability) can be effectively characterized, witha more straightforward proof compared to the previous approaches [1, 53].

• We characterize accessibility of the Lindblad-Kossakowski master equation(open quantum systems) using the classification of (matrix) Lie groupswhich act transitively on Rd \ 0 [11, 12, 18, 44]. In particular, we derive

4 Introduction

necessary and sufficient conditions for accessibility of the unital Lindblad-Kossakowski master equation and sufficient conditions for accessibility ofthe non-unital case. For the special case of n-coupled spin-1/2 systems, weobtain a remarkably simple characterization for accessibility. Our resultscorrect some earlier statements appeared in the quantum control literature[6, 7, 8].

• We prove that accessibility of the Lindblad-Kossakowski master equationis a generic property. This holds for both the unital and non-unital case.The genericity results are basically obtained by utilizing the structure ofsemisimple Lie algebras [33, 34]. Note that the genericity statements havealready appeared in the literature [6, 7, 8], but the proofs are either missingor false.

4. We develop an algorithm to decide whether a unital open quantum system is ac-cessible or not, given an explicit Lindblad-Kossakowski master equation. Whenthe system is accessible, the algorithm also decides what type of system Liealgebra (up to conjugation) is generated by the Lindblad-Kossakowski masterequation. Although the algorithm is a straightforward adaptation from [12], itrequires some modifications due to our theoretical results on accessibility. Forthe special case of n-coupled spin-1/2 systems, the algorithm becomes very sim-ple. We also provide a complete explanation of the algorithm since the originalone in [12] contains some hidden subtleties and minor mistakes.

The outline of the thesis is as follows. The first chapter summarizes basic notionsand definitions about quantum states, quantum operations and completely positiveoperators. We focus on reviewing a quantum operation as a completely positive andtrace preserving (CPTP) linear map. One important fact of completely positive linearoperator is the equivalence to a Kraus operator. We explain how the Kraus operatorarises in the context of open quantum systems. Another interesting characterization isthe fact that complete positivity of a linear map can be effectively decided via checkingpositive semidefiniteness of the so-called Choi-matrix. This remarkable property canbe used to derive the explicit form of the infinitesimal generator of one-parametersemigroup of completely positive linear maps. We also recall other properties such asconvexity and transitivity of CPTP linear maps on the set of all density operators.We basically follow [15] in summarizing most of these properties.

Chapter 2 reveals the structure of the Lindblad-Kossakowski generator L. First,we introduce quantum dynamical semigroups as strongly continuous one-parametersemigroups of CPTP linear maps. Then, we recall Kossakowski’s result [43] andpresent a new proof using the linear invariance theorem of closed convex cones [29].Moreover, we restate a result from Gorini-Kosasakowski-Sudarshan (GKS) [24], whichexplicitly characterizes the infinitesimal generator of a quantum dynamical semigroup,and sketch the proof that basically relies on the Kossakowski’s result and the Choi-matrix characterization of completely positive maps. We also discuss various forms

Introduction 5

of the infinitesimal generator, e.g. realizing that the form proposed by Lindblad [50](originally for the infinite dimensional case) is a “diagonal version” of the GKS form.Basic properties of the Lindblad-Kossakowski generator such as negative definitenessof the trace, self- and skew-adjoint decomposition and dimensional aspects are in-vestigated. These properties play a central role to derive accessibility results in thesubsequent chapters. Finally, we describe in detail the Lindblad-Kossakowski masterequation for the vector of coherence representation (of density operators).

In Chapter 3, we study controllability aspects of finite dimensional (N-level)quantum systems. It covers both controllability results for the Liouville master equa-tion and accessibility results for the Lindblad-Kossakowski master equation. To obtaina unified approach, we discuss first some control-theoretical ideas for bilinear controlsystems evolving on Lie groups [35]. Well-known results on controllability and acces-sibility on Lie groups are described in some detail and connections to transitive Liegroup actions on homogenous spaces are discussed. The Liouville master equation onthe unitary orbit can be regarded as an induced system of a bilinear control system onthe Lie group SU(N). Consequently, controllability can be restated as transitive Liegroup actions on a unitary orbit. Then, recent results on transitive Lie group actionson Grassmann manifolds [18, 60] provide sufficient and necessary conditions for con-trollability. This allows us to clarify the different notions of controllability (pure state,projective state, pure state like and density operator controllability) for the Liouvillemaster equation. For open quantum systems, the unital Lindblad-Kossakowski masterequation can be seen as an induced bilinear control system on Rd after “lifting” to theLie group GL(d,R). Accessibility for the unital case then is equivalent to transitiveLie group actions on Rd \ 0 for which necessary and sufficient conditions are known[11, 12, 44]. For the non-unital case, one has to lift the system to the semidirectproduct GL(d,R) ⋊ Rd. In this case, we derive sufficient conditions for accessibil-ity by using transitivity results on Rd \ 0, provided some additional assumptionsare satisfied. In Chapter 3, we also discuss why controllability does not hold for theLindblad-Kossakowski master equation. We therefore introduce two weaker notions ofcontrollability (h-controllability and wh-controllability) and demonstrate how thesenotions help to correct previous results on the reachable set of two-level open quantumsystems [6, 7, 8].

The first part of Chapter 4 studies the genericity property of the accessibil-ity of the Lindblad-Kossakowski master equation, both for the unital and non-unitalcase. This is motivated by a result in [33], which states that accessibility of bilin-ear control systems on semisimple Lie algebras is generic. However, in our case forquantum control applications, we need to modify the proof since we are restrictedto the admissible set of the Lindblad-Kossakowski generators. Here, the observationin Chapter 2 stating that the admissible set of the Lindblad-Kossakowski generatorsforms a closed convex cone in gl(d,R) with non-empty interior plays a crucial role.The main ingredient of the genericity proof utilizes the structure of semisimple Liealgebras [33] and a standard idea from linear system theory. The second part of

6 Introduction

the chapter deals with an algorithm to determine accessibility of the unital Lindblad-Kossakowski master equation. Given an explicit Lindblad-Kossakowski generator, thealgorithm decides if the system is accessible or not. If it is accessible, the algorithmalso determines what type of system Lie algebra (up to conjugation) generated by theunital Lindblad-Kossakowski master equation. We straightforwardly adapt algorithmin [12], applied specifically to the Lindblad-Kossakowski master equation. We correctsome minor mistakes on the original algorithm and give more explanations to clarifysome subtleties.

Chapter 1

Open Quantum Systems andCompletely Positive Operators

1.1 Quantum States

We begin by recalling some basic concepts and notations from quantum mechan-ics. Specifically, we consider a finite dimensional setting where all objects can berepresented explicitly as matrices and column vectors. To a finite dimensional quan-tum system, there corresponds a complex Hilbert space H := CN , endowed with thestandard Hermitian inner product 〈 , 〉, the state space of the system. A normalizedelement of H = CN called a state vector

ψ ∈ S(CN) :=ϕ ∈ CN

∣∣ ||ϕ||2 := 〈ϕ, ϕ〉 = 1,

represents the state of a closed quantum system; a term commonly used in quantummechanics to describe an isolated quantum system.

In order to encounter more general scenarios in quantum mechanics, one hasto extend the notion of quantum states. Such a general scenario arises, for example,when one has to work with quantum ensembles which are statistical collections of alarge number of identical non-interacting quantum systems. Another widely encoun-tered situation are open quantum systems; quantum systems which interact with theenvironment. In both cases, the concept of quantum states has to be generalized fromthe state vector to the density matrix or density operator formalism. In this setting,the states of finite dimensional N -level quantum systems are completely describedby density operators ρ, i.e. the state space is the convex set of all non-negative selfadjoint linear maps on H = CN with trace one,

P :=ρ : CN → CN | ρ = ρ† ≥ 0 , Tr(ρ) = 1

. (1.1)

Here we use (·)† to denote the conjugate transpose of a matrix (or linear map) whilelater on (·)⊤ is used for transposing a matrix without conjugation. A quantum state

7

8 Chapter 1. Open Quantum Systems and Completely Positive Operators

is said to be a pure state if the density operator is a rank one projector, i.e. ρ can bewritten in the form

ρ = ψψ† , ψ ∈ S(CN). (1.2)

Pure states are characterized by the equality

Tr(ρ2) = Tr(ρ) = 1. (1.3)

Otherwise, it is called a mixed state. One can easily conclude that the set of all purestates is diffeomorphic to the complex projective space CPN−1 since ψ = eiϕψ implies

ψψ† = ψψ†.

Any element ρ ∈ P can be decomposed by the spectral decomposition withrespect to an orthonormal basis ψ1, . . . , ψN of CN as

ρ =N∑

i=1

λiψiψ†i ,

where λi are the eigenvalues of ρ. The unit trace condition Tr(ρ) = 1 and positivesemidefiniteness ρ ≥ 0 imply

∑Ni=1 λi = 1 and λi ≥ 0, respectively. This implies

Tr(ρ2) =

N∑

i=1

λ2i ≤ 1, (1.4)

where equality holds only for pure states. As previously noted, one example of amixed state is a statistical ensemble of identical quantum system (quantum ensemble)characterized by a finite number of pure states ρi and their relative frequencies ωi inthe ensemble. The associated density operator is given by

ρ =∑

i

ωiρi,

where∑

i ωi = 1 and ωi ≥ 0, i.e. a mixed state ρ as a convex combination of purestates ρi.

1.2 Completely Positive Operators

According to the second postulate of quantum mechanics, the evolution of a closedquantum system is described by a unitary transformation U of the state vectors as

ψ = Uψ, (1.5)

1.2. Completely Positive Operators 9

while the corresponding density operator transforms as

ρ = UρU †. (1.6)

However, a closed quantum system which does not allow any interactions with theenvironment is in principle just an idealized model. In reality, quantum systemsare likely to suffer from such interactions with the environment. Therefore, anotherconcept is required to properly describe the evolution of open quantum systems. Theso-called completely positive operators [15, 45, 46] provide a tool to deal with suchopen quantum systems and are able to address widely encountered circumstancesof real-world quantum systems, e.g. interactions between system and environment,relaxation and dissipation phenomena, etc.

Before we introduce completely positive operators and clarify some of their prop-erties, we fix some notations. LetMp(C

N×N) denote the collection of all p×p block ma-trices with N×N complex matrices as entries. We can identify the (pN)2-dimensionalvector space Mp(C

N×N) with the tensor product vector space Cp×p ⊗ CN×N via themap

Cp×p ⊗ CN×N −→Mp(CN×N) ,

p∑

j,k=1

eje⊤k ⊗Ajk 7→ A := [Ajk].

Note that for p = 1, we identify Mp(CN×N) with MN (C) := CN×N , the vector space

of all complex matrices of size N , having a basis eje⊤k where ei is a unit column vector

with 1 at the i-th position and zero elsewhere.

Therefore, A ∈ Mp(CN×N) is viewed as a p× p block matrix with block entries

Ajk ∈MN (C), i.e.

A =

A11 A12 . . . A1p

A21. . .

......

...Ap1 . . . . . . App

.

A matrix X ∈ MN(C) is called positive semidefinite (X ≥ 0) if X is Hermitian(X† = X) and v†Xv ≥ 0 for all v ∈ CN . Correspondingly, A ∈ Mp(C

N×N) is positive

semidefinite if A is Hermitian as pN × pN matrix (A†jk = Akj) and y†Ay ≥ 0 for all

y ∈ CpN .

Definition 1.1. A linear map Λ : MN (C) →MN (C) is called positive if and only ifΛ(X) ≥ 0 for all X ≥ 0, X ∈MN (C).

The linear map Λ can be extended to linear tensor product map (Ip⊗Λ) onMp(CN×N)

as(Ip ⊗ Λ) : Mp(C

N×N ) −→ Mp(CN×N)

A = [Ajk] 7−→ (Ip ⊗ Λ)([Ajk]) := [Λ(Ajk)].(1.7)


It is obvious that the matrix (Ip ⊗ Λ)(A) = (Ip ⊗ Λ)([Ajk]) := [Λ(Ajk)], 1 ≤ j, k ≤ p,is again a p × p block matrix where the corresponding block entry is the matrixΛ(Ajk) ∈MN(C) at the j-th block row and the k-th block column. Written explicitely,

(Ip ⊗Λ) :

A11 A12 . . . A1p

A21. . .

......

...Ap1 . . . . . . App

7−→

Λ(A11) Λ(A12) . . . Λ(A1p)

Λ(A21). . .

......

...Λ(Ap1) . . . . . . Λ(App)

= [Λ(Ajk)].

Definition 1.2. A linear map Λ is called completely positive (CP) if and only ifthe maps (Ip ⊗ Λ) are positive for all positive integers p, i.e (Ip ⊗ Λ)(A) ≥ 0 for allA ≥ 0, A ∈ Mp(C

N×N) and all p ∈ N.

Note, it is not true that (Ip ⊗ Λ) is positive if Λ is positive.

Lemma 1.3. A linear map Λ is trace preserving if and only if (Ip ⊗ Λ) is tracepreserving for all p ∈ N.

Proof. For any X = [Xjk] ∈Mp(CN×N), the following holds

Tr((Ip ⊗ Λ)X) =

p∑

i=1

Tr(Λ(Xii)) =

p∑

i=1

Tr(Xii) = Tr(X),

if and only if Λ is trace preserving.

Let us write X = [x1 x2 . . . xN ] ∈ MN (C) where xi ∈ CN denotes the i-thcolumn vector of X. We define the C-linear isomorphism

Vec : MN(C) −→ CN2

X 7−→ x = Vec(X) := [x⊤1 x⊤2 . . . x⊤N ]⊤

as the standard Vec operator which stacks each column vectors of X concatenately.Then, for Y = [y1 y2 . . . yN ] ∈MN(C), we have

Vec(X)Vec(Y )† = xy† =

x1y†1 x1y

†2 . . . x1y

†N

x2y†1

. . ....

......

xNy†1 . . . . . . xNy

†N

∈MN (CN×N).

Since

xjy†k = [x1 x2 . . . xN ](eje

⊤k )

y†1y†2...

y†N

= X(eje

⊤k )Y † , 1 ≤ j, k ≤ N,


this shows that

xy† = [X(eje⊤k )Y †] =

X(e1e⊤1 )Y † X(e1e

⊤2 )Y † . . . X(e1e

⊤N)Y †

X(e2e⊤1 )Y † . . .

......

...X(eNe

⊤1 )Y † . . . . . . X(eNe

⊤N )Y †

, (1.8)

for x = Vec(X) and y = Vec(Y ). Denote by gl(MN (C)) the set of all C-linearoperators on MN (C). We define a map

Φ : gl(MN (C)) −→ MN (CN×N)Λ 7−→ Φ(Λ) := (IN ⊗ Λ)(E),

(1.9)

where

E =N∑

j,k=1

eje⊤k ⊗ eje

⊤k =

e1e⊤1 e1e

⊤2 . . . e1e

⊤N

e2e⊤1

. . ....

......

eNe⊤1 . . . . . . eNe

⊤N

∈MN (CN×N).

Note that Φ(Λ) = [Λ(eje⊤k )]Nj,k=1 is known as the Choi-matrix [15]. We have the

following lemma

Lemma 1.4. 1. The map Φ is a C-linear isomorphism.

2. Any linear operator Λ ∈ gl(MN(C)) can be expressed as

X 7→ Λ(X) =N2∑

j,k=1

cjkMjXM†k , cjk ∈ C, (1.10)

where MjN2

j=1 is any orthonormal basis of MN(C) with respect to the innerproduct 〈X, Y 〉 = Tr(X†Y ).

Proof. Note that we have identified MN(CN×N) with CN×N ⊗CN×N , which is isomor-phic to CN2×N2

. As a vector space, the dimension of gl(MN(C)) is equal to N4. Sincethe linear map Φ is injective, we have the first statement. Using the singular valuedecomposition, we can write the matrix Φ(Λ) as

Φ(Λ) =

N2∑

j=1

αjmjw†j , mj, wj ∈ CN2

, αj ∈ R+0 ,

where mjN2

j=1 (and also wjN2

j=1) are orthogonal. Hence, wj can be written as somecomplex linear combination of mj such that

N2∑

j=1

αjmjw†j =

N2∑

j=1

αjmj

N2∑

k=1

γjkm†k =

N2∑

j,k=1

cjkmjm†k , cjk := αjγjk ∈ C.


By Eq.(1.8) and writing mj = Vec(Mj), we have

Φ(Λ) = [Λ(ere⊤s )] =

N2∑

j,k=1

cjkmjm†k =

N2∑

j,k=1

cjk[Mj(ere⊤s )M †

k ] ∈MN (CN×N),

for r, s = 1, . . . , N . Note again that [Ars] ∈MN (CN×N) is an N×N block matrix withblock entries Ars ∈ CN×N . Since Φ is a C-linear isomorphism, the second statementfollows.

The isomorphism Φ provides crucial information about certain properties of Λ,which can be read off from the Choi-matrix Φ(Λ). Two such particular properties areHermitian preserving and complete positivity, respectively, which are captured in thefollowing two results.

Lemma 1.5. For a linear map Λ : MN (C) →MN (C), the following are equivalent.

1. Λ(X) = Λ(X)†, for all X = X† ∈MN(C), i.e. Λ is Hermitian preserving.

2. The matrix Φ(Λ) is Hermitian.

3. The map Λ is of the form

X 7→ Λ(X) =N2∑

j,k=1

cjkMjXM†k , cjk = ckj ∈ C,

where MjN2

j=1 is any orthonormal basis of MN (C).

Proof. To show (1) ⇒ (2), we take X := 12(eje

⊤k + eke

⊤j ) and Y := i

2(eje

⊤k − eke

⊤j )

which are Hermitian, and by assumption we obtain

12

(Λ(eje

⊤k ) + Λ(eke

⊤j ))

= Λ(X) = Λ(X)† = 12

(Λ(eje

⊤k )† + Λ(eke

⊤j )†)

i2

(Λ(eje

⊤k ) − Λ(eke

⊤j ))

= Λ(Y ) = Λ(Y )† = i2

(−Λ(eje

⊤k )† + Λ(eke

⊤j )†)

respectively. Hence, multiplying the first equation with i and subtracting with thesecond gives Λ(eje

⊤k )† = Λ(eke

⊤j ), and we conclude the desired implication. Next, let

Φ(Λ) is Hermitian and write

Φ(Λ) =

N2∑

i=1

αidid†i , di ∈ CN2

, αi ∈ R.

Expressing di as a complex linear combination of an orthonormal basis mjN2

j=1 of

CN2

, we obtain

N2∑

i=1

αidid†i =

N2∑

i=1

αi

N2∑

j,k=1

λijmjλikm†k =

N2∑

j,k=1

cjkmjm†k,


where cjk =∑N2

i=1 αiλijλik ∈ C and clearly cjk = ckj. Repeating the same argumentfor the second statement of Lemma 1.4, we conclude that (2) ⇒ (3). Finally, (3) ⇒ (1)is immediate.

Theorem 1.6. For a linear map Λ : MN (C) → MN(C), the following are equivalent.

1. The map Λ is completely positive.

2. The matrix Φ(Λ) is positive.

3. The map Λ is of the form

X 7→ Λ(X) =

N2∑

j=1

VjXV†j , Vj ∈MN (C).

Proof. The implication (1) ⇒ (2) follows by definition of completely positive,

Φ(Λ) = (IN ⊗ Λ)(E) ≥ 0,

since E = Vec(e)Vec(e)⊤ ≥ 0, with e = [e1 e2 . . . eN ]. Next, suppose Φ(Λ) ≥ 0 andrecall that any positive semidefinite matrix can be decomposed as the sum of rank-1matrices. Thus, we have

0 ≤ Φ(Λ) := [Λ(ere⊤s )] =

N2∑

j=1

vjv†j =

N2∑

j=1

[Vj(ere⊤s )V †

j ],

where the last equality follows from Eq.(1.8) by setting vj = Vec(Vj). Therefore,

Λ(ere⊤s ) =

∑N2

j=1 Vj(ere⊤s )V †

j and hence Λ(X) =∑N2

j=1 VjXV†j . This shows (2) ⇒ (3).

Finally, let Λ(·) = V (·)V † for some V ∈ MN (C). For arbitrary p ∈ N and 0 ≤ A =[Ajk] ∈Mp(C

N×N), we have

(Ip ⊗ Λ)([Ajk]) = [V (Ajk)V†] =

V A11V† V A12V

† . . . V A1pV†

V A21V† . . .

......

...V Ap1V

† . . . . . . V AppV†

=

VV

. . .

V

A11 A12 . . . A1p

A21. . .

......

...Ap1 . . . . . . App

V †

V †

. . .

V †

= (Ip ⊗ V )A(Ip ⊗ V )† ≥ 0,

since A ≥ 0. So Λ is completely positive and moreover, the sum of such positiveoperators is also positive and thus we have shown (3) ⇒ (1).


Theorem 1.6 shows that complete positivity, which demands positivity of (Ip⊗Λ)for all p ∈ N, reduces only to check positivity for p = N , where N is the dimensionof the Hilbert space of the underlying system. In particular, the matrix Φ(Λ) =(IN ⊗ Λ)(E) encodes full information about complete positivity of Λ. Thus, it is notnecessary to check positivity of (Ip ⊗ Λ) for all positive integer p. This observationimplies that the set of completely positive operators is a semialgebraic object, i.e.complete positivity can be decided via the set of polynomial inequalities arising fromdetermining positive semidefiniteness of the Choi-matrix Φ(Λ).

We note that the explicit form of a completely positive linear map in Theorem1.6(3) is well-known as the so-called Kraus representation or Kraus map [15, 45, 46].The Kraus map (or operator) is widely used to model general transformation of quan-tum states; i.e. quantum operation. We briefly give an example how Kraus operatorsemerge from a particular situation of open quantum systems, when a quantum systeminteracts with the environment. This is known as the description of reduced systemsusing the partial trace operation, see e.g.[16, 52].

Suppose we have a finite dimensional composite quantum system consisting ofthe so-called reduced system (or the principal system) and the environment. Theassociated Hilbert space of this composite quantum system is H = H1 ⊗ H2 whereH1 := CN and H2 := Cm are the corresponding Hilbert space of the principal systemand the environment, respectively. We define the partial trace TrE as follows.

Definition 1.7. Let Tr : Cm×m → C be the trace function on Cm×m. Then,

TrE : MN (Cm×m) −→ MN (C)X = [Xjk] 7−→ TrE(X) := [Tr(Xjk)].

Using the standard unit vectors ei ∈ Cm as a basis for the Hilbert space Cm andusing the following notation

〈φ,Xψ〉E := [φ†Xjkψ] ∈MN (C), (1.11)

for any φ, ψ ∈ Cm and X = [Xjk] ∈ MN(Cm×m), the partial trace can be calculatedas

TrE(X) =

m∑

i=1

〈ei, Xei〉E .

Lemma 1.8. Consider a composite quantum system with the Hilbert space H =CN ⊗ Cm. Define the map ρ → ρ := ρ ⊗ ρe for ρ ∈ P ⊂ CN×N and ρe ∈ Pm := ρ ∈Cm×m | ρ† = ρ ≥ 0 , Tr(ρ) = 1 ⊂ Cm×m, and

Λ(ρ) := TrE(U(ρ⊗ ρe)U†) = TrE(UρU †),

where U = [Ujk] ∈MN(Cm×m) is unitary. Then Λ is completely positive.


Proof. Starting with a pure state ψeψ†e ∈ Pm and using Eq.(1.11), we have

Λe(ρ) := TrE(U(ρ⊗ ψeψ†e)U

†) =m∑

i=1

〈ei, U(ρ⊗ ψeψ†e)U

†ei〉E

=

m∑

i=1

〈ei, U(ρ⊗ ψeψ†eψeψ

†e)U

†ei〉E

=

m∑

i=1

〈ei, U(ρ⊗ ψeψ†e)ψe〉E〈ψe, U

†ei〉E,

where the last equality follows from a straightforward but tedious calculation. More-over, since U(ρ ⊗ ψeψ

†e) =

[∑l Ujlρlkψeψ

†e

]∈ MN (Cm×m) with Ujl ∈ Cm×m and

ρlk ∈ C, we obtain

〈ei, U(ρ⊗ ψeψ†e)ψe〉E =

[e†i

N∑

l=1

Ujlρlkψeψ†eψe

]=[ N∑

l=1

e†iUjlψeρlk

]

= 〈ei, Uψe〉Eρ ∈MN (C).

Hence, Λe is of the Kraus form,

Λe(ρ) = TrE(U(ρ⊗ ψeψ†e)U

†) =m∑

i=1

ViρV†i ,

with

Vi = 〈ei, Uψe〉E ∈MN(C) , V †i = 〈ψe, U

†ei〉E ∈MN(C). (1.12)

By Theorem 1.6, Λe is completely positive. Since any quantum state ρe ∈ Pm can bewritten as a convex combination of pure states, we obtain Λ as a sum of completelypositive operators which is completely positive. The result follows.

In the construction of Lemma 1.8, the composite system can be considered asa closed quantum system as a whole and the evolution of the total system is thendescribed by a unitary transformation. In constrast, the evolution of the principalsystem alone, obtained by the partial trace operation after the unitary transformation,is no longer unitary and is described by a completely positive operator Λ. In this way,we see that the partial trace operation describes the evolution of the principal systemregarded as an open quantum system interacting with the environment. We remarkthat the derivation of Lemma 1.8 crucially depends on the fact that the initial stateadmits the product structure.


1.3 Properties of Completely Positive Operators

In this section, we discuss some basic geometrical properties of completely positiveoperators. The decomposition of the Choi-matrix

0 ≤ Φ(Λ) =

d∑

i=1

viv†i ∈MN(CN×N ) , vi ∈ CN2

,

suggests that the maximum number of required Kraus operators Vi, to represent acompletely positive operator Λ, is equal to the rank of Φ(Λ) and thus, the summandsin the Kraus representation can be bounded byN2. Therefore, any completely positiveoperator takes the form

Λ(·) =

d≤N2∑

i=1

Vi(·)V †i , Vi ∈MN (C).

We note that the above decomposition of a positive semidefinite matrix is notunique and hence, the set of Vi is also not unique. The following result is a straight-forward consequence of the existence and uniqueness property of the Cholesky factor-ization of Φ(Λ), see [15].

Lemma 1.9. (Canonical Kraus Form) Let a completely positive operator be repre-sented by a Kraus form Λ(·) =

∑di=1 Vi(·)V †

i where Vi is linearly independent. Then,

Λ also has the expression Λ(·) =∑d

j=1Wj(·)W †j for another linearly independent Wj

if and only if there exists a unitary matrix U = [ujk] such that Wj =∑d

i=1 ujiVi.

Let CPN be the set of all completely positive operator acting on MN(C). Wedefine

CPTPN :=Λ ∈ CPN | Tr(Λ(X)) = Tr(X), for all X ∈MN (C)

as the set of all completely positive trace preserving operator on MN(C). In term ofKraus representation, the trace preserving condition can be translated as

∑i V

†i Vi =

IN . Note that CPN and CPTPN are closed with respect to the composition of maps.Thus, CPN and CPTPN are semigroups. We recall some basic geometric propertiesof completely positive trace preserving maps such as convexity, see e.g. [15] andtransitivity on the set of all density operators P.

Theorem 1.10. (Convexity of CPTPN) [15]

(i) CPTPN is a compact convex semigroup.

(ii) Let Λ(·) =∑d

j=1 Vj(·)V †j denote the Kraus form with

∑dj=1 V

†j Vj = IN . Then,

Λ ∈ CPTPN is an extreme point if and only if the set V †j Vi1≤i,j≤d is linearly

independent.

1.3. Properties of Completely Positive Operators 17

Proof. (i) It is straightforward to check that the map Λ =∑

i piΛi, where Λi ∈ CPTPN

and∑

i pi = 1, pi ≥ 0, is clearly CPTPN . For (ii), see e.g. [15].

Suppose Λ is an extreme point of CPTPN expressed in canonical Kraus form∑di=1 Vi(·)V †

i . By Theorem 1.10, V †j Vi1≤i,j≤d forms d2 linearly independent matrices.

Now, V †j Vi ∈ MN (C), where the dimension of MN (C) is N2. Since the number of

linearly independent elements can not exceed the dimension of the underlying vectorspace, we have the bound on the number of required Kraus operators for an extremepoint Λ ∈ CPTPN , i.e. d2 ≤ N2 or d ≤ N .

Theorem 1.11. (Transitivity of CPTPN on P). Given any two arbitraryρ1, ρ2 ∈ P, there exists Λ ∈ CPTPN such that ρ2 = Λ(ρ1), i.e. the semigroup CPTPN

acts transitively on the set of all density operators P.

Proof. First consider the set of pure states, i.e. ρ ∈ P with rank-1. We recall thewell-known fact that any pure state ρ1 can be transferred to any other pure state ρ2

via unitary evolution ρ1 7→ ρ2 = Uρ1U†, which is clearly a completely positive and

trace preserving map, i.e. the set of unitary matrix U(N) acts transitively on the setof pure states via conjugation U : ρ 7→ UρU †. To prove the theorem, then we just needto show that: (i) any mixed state can be transferred to a specific pure state; and (ii)the converse, a specific pure state can be transferred to any mixed state; respectivelyby choosing a particular completely positive trace preserving map.

Part (i). Choose a particular pure state ρF = e1e⊤1 . Assume a mixed state ρ1 as

a diagonal matrix with respect to the standard unit vectors ei,

ρ1 =

N∑

i=1

dieie⊤i ,

N∑

i=1

di = 1 , di ≥ 0.

For mixed states which do not appear as diagonal forms, a unitary transformation willdo the diagonalization and this map is obviously completely positive and trace preserv-ing by the above consideration. Now, we choose the Kraus form Λ(·) =

∑j Vj(·)V †

j ,

where Vj := e1e⊤j . Applying Λ to ρ1, we obtain

Λ(ρ1) =N∑

j=1

Vjρ1V†j =

N∑

j=1

e1e⊤j (

N∑

i=1

dieie⊤i )eje

⊤1

=N∑

j=1

dje1e⊤1 = e1e

⊤1 = ρF ,

since∑N

j=1 dj = 1. We check the trace preserving condition,

N∑

j=1

V †j Vj =

N∑

j=1

(eje⊤1 )(e1e

⊤j ) =

N∑

j=1

eje⊤j = IN .


Hence Λ with the prescribed Vj is a required completely positive trace preserving map.

Part (ii). Choose the same particular pure state ρ1 = e1e⊤1 . We want to transfer

this state to a diagonal mixed state ρF =∑N

j=1 λjeje⊤j with λj ≥ 0,

∑Nj=1 λj = 1. As

noted before, for arbitrary non-diagonal mixed states, the transfer can be continuedby a unitary transformation. For doing the prescribed transfer, consider the Krausform

Λ(ρ) =

N2∑

l=1

VlρV†l =

N∑

j,k=1

WjkρW†jk,

where Wjk :=√λjeje

⊤k for given λj as above. Apply this map to ρ1 = e1e

⊤1 , we obtain

Λ(ρ1) =

N∑

j,k=1

√λjeje

⊤k (e1e

⊤1 )√λjeke

⊤j =

N∑

j=1

λjeje⊤j = ρF .

We check that

N∑

j,k=1

W †jkWjk =

N∑

j,k=1

√λjeke

⊤j

√λjeje

⊤k =

N∑

j=1

λj

N∑

k=1

eke⊤k = IN ,

since∑N

j=1 λj = 1, hence Λ is completely positive and trace preserving. The resultfollows.

Chapter 2

The Lindblad-Kossakowski MasterEquations

2.1 Linear Operators on Matrices

Before we discuss the Lindblad-Kossakowski master equation and its properties, wesummarize some important facts about linear operators which act on the set of ma-trices. In quantum physics, these kind of operators are frequently termed as superop-erators.

Lether(N) = X ∈ CN×N | X† = X

denote the real vector space of Hermitian matrices equipped with the inner product〈X, Y 〉 = Tr(X†Y ) = Tr(XY ). Correspondingly, denote

her0(N) := X ∈ her(N) | Tr(X) = 0.

Definition 2.1. Let gl(her(N)) denote the set of all linear maps from her(N) toitself, i.e.

gl(her(N)) = L ∈ gl(CN×N) | L(her(N)

)⊂ her(N).

1. L† ∈ gl(her(N)) is called the adjoint of L when

〈X,L(Y )〉 = 〈L†(X), Y 〉 , for all X, Y ∈ her(N).

2. L ∈ gl(her(N)) is called skew-adjoint when L† = −L, that is

〈X,L(Y )〉 = −〈L(X), Y 〉 , for all X, Y ∈ her(N).

3. L ∈ gl(her(N)) is called self-adjoint when L† = L, that is

〈X,L(Y )〉 = 〈L(X), Y 〉 , for all X, Y ∈ her(N).

19

20 Chapter 2. The Lindblad-Kossakowski Master Equations

4. With respect to the Hilbert space her(N), we say that a self-adjoint operatorL ∈ gl(her(N)) is positive semidefinite (negative semidefinite), or for short L ≥ 0(L ≤ 0), if and only if

〈X,L(X)〉 ≥ 0(〈X,L(X)〉 ≤ 0

),

for all X ∈ her(N). Equivalently, L ∈ gl(her(N)) is positive semidefinite (neg-ative semidefinite) if and only if all eigenvalues of L = L† are nonnegative(nonpositive).

5. The trace of L ∈ gl(her(N)) is defined as the sum of all eigenvalues of L andcan be computed as

Tr(L) =N2∑

i=1

〈Gi,L(Gi)〉,

where GiN2

i=1 is any orthonormal basis of her(N).

Lemma 2.2. For A,B ∈ her(N), the following linear operators in gl(her(N)) satisfy:

1. CAB : X 7→ iAXB − iBXA, is skew-adjoint,

2. iadA : X 7→ i[A,X] = i(AX −XA), is skew-adjoint,

3. ad2A : X 7→ [A, [A,X], is self-adjoint and positive semidefinite,

4. BA : X 7→ AX +XA is self-adjoint.

Proof. For (1), direct calculation shows

〈X, CAB(Y )〉 = iTr(XAY B −XBY A) = −iTr(AXBY − BXAY )= −〈CAB(X), Y 〉 , for all X, Y ∈ her(N).

(2) follows from (1) for B = IN . For (3), since iadA is skew-adjoint, we have

ad2A = −iadA · iadA = (iadA)†(iadA) ≥ 0.

Finally,〈X,BA(Y )〉 = Tr(XAY +XY A) = Tr(AXY +XAY )

= 〈BA(X), Y 〉 , for all X, Y ∈ her(N)

shows (4).

Now consider gl(CN×N) as the complex vector space of all linear operators onCN×N and define the inner product on gl(CN×N) for Γ1,Γ2 ∈ gl(CN×N) as

〈Γ1,Γ2〉gl(CN×N ) :=

N2∑

λ=1

⟨Γ1(Mλ),Γ2(Mλ)

⟩=

N2∑

λ=1

Tr((Γ1(Mλ))

†Γ2(Mλ)), (2.1)

where MλN2

λ=1 is an orthonormal basis of CN×N .

2.1. Linear Operators on Matrices 21

Lemma 2.3. For any A ∈ CN×N and any orthonormal basis MλN2

λ=1 of CN×N ,

N2∑

λ=1

M †λAMλ = INTr(A).

Proof. Any other orthonormal basis MγN2

γ=1 of CN×N is unitarily equivalent to

MλN2

λ=1, i.e. Mγ :=∑N2

λ=1 uλγMλ, where U = [uλγ ] is unitary. Then,

N2∑

γ=1

Mγ

†AMγ =

N2∑

γ=1

( N2∑

λ=1

uλγMλ

)†A( N2∑

µ=1

uµγMµ

)=

N2∑

λ,µ=1

( N2∑

γ=1

uλγuµγ

)M †

λAMµ

=

N2∑

λ,µ=1

δλµM†λAMµ =

N2∑

λ=1

M †λAMλ.

Thus, choosing the orthonormal basis eje⊤k N

j,k=1 of CN×N , we have

N2∑

λ=1

M †λAMλ =

N∑

j,k=1

(eje⊤k )†Aeje

⊤k =

N∑

j,k=1

eke⊤j Aeje

⊤k

=( N∑

k=1

eke⊤k

)( N∑

j=1

e⊤j Aej

)= INTr(A),

as desired.

Lemma 2.4. Let Bjd=N2−1j=1 be an orthonormal basis of her0(N) and Gjp=N2

j=1 bean orthonormal basis of her(N). Let Xjk,Yjk,Vjk ∈ gl(CN×N) be families of linearoperators on CN×N defined as

Xjk(·) := Bj(·)Bk +Bk(·)Bj,Yjk(·) := Bj, Bk(·) + (·)Bj, Bk,Vjk(·) := iGj(·)Gk − iGk(·)Gj,

with Bj , Bk := BjBk +BkBj. Then, the linear operators on CN×N

1. Xjk, 1 ≤ j ≤ k ≤ d, are mutually orthogonal.

2. Xjk and Yrs, 1 ≤ j, k, r, s ≤ d, are orthogonal.

3. Vjk, 1 ≤ j < k ≤ p are mutually orthogonal.

4. Zjk := Xjk − Yjk, 1 ≤ j ≤ k ≤ d, are linearly independent.


Proof. We check the orthogonality of the linear operators in CN×N via the innerproduct defined in Eq.(2.1) and fixing an orthonormal basis MλN2

λ=1 of CN×N .

(1). Between Xjk and Xrs :

〈Xjk,Xrs〉gl(CN×N ) =

N2∑

λ=1

Tr((BjMλBk +BkMλBj)

†(BrMλBs +BsMλBr))

=N2∑

λ=1

Tr(BjM

†λBkBrMλBs +BjM

†λBkBsMλBr

+BkM†λBjBrMλBs +BkM

†λBjBsMλBr

)

= 2Tr(BkBr)Tr(BjBs) + 2Tr(BkBs)Tr(BjBr)

= 2δkrδjs + 2δksδjr,

where we have used Lemma 2.3 for the third equality and the fact that BjN2−1j=1 is

an orthonormal basis of her0(N). This shows the orthogonality of Xjk and Xrs, forj ≤ k and r ≤ s.

(2). Between Xjk and Yrs :

〈Xjk,Yrs〉gl(CN×N ) =

N2∑

λ=1

Tr((BjMλBk +BkMλBj)

†(Br, BsMλ +MλBr, Bs))

=

N2∑

λ=1

Tr(BjM

†λBkBr, BsMλ +BjM

†λBkMλBr, Bs

+BkM†λBjBr, BsMλ +BkM

†λBjMλBr, Bs

)

= 0,

where we have used again Lemma 2.3 and the fact that Tr(Bj) = 0 for all j. Thisshows the orthogonality of Xjk and Yrs for all j, k, r, s.

(3). Between Vjk and Vrs :

〈Vjk,Vrs〉gl(CN×N ) =N2∑

λ=1

Tr((iGjMλGk − iGkMλGj)

†(iGrMλGs − iGsMλGr))

=

N2∑

λ=1

Tr(−GjM

†λGkGrMλGs +GjM

†λGkGsMλGr

+GkM†λGjGrMλGs −GkM

†λGjGsMλGr

)

= −2Tr(GkGr)Tr(GjGs) + 2Tr(GkGs)Tr(GjGr)

= −2δkrδjs + 2δksδjr,

2.1. Linear Operators on Matrices 23

where we have used the fact that GjN2

j=1 is an orthonormal basis of her(N). Thisshows the orthogonality of Vjk and Vrs, for j < k and r < s.

(4). We want to show that Zjk := Xjk −Yjk1≤j≤k≤d are linearly independent.Suppose they are not. Then, one of them can be written as a linear combination ofthe others, i.e. there are complex numbers ajk with

Zrs =∑

1≤j 6=r≤k 6=s≤d

ajkZjk.

Taking the inner product of above equation with Xrs and using (1) and (2) yields

〈Xrs,Xrs〉 = 〈Xrs,Xrs − Yrs〉 =∑

1≤j 6=r≤k 6=s≤d

ajk

(〈Xrs,Xjk〉 − 〈Xrs,Yjk〉

)= 0,

which is a contradiction. Hence, we conclude the statement.

Remark 2.5. (a). In Lemma 2.4 we consider Xjk,Yjk and Vjk as linear operatorson CN×N . By restricting to the real vector space her(N), it is not difficult tosee that Xjk,Yjk,Vjk ∈ gl(her(N)). Moreover, orthogonality and linear inde-pendence property as stated in Lemma 2.4 for Xjk,Yjk,Vjk considered as linearoperators on her(N) are still valid. This can be seen directly from the proofof Lemma 2.4 where now we have to calculate the inner product Eq.(2.1) usingan orthonormal basis of her(N). Note that since CN×N is the complexificationof her(N), any orthonormal basis of her(N) can be considered as an orthonor-mal basis of CN×N . Therefore, the calculations remain in force and Lemma 2.4remains true for Xjk,Yjk,Vjk ∈ gl(her(N)).

(b). Consider a finite dimensional real Euclidean vector space V and define

so(V ) := L ∈ gl(V ) | L skew-adjointsym(V ) := L ∈ gl(V ) | L self-adjoint

such that so(V ) ⊕ sym(V ) = gl(V ). Then, it is straighforward to check thatXjk,Yjk ∈ sym(her(N)) and Vjk ∈ so(her(N)), since for all A,B ∈ her(N), onehas

〈A,Xjk(B)〉 = 〈Xjk(A), B〉 , 〈A,Yjk(B)〉 = 〈Yjk(A), B〉,〈A,Vjk(B)〉 = −〈Vjk(A), B〉.

(c). Consider the real span

span〈Vjk | 1 ≤ j < k ≤ p〉 ⊂ so(her(N).

By Lemma 2.4, there are p(p− 1)/2 mutually orthogonal Vjk. Therefore,

span〈Vjk | 1 ≤ j < k ≤ p〉 = so(her(N)), (2.2)

since the dimension of so(her(N)) is exactly p(p− 1)/2.


2.2 Quantum Dynamical Semigroup

In order to have a mathematically convenient and also physically widely applicabledescription of a continuous time evolution of open quantum systems, we introduce thenotion of quantum dynamical semigroup.

Definition 2.6. A Quantum Dynamical Semigroup (QDS) is a strongly con-tinuous one-parameter family Λt of completely positive trace preserving linear mapssatisfying the semigroup conditions

Λ0 = id , ΛtΛs := Λt Λs = Λt+s , for all s, t ≥ 0,

where denotes the composition map.

It is well-known that there exists a linear operator L : MN (C) → MN(C) suchthat Λt satisfies a differential equation

Λt = L(Λt) , Λ0 = id , t ≥ 0. (2.3)

Such a linear operator L is called the infinitesimal generator of QDS. Since we onlydeal with finite dimensional systems, we can express the solution of Eq.(2.3) via theexponential series

Λt = exp(tL) :=∞∑

n=0

tn

n!Ln , t ≥ 0.

A QDS then describes the time evolution of open quantum systems on the set ofdensity operator P via

ρ(t) = Λt(ρ0) = exp(tL)(ρ0) , t ≥ 0.

Equivalently, the dynamics of open quantum systems satisfies the following ordinarydifferential equation on the set of density operator P

ρ(t) = L(ρ(t)) , ρ0 = ρ(0), ρ(t) ∈ P. (2.4)

By regarding the density operator as a kind of probability distribution, Eq.(2.4) canbe seen as an analog of a Markov process in classical system; that is, considering s isthe present time, the state of the system at t units of time later depends only on thepresent state ρ(s) and ignores the past. Therefore, Eq.(2.4) is often called a Markovianmaster equation which is considered as an important class of master equation becauseof its wide applicability to model a large class of quantum physical processes.

In what follows, we will describe how to obtain a very explicit and generalformula of L as the infinitesimal generator of a QDS which is well-known as theLindblad-Kossakowski generator [24, 50]. We begin with the following definition.

2.2. Quantum Dynamical Semigroup 25

Definition 2.7. A Positive Dynamical Semigroup is a strongly continuous one-parameter semigroup of positive trace preserving linear maps Γt : MN (C) → MN(C),t ≥ 0, Γ0 = id.

Note that in contrast to QDS, Definition 2.7 only requires that the correspondingdynamical semigroup is positive, not necessarily completely positive. Again, thereexists an infinitesimal generator L : MN (C) →MN (C) of Γt such that

Γt = L(Γt) , t ≥ 0.

To the best of our knowledge, Kossakowski [43] was the first who provided generalnecessary and sufficient conditions for a linear operator L to be the infinitesimal gen-erator of a positive dynamical semigroup. In [43], Kossakowski introduced the termQDS in an infinite dimensional setting to mean a positive dynamical semigroup as inthe Definition 2.7, not a completely positive one. Later on, after the importance ofcomplete positivity is recognized in quantum physics, the term QDS refers to com-pletely positive dynamical semigroups. Here, we recall Kossakowski’s result in a finitedimensional version.

Theorem 2.8. (Kossakowski [43]) Let P = PiNi=1 ∈MN(C) such that

PiPj = δijPi , P †i = Pi ,

N∑

i=1

Pi = I,

i.e. a set of orthogonal one-dimensional self-adjoint projections in MN (C) which formsa resolution of identity. A linear map L : MN (C) → MN (C) is the infinitesimalgenerator of a positive dynamical semigroup if and only if

(i) Tr[PrL(Ps)] ≥ 0, r, s = 1, 2, . . . , N, r 6= s (positivity)

(ii)∑N

r=1 Tr[PrL(Ps)] = 0, s = 1, 2, . . . , N (trace preservation)

hold for all such families P.

We will demonstrate that Theorem 2.8 can be obtained as a special case of theso-called linear invariance theorem for wedges, cf. Theorem I.5.27 in [29]. Beforeproceeding further, we recall some notions about the geometry of wedges [29].

A wedge is a subset W of a real vector space V which satisfies the followingcondition

(i) W +W ⊆ W ,

(ii) R+0 ·W ⊆W ,

(iii) W = W , i.e. W is topologically closed in V .


So a wedge is a closed convex cone. The edge of the wedge defined by H(W ) :=W ∩ −W is the largest vector subspace contained in W . A pointed wedge is a wedgewhose edge is trivial i.e. H(W ) = 0. A generating wedge is a wedge which satisfiesV = W −W . For a finite dimensional real vector space, W is a generating wedge ifand only if W has non-empty interior in V .

Let 〈·, ·〉 be a scalar product on a finite dimensional real vector space V . Definethe dual wedge W ∗ of W in V ,

W ∗ := X ∈ V | 〈X, Y 〉 ≥ 0, for all Y ∈W. (2.5)

For any subset M ⊆ V , we call M⊥ the annihilator of M in V , i.e.

M⊥ := X ∈ V | 〈X, Y 〉 = 0, for all Y ∈M. (2.6)

So the annihilator M⊥ is simply the set of all vectors which are perpendicular to allvectors in M . For any subset M ⊆ V , we define the opposite wedge of M with respectto the wedge W as

opW (M) := M⊥ ∩W ∗. (2.7)

We also define the subtangent wedge of W at a point X as

T sX(W ) := opW (X)∗. (2.8)

A subset F of a wedge W is called a face of W if it is itself a wedge and inaddition, the relation X + Y ∈ F , where X, Y ∈W implies X, Y ∈ F . A face F of awedge W is called an exposed face if

F = opW (F )⊥ ∩W = (F⊥ ∩W ∗)⊥ ∩W. (2.9)

A nonzero point X ∈W is called an extreme point if R+0 ·X is a face, and it is called

an exposed point if R+0 ·X is an exposed face, i.e.

R+0 ·X = (X⊥ ∩W ∗)⊥ ∩W = opW (X)⊥ ∩W. (2.10)

A point X ∈ W is called an E1-point if R+0 ·X +H(W ) is an exposed face. The set

of all E1-points in W is called E1(W ). Note that for a pointed wedge W , E1(W ) isexactly the set of all exposed points since H(W ) = 0.Theorem 2.9. (The Linear Invariance Theorem for Wedges [29]). Let W be

a generating wedge in a finite dimensional vector space V . A linear map L : V → Vsatisfies the invariance condition

exp(tL)W ⊆W,

for all t ∈ R+0 , if and only if

L(X) ∈ T sX(W ),

for all X ∈ E1(W ).


At this point, we are ready to give a new proof of Kossakowski’s positivity resultin Theorem 2.8 using the linear invariance theorem for wedges. In particular, we areonly interested in proving the positivity result Theorem 2.8(i). We start with thefollowing lemma.

Lemma 2.10. Let V := her(N) = X ∈ CN×N | X† = X be the real vector spaceof all Hermitian matrices with the inner product 〈A,B〉 := Tr(A†B) for A,B ∈ V .Let

W := her+(N) := X ∈ her(N) | X ≥ 0be the pointed wedge of all positive semidefinite Hermitian matrices in V . Then,

(i) The wedge W is self-dual, i.e. W ∗ = W ,

(ii) The set of all extreme points of W is λuu† | λ ≥ 0 , ||u|| = 1,

(iii) The set of all exposed points in W coincides with the set of all extreme points.

Proof. (i). Any element A ∈W can be written as a non-negative linear combination

A =N∑

i=1

diuiu†i , di ≥ 0,

for some unitary matrix U = [u1|u2| . . . |uN ] and similarly, any element B ∈ V can beexpressed as

B =

N∑

j=1

bjzjz†j , bj ∈ R,

for some unitary matrix Z = [z1|z2| . . . |zN ]. Then, the dual wedge W ∗ is given by theset of all B ∈ V such that the following condition

⟨ N∑

j=1

bjzjz†j ,

N∑

i=1

diuiu†i

⟩=

N∑

i,j=1

bjdi |z†jui|2 ≥ 0

holds, which is equivalent to bj ≥ 0 for all j. This shows that W ∗ = W , i.e. W is selfdual.

For (ii), we refer to [31] that a ray λA ∈W | λ ≥ 0 is a face if and only if A hasrank 1. For (iii), note that by definition, any exposed point is an extreme point butthe converse is not necessarily true in general. We will show that for the particularwedge W = her+(N) the converse is also true. Suppose we take a particular extreme

point X = e1e†1, where e1 is a standard unit vector. Then the set X⊥ is given by the

set of all matrices B ∈ V such that

〈B,X〉 = Tr(Be1e†1) = e†1Be1 = 0.


So we have

X⊥ =B ∈ V | B =

[0 bb† B′

].

Using Eq.(2.7) and the fact that W is self dual, it turns out

opW (X) = X⊥ ∩W ∗ = X⊥ ∩W=

B ∈ W | B =

[0 00 B′

],

since positive semidefiniteness on the set X⊥ forces b equal to zero. It follows that

opW (X)⊥ =B ∈ V | B =

[a bb† 0

],

and again with the similar argument of positive semidefiniteness on opW (X)⊥, weobtain

opW (X)⊥ ∩W = ae1e†1 | a ≥ 0,such that

(X⊥ ∩W ∗)⊥ ∩W = (opW (X))⊥ ∩W = R+0 ·X.

Thus by Eq.(2.10), X = e1e†1 is an exposed point. Now, consider an arbitrary extreme

point of W which is of the form

Y = au1u†1 = aUe1e

†1U

† , a ≥ 0,

where u1 = Ue1 and U is unitary. The set Y ⊥ is given by the set of all matrices B ∈ Vsuch that

〈B, Y 〉 = aTr(BUe1e†1U

†) = ae†1U†BUe1 = 0,

and therefore, we obtain the equality

Y ⊥ = UX⊥U † , X = e1e†1.

Since W is invariant under unitary conjugation, we furthermore have

opW (Y ) = Y ⊥ ∩W = (UX⊥U †) ∩W = U(X⊥ ∩W )U †

= UopW (X)U †,

andopW (Y )⊥ = UopW (X)⊥U †,

and therefore,opW (Y )⊥ ∩W = U(opW (X)⊥ ∩W )U †

= R+0 Ue1e

†1U

† = R+0 u1u

†1.

So any extreme point is also an exposed point and we have equality between the setof extreme points and exposed points in W = her+(N).


Proof. (Theorem 2.8(i)). Observe that in Lemma 2.10, W is a generating wedge in V .Recall that W is also a pointed wedge. Thus, the set E1(W ) coincides with the setof all exposed points, which is by Lemma 2.10, equals to the set of all extreme pointsin W . Therefore, by Theorem 2.9, the invariance condition on the wedge W

exp(tL)W ⊆W , for all t ∈ R+0 ,

reduces to checking whetherL(X) ∈ T s

X(W )

holds for all extreme points X ∈ W . Without loss of generality, take any extremepoint of the form X = u1u

†1, with ||u1|| = 1. Recall

opW (X) := X⊥ ∩W ∗ = X⊥ ∩W= A ∈W | 〈A,X〉 = 0.

Since any A ∈W can be written as

A =N∑

i=1

diziz†i , di ≥ 0,

for some unitary matrix Z = [z1|z2| . . . |zN ], the set opW (X) is given by the matricesA ∈W satisfying the condition

Tr(

N∑

i=1

diziz†iu1u

†1) =

N∑

i=1

di|z†iu1|2 = 0.

Hence for X = u1u†1,

opW (X) =

N∑

i=2

diuiu†i

∣∣∣ uiNi=2 any orthonormal basis of u⊥1

.

Now, using Eq.(2.8) we have

T sX(W ) = opW (X)∗,

and by Eq.(2.5) for the dual wedge, the condition L(X) ∈ T sX(W ) is equivalent to the

condition⟨L(u1u

†1) ,

N∑

i=2

diuiu†i

⟩≥ 0 , for all di ≥ 0,

which is equivalent to

⟨L(u1u

†1), uiu

†i

⟩= Tr(uiu

†iL(u1u

†1)) ≥ 0, (2.11)


for i = 2, . . . , N . We then recall that the set P = Pi := uiu†i , i = 1, 2, . . . , N forms

the set of orthogonal one-dimensional self-adjoint projections

PiPj = δijPi , P †i = Pi ,

N∑

i

Pi = I,

such that the condition in Eq.(2.11) must hold for all families P. Therefore, we obtainthe desired result for positivity as in Theorem 2.8(i).

The problem of determining an explicit form of the infinitesimal generator ofa positive dynamical semigroup is open and no complete characterization has beenobtained up to now. However, if we restrict the positive dynamical semigroup to becompletely positive, then a very explicit form of the generator of a completely positivedynamical semigroup can be derived. By the positivity result of Theorem 2.8 andthe remarkable semialgebraic characterization of completely positive maps Λ usingthe positivity of the map (IN ⊗ Λ) (cf. Thereom 1.6 and discussion therein), Gorini,Kossakowski and Sudarshan (GKS) [24] provided an explicit form of the generatorsof completely positive dynamical semigroups (QDS) in the finite dimensional setting,cf. Theorem 2.11. The same result for the infinite dimensional case was obtainedindependently by Lindblad [50] using a different approach.

Theorem 2.11. (Gorini, Kossakowski and Sudarshan [24]) A linear mapL : MN (C) → MN (C) is the generator of completely positive dynamical semigroup(QDS) if and only if it is of the form

L(X) = −i[H,X] +1

2

N2−1∑

j,k=1

ajk

([Bj, XB

†k] + [BjX,B

†k]), (2.12)

where H is a Hermitian matrix with trace zero and BjN2−1j=1 is an orthonormal set of

complex matrices with trace zero, and the so-called GKS matrix A := [ajk] ∈MN2−1(C)is positive semidefinite. Here, [·, ·] denotes the matrix commutator [A,B] = AB−BA.

Proof. (Sketch). From Lemma 1.5, any Hermitian preserving linear operator Γ :MN (C) → MN(C) is of the form

Γ(X) =

N2∑

j,k=1

ajkMjXM†k , ajk = akj ∈ C, (2.13)

where MjN2

1 is an orthonormal basis of MN (C). Choose MN2 = 1√NIN and the

remaining elements of the basis MjN2−1j=1 := BjN2−1

j=1 , where all Bj have zero trace.

2.3. Various Forms of the Infinitesimal Generators 31

Then, by writing the summands of Eq.(2.13) which contain the element MN2 sepa-rately from the summation, we obtain

Γ(X) =1

NaN2N2X +

1√N

N2−1∑

i=1

(aiN2BiX + aN2iXB

†i

)+

N2−1∑

j,k=1

ajkBjXB†k

= −i[H,X] + G,X +N2−1∑

j,k=1

ajkBjXB†k , G,X := GX +XG,

whereH = 12i

(K†−K) and G = 12NaN2N2IN + 1

2(K†+K), withK = 1√

N

∑N2−1i=1 aiN2Bi.

The trace condition Tr(Γ(X)) = 0 for all X ∈ MN(C) (to ensure that exp(tΓ) is tracepreserving),

0 = Tr(Γ(X)) = Tr

((2G+

N2−1∑

j,k=1

ajkB†kBj

)X

),

implies that G = −12

∑N2−1j,k=1 ajkB

†kBj . Hence, the linear map (2.13) which generates

trace preserving flow exp(tΓ) can be written as

Γ(X) = −i[H,X] +1

2

N2−1∑

j,k=1

ajk

([Bj , XB

†k] + [BjX,B

†k]), (2.14)

where H = H†, Tr(H) = 0, Tr(Bj) = 0, and A† = A = [ajk] for 1 ≤ j, k ≤ N2 − 1.

From Theorem 1.6 and Lemma 1.3, we know that Λt is a completely positivedynamical semigroup if and only if (IN ⊗ Λt) is a positive dynamical semigroup.Suppose Γ generates Λt, then (IN ⊗ Γ) is the generator of (IN ⊗ Λt). Hence, toimpose the conditions for a linear map Γ to be the generator of a completely positivedynamical semigroup Λt, it is enough to check whether the map (IN ⊗ Γ) is thegenerator of a positive dynamical semigroup on MN (CN×N). By constructing the setof orthogonal one-dimensional self-adjoint projections P = PαN2

α=1 on MN(CN×N)and applying Theorem 2.8(i) to (IN ⊗ Γ), we have that

Tr[Pα(IN ⊗ Γ)Pβ] ≥ 0 ⇐⇒ A := [ajk] ≥ 0,

for α, β = 1, . . . , N2, α 6= β, which yields the desired result, see [24].

2.3 Various Forms of the Infinitesimal Generators

The dynamical equation on the set of density operators P

ρ(t) = L(ρ(t)) , ρ0 = ρ(0), ρ(t) ∈ P


where L is the generator of QDS admitting the form of Eq. (2.12) is called theLindblad-Kossakowski master equation. For convenience, we name such form of Lthe GKS form. This class of Markovian master equation is widely used in quantumphysics to model the dynamics of open quantum systems.

Markovian master equations can also be derived using physical assumptions dueto some specific conditions on the underlying quantum systems. This is called theconstructive approach from the first principle, see e.g. [4, 5]. Some of those Markovianmaster equations, even though they appear in a different form than the GKS form,are actually within the class of the Lindblad-Kossakowski master equation. On theother hand, we note that there are Markovian master equations, which similarly lookslike Eq.(2.12), but they are in fact contradicting the Lindblad-Kossakowski masterequation, i.e. they can not be written in the GKS form and hence, they do notcomply with the complete positivity. Markovian master equations of these types arefor example the Redfield/Agarwal master equation and Caldeira-Leggett/Oppenheim-Romero-Rochin master equation, see e.g. [5, 42].

Due to the fact that Markovian master equations can appear in a different formthan the GKS form, it is of interest to write the GKS form in different ways. Thiswill help to recognize whether a particular Markovian master equation is within theclass of the Lindblad-Kossakowski master equation.

2.3.1 Diagonal Lindblad Forms

We consider the Lindblad-Kossakowski master equation in the GKS form

ρ = L(ρ) = LH(ρ) + LD(ρ)

= −i[H, ρ] +1

2

N2−1∑

j,k=1

ajk

([Bj , ρB

†k] + [Bjρ,B

†k]),

(2.15)

where H = H†, Tr(H) = 0, A := [ajk] ≥ 0 and BjN2−1j=1 is an orthonormal basis of

sl(N,C) := X ∈ CN×N | Tr(X) = 0,

i.e. Tr(BjB†k) = δjk. We show that a change of basis does not change the GKS

structure of complete positivity.

Lemma 2.12. Consider the GKS master equation (2.15). For any unitary matrixV = [vjk] ∈ U(N2 − 1,C),

Fr :=

N2−1∑

j=1

vjrBj , r = 1, . . . , N2 − 1,


defines an orthonormal basis of sl(N,C). Then, with respect to the new basis FrN2−1r=1 ,

the GKS form reads

LD(ρ) =1

2

N2−1∑

r,l=1

brl

([Fr, ρF

†l ] + [Frρ, F

†l ]),

with B := [brl] = V †AV ≥ 0.

Proof. By direct calculation of the following term (while the other terms follow asimilar calculation), we have

N2−1∑

r,l=1

brlFrρF†l =

N2−1∑

r,l=1

brl

(N2−1∑

j=1

vjrBj

)ρ

(N2−1∑

k=1

vklB†k

)

=N2−1∑

j,k=1

(N2−1∑

r,l=1

brlvjrvkl

)BjρB

†k

=

N2−1∑

j,k=1

ajkBjρB†k,

with A := [ajk] = V BV † ≥ 0. The second statement follows. Checking for the newbasis of sl(N,C), we have Tr(Fr) = 0 and

Tr(FrF†l ) = Tr

(∑

j

vjrBj

∑

k

vklB†k

)

=∑

j,k

vjrvklTr(BjB†k) =

∑

j,k

vjrvklδjk

=∑

j

vjrvjl = δrl,

the orthonormality condition.

According to Lemma 2.12, we obtain the so-called diagonal Lindblad form bychoosing an arbitrary unitary matrix W that diagonalizes the GKS matrix A,

D = W †AW = Diag(γ1, γ2, . . . , γN2−1) , γi ≥ 0. (2.16)

Transforming the basis Fi =∑N2−1

j=1 wjiBj , i = 1, . . . , N2 − 1, the GKS form thentransforms to the diagonal Lindblad form [50], which can be written as

LD(ρ) =1

2

N2−1∑

i,l=1

dil

([Fi, ρF

†l ] + [Fiρ, F

†l ])

=1

2

N2−1∑

i=1

γi

([Fi, ρF

†i ] + [Fiρ, F

†i ])

=

N2−1∑

i=1

γi

(FiρF

†i − 1

2F †

i Fiρ−1

2ρF †

i Fi

).

(2.17)


In the diagonal Lindblad form Eq.(2.17), we impose that FiN2−1i=1 must have zero

trace and forms an orthonormal basis of sl(N,C). However, suppose that we considerthe apparently more general master equation

ρ = L(ρ) = −i[H, ρ] +

NA∑

i=1

pi

(ViρV

†i − 1

2V †

i Viρ−1

2ρV †

i Vi

), (2.18)

where H† = H , but not necessary with zero trace, and Vi ∈ CN×N are any complexmatrices, also not necessarily having zero trace. The number of summands NA isarbitrary but finite and pi > 0. We show that the master equation (2.18) is alsowithin the class of the Lindblad-Kossakowski master equation. First, note that Hand V can be written as H = H + kI, k ∈ R and V = V + cI, c ∈ C, whereTr(H) = 0 and Tr(V ) = 0, respectively. Plugging H and V in Eq. (2.18) andconsidering only one summand, we have

LH(ρ) = −i[H + kI, ρ] = −i[H, ρ] = LH(ρ)

LD(ρ) = 2(V + cI)ρ(V † + cI) − (V † + cI)(V + cI)ρ− ρ(V † + cI)(V + cI)

= 2V ρV † − V †V ρ− ρV †V + (cV − cV †)ρ+ ρ(cV † − cV ).

Note that the last two terms (cV − cV †) and (cV †− cV ) are skew Hermitian and havezero trace such that they can be absorbed as additional Hamiltonian term togetherwith −iH . So we see that the trace of H and Vi do not really matter. Now, consideringagain Eq.(2.18), it is easily seen that the flow is trace preserving since Tr(L(ρ)) = 0and it is indeed of the GKS form. Precisely, consider the dissipative term

LD =

NA∑

i=1

pi

(ViρV

†i − 1

2V †

i Viρ−1

2ρV †

i Vi

),

where now Tr(Vi) = 0. Choose any orthonormal basis BjN2−1j=1 of sl(N,C) and write

each Vi as Vi =∑N2−1

j=1 cijBj. For each summation index i, we obtain

ViρV†i − 1

2V †

i Viρ−1

2ρV †

i Vi =1

2

N2−1∑

j,k=1

aijk([Bj , ρB

†k] + [Bjρ,B

†k]),

with Ai := [aijk] = cic

†i ≥ 0, where c⊤i = [ci1 c

i2 . . . ciN2−1]. Therefore, Eq.(2.18)

assumes the GKS form Eq.(2.12) with GKS matrix

A =

NA∑

i=1

piAi ≥ 0.

Next, we give some examples of the Lindblad-Kossakowski master equationswhich are in the double-commutator forms.


Example 2.13. (Double-Commutator Equation, Vi Hermitian)

Consider the dissipative term of the Lindblad-Kossakowski master equation (2.18)

LD(ρ) =∑

i

ViρV†i − 1

2V †

i Viρ−1

2ρV †

i Vi,

with Vi = V †i Hermitian. Then,

LD(ρ) = −1

2

∑

i

V 2i ρ− 2ViρVi + ρV 2

i = −1

2

∑

i

[Vi, [Vi, ρ]],

and therefore, the Lindblad-Kossakowski master equation with Vi Hermitian can bewritten in the double-commutator form

ρ = −i[H, ρ] − 1

2

∑

i

[Vi, [Vi, ρ]]. (2.19)

This type of equation frequently appears in the area of nuclear magnetic resonance(NMR), see e.g. [23, 38, 39, 40], to model the dynamics of n-coupled spin-1/2 sys-tems under the presence of relaxation, which is particularly important in NMR spec-troscopy. The double commutator form Eq.(2.19) can be also deduced from the GKSform Eq.(2.12) as follows. Suppose the GKS matrix A is real symmetric and chooseBj to be Hermitian. Then, the diagonalizing matrix W in Eq.(2.16) is orthogonal with

F †i = Fi in Eq.(2.17) and we obtain the desired double-commutator form Eq.(2.19).

Example 2.14. (Double-Commutator Equation, Vi Normal)

More generally, we consider another version of the double-commutator equation,

LD(ρ) = −1

2

∑

i

[V †i , [Vi, ρ]], (2.20)

where Vi is assumed to be normal, i.e. [Vi, V†i ] = 0. The dissipative term Eq.(2.20)

appears in applications of NMR spectroscopy, see e.g. [23, 57], represented by coupledspin systems in the presence of longitudinal and transverse relaxation. At a firstglance, Eq.(2.20) does not look like in the Lindblad-Kossakowski form. However, astraightforward calculation using V †

i Vi = ViV†i shows that

LD(ρ) =1

2

∑

i

V †i Viρ− V †

i ρVi − ViρV†i + ρViV

†i

=1

2

∑

i

(1

2V †

i Viρ+1

2ρV †

i Vi − ViρV†i

)+

1

2

∑

i

(1

2ViV

†i ρ+

1

2ρViV

†i − V †

i ρVi

),

and therefore, it is of the form (2.18).


2.3.2 Decompositions of the Dissipative Terms

We recall the dissipative term of the Lindblad-Kossakowski master equation in thediagonal form

LD(ρ) :=∑

j

VjρV†j − 1

2V †

j Vjρ−1

2ρV †

j Vj , (2.21)

with Vj ∈ CN×N . We decompose each Vj into Hermitian and skew-Hermitian part (bydropping the subscript)

V = C + iD,

where C and D are Hermitian. Calculating each term of LD(ρ) in Eq.(2.21),

V ρV † = CρC − iCρD + iDρC +DρD−1

2ρV †V = −1

2ρC2 − i

2ρCD + i

2ρDC − 1

2ρD2

−12V †V ρ = −1

2C2ρ− i

2CDρ+ i

2DCρ− 1

2D2ρ,

and summing up, we have another equivalent form of the Lindblad-Kossakowski mas-ter equation,

LD(ρ) = −1

2

∑

j

([Cj , [Cj, ρ]] + [Dj , [Dj, ρ]]+

2iCjρDj − 2iDjρCj − i[Dj , Cj]ρ− iρ[Dj , Cj]).

(2.22)

Now we can use this equivalent form Eq.(2.22) to reinspect the previous examples.

1. Example 2.13. (Double-commutator equation with Vj Hermitian).

In Eq.(2.21) when Vj = V †j , we immediately see that the double-commutator

term in Eq.(2.19) is a special form of Eq.(2.22) with Dj = 0,

LD(ρ) = −1

2

∑

j

[Vj , [Vj, ρ]] = −1

2

∑

j

[Cj, [Cj , ρ]]. (2.23)

2. Example 2.14. (Equation (2.20) with Vj normal).By using V = (C + iD), the LD part in Eq.(2.20) reads

LD(ρ) = −1

2

∑

j

[V †j , [Vj, ρ]]

= −1

2

∑

j

[Cj, [Cj, ρ]] −1

2

∑

j

[Dj, [Dj , ρ]].(2.24)

We see that the double-commutator term [V †j , [Vj, ρ]] in Equation (2.20) is a

special form of Eq.(2.22) with [Dj , Cj] = 0 and CjρDj = DjρCj . Note that thecondition CjρDj = DjρCj , for all ρ ∈ P, is equivalent to Cj = kDj for somek ∈ C.

2.4. Properties of the Lindblad-Kossakowski Generator 37

3. Equation (2.21) with Vj normal.The normality of Vj in Eq.(2.21) implies (dropping the subscript j)

0 = [V, V †] = 2i[D,C].

In this case, the Lindblad form Eq.(2.21) becomes (2.22) with [Dj , Cj] = 0, i.e.

LD(ρ) = −1

2

∑

j

([Cj , [Cj, ρ]] + [Dj , [Dj, ρ]] + 2iCjρDj − 2iDjρCj

). (2.25)

2.4 Properties of the Lindblad-Kossakowski Gen-

erator

The first part of this section concerns about the unitality of the Lindblad-Kossakowskimaster equation and its relation to the purity of quantum states. The results on uni-tality and purity are indispensable when we discuss controllability issues of open quan-tum systems in Section 3.2.2. Moreover, as we will see later, the accessibility resultsof open quantum systems are stated with respect to the unital and non-unital case.The second part deals with the trace and decomposition of the Lindblad-Kossakowskigenerator. The dimensional aspect of the Lindblad-Kossakowski generator is studiedin the last part of this section. Precisely, we investigate the set of admissible Lindblad-Kossakowski generators. These results play crucial role when we derive conditions foraccessibility of the Lindblad-Kossakowski master equation and to obtain genericityresults, cf. Section 3.3 and Section 4.2, respectively.

We recall the dissipative part of the Lindblad-Kossakowski master equation inthe GKS form (2.12)

LD(ρ) =1

2

N2−1∑

j,k=1

ajk ([Bj , ρBk] + [Bjρ,Bk]) , (2.26)

where now we choose a particular orthonormal basis BjN2−1j=1 of sl(N,C) such that

B†j = Bj, i.e. each Bj is Hermitian. Note that their complex span forms the complex

vector space of sl(N,C) while the real span of BjN2−1j=1 yields the real vector space

of her0(N).

In what follows, we consider interchangeably the dissipative part LD of theLindblad-Kossakowski master equation in the diagonal Lindblad form Eq.(2.21) andthe GKS form Eq.(2.26). Both forms have been proven to be equivalent to the originalGKS form (2.12) in Theorem 2.11. In particular, when we deal with the GKS form,

we will always use the above prescribed orthonormal basis BjN2−1j=1 of sl(N,C), with

B†j = Bj .


2.4.1 Unitality and Purity

Definition 2.15. The Lindblad-Kossakowski master equation is called unital ifL(I) = 0.

It is obvious that the unital Lindblad-Kossakowski master equations leave theso-called completely mixed state IN/N unchanged, i.e. the completely mixed state isa fixed point or an equilibrium point of the unital master equation.

Lemma 2.16. (1) The Lindblad-Kossakowski master equation is unital (for theGKS form (2.26)), if and only if

N2−1∑

j<k

Im(ajk)[Bj, Bk] = 0.

In particular, unitality is satisfied when the GKS matrix A is real. However, theGKS matrix A being real is equivalent to unitality only for N = 2.

(2) The Lindblad-Kossakowski master equation is unital (for the diagonal Lindbladform (2.21)), if and only if ∑

j

[Vj, V†j ] = 0.

In particular, unitality is satisfied when each Vj is a normal matrix, i.e. VjV†j =

V †j Vj for all j.

Proof. (1). Since ajk = akj and [Bj , Bk] = −[Bk, Bj ], then

L(I) =

N2−1∑

j,k=1

ajk[Bj , Bk] =

N2−1∑

j<k

(ajk − akj)[Bj , Bk]

= 2iN2−1∑

j<k

Im(ajk)[Bj , Bk].

When Im(A) = 0, obviously L(I) = 0. For two-level systems N = 2, recall thatBj3

j=1 is given by the Pauli matrices

B1 = σx =1√2

[0 11 0

], B2 = σy =

1√2

[0 −ii 0

], B3 = σz =

1√2

[1 00 −1

]

with the commutation relation [σx, σy] =√

2iσz, [σy, σz] =√

2iσx and [σz, σx] =√

2iσy.Hence,

∑3j<k cjk[Bj , Bk] = 0, implies cjk := Im(ajk) = 0, for 1 ≤ j < k ≤ 3. For the

higher dimensional case N > 2, there always exist unital master equations where theGKS matrices are not necessarily real. We refer to Example 2.18.


(2). For the diagonal form (2.21), we have

L(I) =∑

j

VjV†j − 1

2V †

j Vj −1

2V †

j Vj =∑

j

[Vj, V†j ].

The condition VjV†j = V †

j Vj for all j immediately implies L(I) = 0.

Note that in the Lindblad form Eq.(2.22), the condition of unitality for Vj =Cj + iDj , with Cj , Dj ∈ her(N) easily translates into

∑j [Dj , Cj] = 0.

Example 2.17. (Unital equations). It is trivial to see that the Lindblad-Kossakowski master equations in the double-commutator form e.g,

(i) LD(ρ) = −1

2

∑

j

[Vj , [Vj, ρ]] , Vj ∈ her(N)

(ii) LD(ρ) = −1

2

∑

j

[Vj, [V†j , ρ]] = −1

2

∑

j

[Cj, [Cj , ρ]] + [Dj, [Dj , ρ]],

are unital. Moreover, the dissipative term of the form

LD(ρ) = −1

2

∑

j

[Cj , [Cj, ρ]] + [Dj , [Dj, ρ]] + 2iCjρDj − 2iDjρCj,

where [Dj , Cj] = 0, also yields a unital equation.

Example 2.18. (Unital equation with complex GKS matrix). We considera three level system N = 3. An orthonormal basis Bj8

j=1 of her0(3) is given by theso-called Gell-Mann matrices which are the generalization of the Pauli matrices toN = 3, see Appendix C.1. This Gell-Mann basis can be easily extended to arbitraryhigher dimension N . Now suppose we have a complex GKS matrix A ∈ her(8) suchthat

Im(a17) = −Im(a26) , Im(a16) = Im(a27) , Im(a15) = Im(a24),

and the remaining imaginary parts Im(ajk) = 0. The real parts of the GKS matrixRe(ajk) are assumed that A is positive semidefinite, to guarantee a valid Lindblad-Kossakowski master equation. This is always possible since, for example, the realdiagonal elements of A can be chosen to be very large majorizing the off-diagonalelements, i.e. ensuring positive semidefiniteness of A. Then, by the commutationrelations in Appendix C.1, it is immediate to see that

8∑

j<k=1

Im(ajk)[Bj , Bk] = − 1√2B4(Im(a17) + Im(a26)) +

1√2B5(Im(a16) − Im(a27))

+1√2B6(Im(a24) − Im(a15)) = 0.


Thus, the master equation is unital while the GKS matrix A is not necessarily real.Note that there are infinitely many ways of choosing the elements of Im(ajk) suchthat the condition of unitality is satisfied.

The following definition deals with the notion of purity of quantum states.

Definition 2.19. The function P : P → R, P (ρ) := Tr(ρ2) is called the purity of aquantum state. Note that 1/N ≤ P (ρ) ≤ 1.

The purity function is commonly used to measure the degree of mixture of aquantum state. It is also used to quantify the amount of information contained in aquantum state. For pure states, P (ρ) = Tr(ρ2) = Tr(ρ) = 1. Hence, it corresponds tothe fact that pure states carry maximal amount of quantum information. In contrast,the completely mixed state IN/N contains the least amount of quantum informationas P (ρ) = 1

N2 Tr(IN) = 1/N . In the evolution of open quantum systems governed bythe Lindblad-Kossakowski master equation, it is interesting to discover how the purityof the states evolves. In particular, it is instructive to show how purity is related tounitality.

Lemma 2.20. (Monotonically Decreasing Purity). Any unital Lindblad-Kossakowski master equation monotonically decreases the purity of quantum states.

Proof. Consider the Lindblad-Kossakowski master equation

ρ = L(ρ) = LH(ρ) + LD(ρ) , ρ ∈ P,

where the dissipative part LD is of the form Eq.(2.22). Note that the unitality condi-tion reduces to

∑j[Dj , Cj] = 0. Then, the unital dissipative part reads

LD(ρ) = LD1(ρ) + LD2(ρ) , withLD1(ρ) = −1

2

∑j[Cj , [Cj, ρ]] + [Dj , [Dj, ρ]] , LD2(ρ) = −1

2

∑j 2iCjρDj − 2iDjρCj.

The purity of the states evolves according to

P (ρ) =d

dtTr(ρ2) = 2Tr(ρρ) = 2〈ρ,L(ρ)〉 = 2〈ρ,LD(ρ)〉, (2.27)

since 〈ρ,LH(ρ)〉 = −iTr(ρ[H, ρ]) = 0. Now, by Lemma 2.2(3) and (1), we have thedouble-commutator LD1 is negative semidefinite such that

〈ρ,LD1(ρ)〉 ≤ 0 , for all ρ ∈ P,

and LD2 is skew-adjoint which implies

〈ρ,LD2(ρ)〉 = 0 , for all ρ ∈ P,

respectively. Thus we prove that P (ρ) ≤ 0, for all ρ ∈ P, and hence the purity ismonotonically decreasing.


Note that for closed quantum systems where LD = 0, the purity does not changeduring time evolution and the degree of mixture will be preserved. Lemma 2.20 pro-vides an easy proof that the unitality is a sufficient condition for monotonically de-creasing purity. Another proof using the Cauchy-Schwartz inequality can be foundin [49]. There it is shown that unitality is also necessary to guarantee monotoni-cally decreasing purity, in the finite dimensional case. Generally, for the non-unitalcase, purity can increase during time evolution depending on the initial state, since〈ρ,LD(ρ)〉 can not be guaranteed to be less than zero for all possible ρ ∈ P. Thisphenomenon in open quantum systems is sometimes referred as the purification ofmixed states toward the pure states.

2.4.2 Trace and Decomposition

Proposition 2.21. (Trace of L). Consider the linear operator

L : her(N) → her(N) , L = LH + LD , LD 6= 0

where LH(·) = −i[H, (·)], H ∈ her(N), and LD(·) =∑

j Vj(·)V †j − 1

2V †

j Vj(·)− 12(·)V †

j Vj,

Vj ∈ CN×N . Then the trace of linear operator L is strictly negative, i.e. Tr(L) < 0.

Proof. From Lemma 2.2(2), LH is skew-adjoint such that Tr(LH) = 0. Now writeVj = Cj + iDj 6= 0 and decompose LD = LD1 + LD2 + LD3 as in Eq.(2.22),

LD1((·)) = −1

2

∑

j

[Cj , [Cj, (·)]] + [Dj , [Dj, (·)]] 6= 0,

LD2((·)) = −∑

j

iCj(·)Dj − iDj(·)Cj , LD3((·)) =i

2

∑

j

[Dj, Cj](·) + (·)[Dj, Cj].

From Lemma 2.2(3), LD1 is self-adjoint and LD1 ≤ 0. Since LD1 6= 0, we obtainTr(LD1) < 0. By Lemma 2.2(1), LD2 is skew-adjoint which implies Tr(LD2) = 0.Again by Lemma 2.2(4), LD3 is self-adjoint since i[Dj , Cj] ∈ her(N) for Dj, Cj ∈her(N). However, LD3 is neither positive nor negative semidefinite. Thus, we want todirectly calculate

Tr(LD3) =N2∑

i=1

〈Gi,LD3(Gi)〉,

where GiN2

i=1 is an orthonormal basis of her(N). Use the fact that 2i[Dj , Cj] =

[Vj, V†j ] and suppose, without loss of generality, that LD3 has only one single V term,

(∗) Tr(LD3) =1

4

N2∑

i=1

〈Gi, [V, V†]Gi〉 + 〈Gi, Gi[V, V

†]〉

=1

2

N2∑

i=1

Tr(Gi[V, V†]Gi) =

1

2

N2∑

i=1

Tr(Gi(V V† − V †V )Gi).


Decompose V by the singular value decomposition as V = UΣZ†, where U , Z areunitary matrices and Σ is a real diagonal matrix. For the unitary matrix X := U−1Z,we obtain

V V † = UΣ2U † , V †V = ZΣ2Z† = UXΣ2X†U †

V V † − V †V = U(Σ2 −XΣ2X†)U † = UY U †,

where Y = Σ2 −XΣ2X†. We choose the orthonormal basis GiN2

i=1 of her(N), whereTr(GiGj) = δij , as follows :

Gi =

(i). uju†j (N terms)

(ii). 1√2(uju

†k + uku

†j), j < k (N2−N

2terms)

(iii). i√2(uju

†k − uku

†j), j < k (N2−N

2terms)

total terms = N2,

where uj denotes the j-th column vector of U = [u1 u2 . . . uN ]. Now calculate (∗)with respect to the basis defined above

(i).N∑

j

Tr(uju†jUY U

†uju†j) =

N∑

j

e⊤j Y ej

(ii).1

2

N∑

j<k

Tr((uju†k + uku

†j)UY U

†(uju†k + uku

†j))

=1

2

N∑

j<k

(Tr(uju

†kUY U

†uku†j) + Tr(uku

†jUY U

†uju†k))

=1

2

N∑

j<k

(e⊤k Y ek + e⊤j Y ej

)

(iii). −1

2

N∑

j<k

Tr((uju

†k − uku

†j)UY U

†(uju†k − uku

†j))

=1

2

N∑

j<k


).

Summing all terms (i), (ii) and (iii) we obtain

(∗) 2Tr(LD3) =N∑

j

e⊤j Y ej +N∑

j<k


)=

1

2

N∑

j,k=1

(e⊤j Y ej + e⊤k Y ek

)

= NTr(Y ) = 0,

since Y = Σ2 − XΣ2X†, where X is unitary. Note, we used the fact that ejNj=1

is a standard basis of CN and Y is a linear operator on CN . So we are done withTr(L) = Tr(LH + LD1 + LD2 + LD3) < 0.

By Lemma 2.20 and Proposition 2.21, we can summarize the properties of L inthe following proposition.


Proposition 2.22. The infinitesimal generator L ∈ gl(her(N)) of QDS can bedecomposed as L = Lher + Lskew, where Lher ∈ sym(her(N)), Lher 6= 0, is the self-adjoint part with Tr(Lher) < 0, and Lskew ∈ so(her(N)) is the skew-adjoint part withTr(Lskew) = 0. Moreover, for any unital generator, Lher ≤ 0.

Regarding the decomposition of LD ∈ gl(her(N)), it is interesting to see howLD, particularly for the GKS form (2.26), decomposes into the self-adjoint and skew-adjoint parts explicitly. This will be important when we further investigate the convexcone structure of the Lindblad-Kossakowski generator, cf. Section 2.4.3. In order todo this, we decompose the GKS form (2.26) according to the elements of the GKSmatrix A as follows,

LD(ρ) =1

2

N2−1∑

j,k=1

ajk ([Bj , ρBk] + [Bjρ,Bk])

= −1

2

N2−1∑

j

ajj[Bj, [Bj , ρ]] −1

2

N2−1∑

j<k

Re(ajk) ([Bj , [Bk, ρ]] + [Bk, [Bj, ρ]])

+i

2

N2−1∑

j<k

Im(ajk) (2BjρBk − 2BkρBj)

+i

2

N2−1∑

j<k

Im(ajk) ([Bj , Bk]ρ+ ρ[Bj , Bk]) .

(2.28)Then, we can identify the self-adjoint and skew-adjoint parts of LD by writing LD =LD1 + LD2 + LD3, with

LD1(ρ) := −1

2

N2−1∑

j

ajj[Bj, [Bj , ρ]] −1

2

N2−1∑

j<k

Re(ajk) ([Bj , [Bk, ρ]] + [Bk, [Bj, ρ]]) ,

(2.29)where LD1 ∈ sym(her(N)) is self-adjoint,

LD2(ρ) :=i

2

N2−1∑

j<k

Im(ajk) (2BjρBk − 2BkρBj) , (2.30)

where LD2 ∈ so(her(N)) is skew-adjoint and

LD3(ρ) :=i

2

N2−1∑

j<k

Im(ajk) ([Bj, Bk]ρ+ ρ[Bj , Bk]) , (2.31)

where LD3 ∈ sym(her(N)) is self-adjoint.


Lemma 2.23. Consider the unital LD in the GKS form Eq.(2.28). Then,

1. LD = LD1 + LD2 ∈ gl(her(N))

2. LD ∈ sym(her(N)), if additionally the GKS matrix A is real.

Proof. From Eq.(2.29)-(2.31), the unitality condition∑N2−1

j<k Im(ajk)[Bj , Bk] = 0 im-plies LD3 = 0. Hence, LD = LD1 + LD2 ∈ gl(her(N)). Moreover, real GKS matriximplies that LD2 and LD3 vanish. Hence, LD = LD1 which is self-adjoint.

Lemma 2.23 reveals a fact that the unitality of LD does not necessarily implyself-adjointness of LD unless the GKS matrix A is being real.

2.4.3 Dimensional Aspect

We introduce the reduced density operator σ ∈ her0(N) as

σ := ρ− IN/N , ρ ∈ P. (2.32)

The Lindblad-Kossakowski master equation (in the GKS form) for the reduced densityoperator reads

σ = ρ = L(ρ) = L(σ + IN/N)

= −i[H, σ] +1

2

N2−1∑

j,k=1

ajk ([Bj , σBk] + [Bjσ,Bk]) +1

N

N2−1∑

j,k=1

ajk[Bj, Bk]

= LH(σ) + LaffD (σ),

(2.33)

withLaff

D (σ) := LD(σ) + ν,

ν :=1

N

N2−1∑

j,k=1

ajk[Bj , Bk] =2i

N

N2−1∑

j<k

Im(ajk)[Bj, Bk].(2.34)

Observe that LaffD is an affine operator on her0(N) instead of a linear one. Hence,

L is an affine operator acting on her0(N). In Eq.(2.33) and (2.34), we have nowLH ∈ gl(her0(N)) and LD ∈ gl(her0(N)), acting on the vector space her0(N) insteadof her(N). In the sequel, we investigate the convex cone of all admissible Lindblad-Kossakowski generators L = LH + LD in gl(her0(N)).

Proposition 2.24. (Dimensional Aspect of L). Consider the Lindblad-Kos-sakowski master equation for the reduced density operator Eq.(2.33) and(2.34).

(a) For the unital case, let Lu denote the set of all admissible Lindblad-Kossakowskigenerators L = LH + LD ∈ gl(her0(N)), where LD is such that the constraint∑N2−1

j<k Im(ajk)[Bj, Bk] = 0 holds.


(b) For the non-unital case, let Ln denote the set of all admissible Lindblad -Kossakowski generators L = LH + LD ∈ gl(her0(N)), without any constrainton LD.

Then, Lu and Ln are both closed convex cones of gl(her0(N)) with non-empty interior,i.e. Lu and Ln are generating cones in gl(her0(N)).

Proof. We start with the unital case (a). The fact that Lu is a convex cone is clear sinceLu +Lu ⊆ Lu and R+

0 Lu ⊆ Lu. It is also closed since the non-negativity constraint on

the GKS matrix A ≥ 0 together with the unitality constraint∑N2−1

j<k Im(ajk)[Bj, Bk] =0 defines a closed subset. Now we show that Lu has non-empty interior.

Consider the decomposition of L = LH + LD, with LD = LD1 + LD2 + LD3,where LD1,LD2 and LD3 are according to Eq.(2.29)-(2.31). Firstly, we consider LD1.It turns out that LD1 ∈ sym(her(N)) is formed by some real linear combinations

LD1 =d∑

j≤k

cjk(Xjk −1

2Yjk) , cjj := ajj, cjk := Re(ajk), j < k,

where Xjk,Yjk ∈ sym(her(N)) are linear operators defined in Lemma 2.4. In Eq.(2.29),we see that LD1 is of double-commutator form such that LD1 also leaves her0(N)invariant, i.e. LD1 ∈ sym(her0(N)). Now, define the real span

LD1 := span⟨Xjk −

1

2Yjk

∣∣∣ 1 ≤ j ≤ k ≤ d⟩⊆ sym(her0(N)).

By Lemma 2.4 and dimension counting, there are d(d + 1)/2 linearly independentXjk − 1

2Yjk such that we conclude

LD1 = sym(her0(N)), (2.35)

since the dimension of sym(her0(N)) is exactly d(d+1)/2. Secondly, we consider LH +LD2 and recall Vjk as defined in Lemma 2.4. We choose IN/

√N and an orthonormal

basis Bjd=N2−1j=1 of her0(N) as an orthonormal basis Gjp=N2

j=1 of her(N). Then, wewrite LH + LD2 as a real linear combination of Vjk as follows,

p=N2∑

j<k

cjkVjk(ρ) = 1√N

d=N2−1∑

l=1

hl(−iBlρ+ iρBl) +d=N2−1∑

j<k

cjk(iBjρBk − iBkρBj)

= −i[H, ρ] + LD2(ρ) = LH(ρ) + LD2(ρ),

since H ∈ her0(N) can be expressed as H = 1√N

∑dl hlBl and cjk := Im(ajk) for

1 ≤ j < k ≤ d. Next, define

VHl (·) := −i[Bl, (·)] ∈ so(her(N))

VDjk(·) := iBj(·)Bk − iBk(·)Bj ∈ so(her(N)).


We know from Lemma 2.4 that VHl and VD

jk are mutually orthogonal. We also definethe following real span

LH := span〈VHl | 1 ≤ l ≤ d〉 ⊂ so(her(N))

LD2 := span〈VDjk | 1 ≤ j < k ≤ d〉 ⊂ so(her(N)).

From Eq.(2.2) in Remark 2.5 we have

LH ⊕ LD2 = span〈Vjk | 1 ≤ j < k ≤ p〉 = so(her(N)). (2.36)

Recall that VHl also leaves her0(N) invariant, so actually we have

LH ⊂ so(her0(N)). (2.37)

However, we note that LD2 ∈ LD2 in general does not leave her0(N) invariant. For theunital case, it is indeed the unitality condition that now makes LD2 ∈ so(her0(N)).Precisely,

LuD2 :=

∑

j<k

cjkVDjk

∣∣∣d∑

j<k

cjk[Bj, Bk] = 0⊂ so(her0(N)) (2.38)

since, by the unitality condition∑d

j<k cjk[Bj , Bk] = 0, one has

Tr( d∑

j<k

cjk(iBjσBk − iBkσBj))

= Tr( d∑

j<k

cjki[Bk, Bj ]σ)

= 0,

for all σ ∈ her0(N). So far, by Eq.(2.37) and Eq.(2.38), we have

LH ⊕ LuD2 ⊂ so(her0(N)). (2.39)

We now claim the converse of above inclusion. Let her(N) = her0(N) ⊕ 〈IN〉. Take

any X ∈ so(her0(N)) regarded as an element X ∈ so(her(N)) as follows,

X∣∣her0(N)

= X , X∣∣〈IN 〉 = 0.

Then, by Eq.(2.36), X ∈ so(her(N)) can be written as

X = X1 + X2 , X1 ∈ LH , X2 ∈ LD2.

Since LH ⊂ so(her0(N)) by Eq.(2.37), this implies that X2 has to be in so(her0(N)).Such X2 =

∑j<k cjkVD

jk with Tr(X2(σ)) = 0 for all σ ∈ her0(N) is exactly given by

the unitality condition∑d

j<k cjk[Bj, Bk] = 0. Hence,

X ∈ LH ⊕ LuD2,

2.5. Matrix Representation of the Master Equation 47

and thus we have shown

LH ⊕ LuD2 = so(her0(N)). (2.40)

Finally, recall the proof of Lemma 2.23 to see that LD3 = 0 for the unital case. Bycombining Eq.(2.35) and Eq.(2.40) and the fact that sym(her0(N)) and so(her0(N))are orthogonal complement to each other in gl(her0(N)), we have

span〈Lu〉 = LD1 ⊕ LH ⊕ LuD2 = sym(her0(N)) ⊕ so(her0(N))

= gl(her0(N)).(2.41)

Since the span of the convex cone of unital Lindblad-Kossakowski generators Lu isthe whole gl(her0(N)), we conclude that Lu has non-empty interior, see [29].

For the non-unital case (b), it is similar (as in the unital case) to show that Ln isa closed convex cone. Moreover, since we do not have any constraint on LD, we haveLu ⊆ Ln. Thus, we conclude that span〈Ln〉 is the whole gl(her0(N)) and therefore Ln

also has non-empty interior. This completes the proof.

2.5 Matrix Representation of the Master Equation

We have seen that the Lindblad-Kossakowski generator L can be considered as anaffine operator acting on her0(N). In this section, we consider a version of theLindblad-Kossakowski master equation where the affine operator on her0(N) is nowrepresented as a linear operator (matrix) acting on Rd, d := dim(her0(N)) = N2 − 1,plus a translation on Rd. Consequently, the properties of the Lindblad-Kossakowskigenerator such as trace, decomposition and dimensional aspect with respect to thismatrix representation follows immediately from Section 2.4, since those properties arecompletely independent from a particular choice of basis. This particular matrix rep-resentation will be useful to state accesssibility results of the Lindblad-Kossakowskimaster equation.

2.5.1 Vector of Coherence Representation

We consider the set of all Hermitian matrices with trace one

her1(N) = X ∈ her(N) | Tr(X) = 1 ⊂ her(N)

as an affine subspace of the real vector space her(N). Since her(N) and the realEuclidean space RN2

are isomorphic as vector spaces, we can identify her1(N) withRd, d = N2 − 1. The set of all density operators P ⊂ her1(N) then defines a convexsubset of her(N). Thus, we can identify P with a closed subset of Rd as follows. Writeρ ∈ P in term of the reduced density operator Eq.(2.32), ρ = IN/N +σ, σ ∈ her0(N).


By defining B0 = IN/√N and v0 = 1/

√N , we can express ρ in term of an orthonormal

basis Bjdj=1 of her0(N) uniquely as

ρ = v0B0 +

d∑

j=1

Tr(ρBj)Bj =

d∑

j=0

vjBj , (2.42)

where vj := Tr(ρBj). Consequently, as an element of the affine vector space her1(N),ρ can be represented as a vector of size d+ 1,

v = [v0 v1 . . . vd]⊤ , v0 = 1/

√N.

Hence, ρ is exactly parameterized by d number of parameters vj , j = 1, 2, . . . , d. Interm of v, the condition Tr(ρ2) ≤ 1 reads

1 ≥ Tr(ρ2) =d∑

j,k=0

vjvkTr(BjBk) = v20Tr(B2

0) +d∑

j=1

v2j =

1

N+

d∑

j=1

v2j , (2.43)

or equivalently,∑d

j=1 v2j ≤ 1−(1/N). Finally, this leads to the identification of ρ with

the so-called Bloch ball

Bd := x ∈ Rd | ||x|| ≤√

1 − 1/N ⊂ Rd

via the mapΦ : P −→ Bd , ρ 7−→ v := [v1 v2 . . . vd]

⊤. (2.44)

Here, vj := Tr(ρBj) and Bjdj=1 is an orthonormal basis of her0(N), e.g. the Pauli

matrices for N = 2, the Gell-Mann matrices for N = 3, the generalized Pauli orGell-Mann matrices for N > 3. For n-coupled spin-1/2 systems (N = 2n), we can usethe product basis as often encountered in NMR application, see e.g. [19, 37].

2.5.2 Positivity and Confinement in the Bloch Ball

In two-level quantum systems (N = 2), the representation of quantum states aselements in the Bloch ball (called the Bloch vector in this particular case) providesa useful tool to understand basics of quantum computing in quantum informationtheory. In this case, the Bloch ball is a three dimensional ball.

In higher dimensional quantum systems (N > 2), one might expect that thevector of coherence generalizes the Bloch vector in the higher dimensional Bloch ball.However, careful consideration must be made since we have not yet carried over thepositivity constraint of ρ to the geometry of the Bloch ball in the higher dimensions.We briefly illustrate this issue by giving some examples.

Lemma 2.25. The positivity of the density operator ρ is equivalent with the con-finement of the vector of coherence v in the Bloch ball only for N = 2. For N > 2,the positivity implies the confinement but the converse does not necessarily hold.


Proof. Consider two-level systems (N = 2) where the density operator can be param-eterized by using the Pauli matrices Bx = σx, By = σy, Bz = σz as follows

ρ = v0B0 + vxBx + vyBy + vzBz =1√2

1√2

+ vz vx − ivy

vx + ivy1√2− vz

.

The condition for positivity simply reads Det(ρ) = 12− v2

z − (v2x + v2

y) ≥ 0, which isequivalent to

v2x + v2

y + v2z ≤ 1 − 1/N =

1

2,

i.e, the vector of coherence v = [vx vy vz]⊤ is inside the Bloch ball of radius 1/

√2. So

any point in the Bloch ball correponds to a physically valid quantum state ρ satisfyingpositivity condition.

For N > 2, the situation changes. We have easily seen that the conditionsTr(ρ) = 1 and ρ ≥ 0 implies the condition Tr(ρ2) ≤ 1, which is equivalent to theconfinement of v in the Bloch ball, that is

Tr(ρ) = 1 , ρ ≥ 0 =⇒ Tr(ρ2) ≤ 1 ⇐⇒d∑

i=1

v2i ≤ 1 − (1/N).

Regarding to the failure of the converse statement, we will only give some easy counterexamples from 3-level systems that the confinement of v in the Bloch ball does notgenerally correspond to the positivity of ρ.

Example 2.26. Now consider parameterization of ρ in 3-level systems using the(3 × 3) Gell-Mann matrices (Appendix C.1) as follows

ρ =1√2

√2

3+ a + b√

3c− id e− if

c+ id√

23− a+ b√

3g − ih

e+ if g + ih√

23− 2b√

3

,

where the alphabet letters a, . . . , h stand for vi8i=1. Now take b = 1/

√3 and set

the others equal to zero such that we have

8∑

i=1

v2i = b2 = 1/3 ≤ 1 − (1/N) = 1 − (1/3) = 2/3,

which shows that v is inside the Bloch ball. Then,

ρ =1√2

√2+13

0 0

0√

2+13

0

0 0√

2−23


is not positive semidefinite, but satisfies the Bloch ball constraint. Hence v does notrepresent a physically valid quantum state. As another example, take a =

√2/24,

b =√

6/24, c =√

10/72, d =√

12/72, and the other parameters are zero. We stillhave

Tr(ρ2) = 35/36 < 1 ,

8∑

i=1

v2i = 23/36 < 2/3 , rank(ρ) = 3.

The vector of coherence is inside the Bloch ball and it looks like a mixed state,but again the eigenvalues of ρ are (−0.0181; 0.25; 0.768) showing that ρ is again notpositive semidefinite.

Moreover, we can also easily find a point on the boundary of the Bloch ball(which is suspected to represent a pure state) which does not represent a valid ρ. Foran example, take b = 1/

√3, c = d = 1/

√6 and the other parameters are zero. Then,

ρ =1√2

√2+13

1−i√6

01+i√

6

√2+13

0

0 0√

2−23

satisfies Tr(ρ) = Tr(ρ2) = 1, i.e.∑n

i=1 v2i = b2 + c2 + d2 = 2/3 = 1 − 1/N . So ρ

corresponds to a point on the boundary of the Bloch ball. But again it is clear thatρ is not positive semidefinite and even rank(ρ) = 3 6= 1, which is inconsistent by thedefinition of pure states as ρ with rank one.

The failure of surjectivity of the map Φ in Eq.(2.44) in this case can be easilyidentified from the positivity constraint of ρ. In view of ρ represented by the Gell-Mann matrices, one of the inequalities guaranteeing positivity of ρ reads

√2

3− 2b√

3≥ 0

and thus b ≤√

66

. But the radius of the Bloch ball is√

1 − 1/N =√

2√3

= 2√

66

, still

greater than the constraint b ≤√

66

, hence leaving many points in the Bloch ballincompatible with the positivity requirement.

Example 2.26 shows that the generalization of Bloch ball for higher dimensionalquantum systems (N > 2) is no longer convenient due to the positivity constraint.More information addressing the intricate geometry of Φ(P) ( Bd and the relatedissues about the correspondence of the unitary orbit of ρ with the submanifolds con-tained in the Bloch ball for N > 2 can be found in [54]. Here, we summarize sometrivial facts about the map Φ in Eq.(2.44).

Proposition 2.27. Consider the map Φ : P −→ Bd, ρ 7−→ v defined in Eq.(2.44).Then, (a) The map Φ is bijective only for N = 2. (b) For N > 2, Φ is injective butnot surjective. Moreover, the image of Φ is not dense in Bd.


Proof. The first point and non-surjectivity of Φ in the second point follow from Lemma2.25. Now note that the map Φ is continous and the set of all density operators Pis closed and bounded in her(N). Under a continuous map, the image Φ(P) of thecompact set P is also a compact set in Bd. Non-surjectivity of Φ and closedness ofΦ(P) imply that the set D := Bd \ Φ(P) is open and non-empty. For every pointvp ∈ D, we can always find an open neighborhood V ⊂ D of vp. Thus, we have shown

that the closure of Φ(P) can never be the whole Bloch ball Bd, i.e. Φ(P) ( Bd andthe proposition follows.

2.5.3 Master Equation for the Vector of Coherence

By the identification of ρ ∈ P with its vector of coherence v ∈ Bd, we obtain aversion of the Lindblad-Kossakowski master equation for the vector of coherence. Inorder to transfer the Lindblad-Kossakowski master equation to the equation for thevector of coherence, we explicitly make use the structure of the commutator and anti-commutator of Bjd

j=1, d = N2−1, as an orthonormal basis of her0(N). Note that upto the imaginary unit, the set her0(N) is equal to su(N), the set of all skew-Hermitianmatrices with trace zero. Hence they are isomorphic as real vector spaces and iBjd

j=1

becomes a basis for su(N),

span〈iB1, iB2, . . . , iBd〉 = su(N) , d = N2 − 1.

Any basis Bjdj=1 of her0(N) defines the following commutation relation

[Bj, Bk] = BjBk −BkBj = i∑

m

fjkmBm, (2.45)

where fjkm are the (commutation) structure constants of su(N), which are real andchanges its sign with respect to the interchange of any two indices, e.g. fjkm = −fkjm =−fmkj . Additionally, the following anti-commutation relation is also defined by

Bj, Bk := BjBk +BkBj =2

NINδjk +

∑

m

gjkmBm, (2.46)

where gjkm are the (anti-commutation) structure constants, which are real and invari-ant with respect to any permutation of indices, e.g. gjkm = gkjm = gkmj.

Using these structure constants, we have the following version of the Lindblad-Kosssakowski master equation in the GKS form for the vector of coherence,

v = LHv + LaffD v , v ∈ Bd

= LHv + LDv + q,(2.47)

where LH and LaffD denote the matrix representations of the linear and affine operator

LH and LaffD on her0(N) of the Lindblad-Kossakowski master equation cf. Eq.(2.33)


and (2.34), respectively. The matrix representation of the linear part LD of LaffD is

then expressed by

LD =

d∑

jk=1

ajkLjk, (2.48)

and the translation part q is expressed by

q =d∑

j,k=1

ajkpjkv0. (2.49)

Here, A = [ajk] ≥ 0 is the corresponding GKS matrix. With respect to a particularorthonormal basis Bjd

j=1 of her0(N), the matrix representation LH and Ljk can bewritten in term of structure constants fjkm and gjkm as follows,

• LH := [LHlr ], with H =

∑dm=1 hmBm and

LHlr =

d∑

m=1

hmflmr. (2.50)

• Ljk := [Ljklr ], pjk and v0,

Ljklr = −1

4

d∑

m=1

fjml(fkmr − igkmr) + fkml(fjmr + igjmr)

pjk =i√N

[fjk1 . . . fjkd]⊤ , v0 = 1/

√N.

(2.51)

For full details on the explicit calculations for LH , Ljk and pjk, we refer to AppendixC.2.

Example 2.28. (The Unital Equation for the Vector of Coherence). Theunital Lindblad-Kossakowski master equation leaves the completely mixed state ρ∗ =IN/N invariant. In the vector of coherence representation, the completely mixedstates ρ∗ translates to the zero vector of coherence, v∗ = 0. Hence, the correspondingunitality condition for the vector of coherence master equation is given by

(LH + LaffD )(0) = 0.

Since LaffD has affine action on v, the unitality is equivalent to the translational part

of LaffD being zero, i.e.

q =d∑

j,k

ajkpjkv0 = 0.


Consequently, the unital master equation for the vector of coherence can be expressedby the linear action of LD on v without the translation part q, i.e.

v = LHv + LDv = LHv +d∑

j,k=1

ajkLjkv. (2.52)

In Lemma 2.20, we have already seen that the unital equations monotonically decreasethe purity Tr(ρ2). In term of the vector of coherence representation, this is equivalentto the monotonically decreasing of ||v||. In the generic case, the unital equations drivesome initial states asymptotically toward the center of the Bloch ball v∗ = 0, which isequivalent to the completely mixed state ρ∗ = IN/N , underlining the phenomenon ofenergy dissipation and relaxation of the systems towards its equilibrium state. For thenon-unital case, the norm ||v|| might increase depending on the initial state. In thiscase, the purification phenomenon happens as the vector of coherence asymptoticallyreach the boundary of the Bloch ball and becomes a pure state.

Decomposition of LD

In the view of the decomposition of LD ∈ gl(her0(N)) into the self-adjoint and skew-adjoint parts, it is instructive to see how the corresponding matrix representationLD ∈ gl(d,R) decomposes into the symmetric and skew-symmetric matrices. Thisformalisms will be indeed useful when we provide some particular explicit examplesdealing with accessibility of N -level quantum systems, cf. Section 3.3.

First, we note that the matrix representation Ljk has some symmetric propertieson its real and imaginary parts with respect to the interchange of the indices j, k, i.e.

Re(Ljk) = Re(Lkj) , Im(Ljk) = −Im(Lkj), (2.53)

as can be seen

Re(Ljklr ) = −1

4

∑m fjmlfkmr + fkmlfjmr = Re(Lkj

lr )

Im(Ljklr ) = 1

4

∑m fjmlgkmr − fkmlgjmr = −Im(Lkj

lr ).

Thus, the matrix representation LD in Eq.(2.48) can be decomposed as

LD =

d∑

j,k=1

ajkLjk

=

d∑

j

ajjLjj + 2

d∑

j<k

Re(ajk)Re(Ljk) − 2

d∑

j<k

Im(ajk)Im(Ljk).

(2.54)

It is clear now that LD ∈ gl(d,R) since Ljj, Re(Ljk) and Im(Ljk) are in gl(d,R).


Likewise for the translational part q in Eq.(2.34), we have

q =

d∑

j,k=1

ajkpjkv0 =

d∑

j

ajjpjjv0 +

d∑

j<k

(ajkpjk + akjpkj)v0

=

d∑

j<k

(ajk − akj)pjkv0 = 2i

d∑

j<k

Im(ajk)pjkv0

= − 2√N

d∑

j<k

Im(ajk)[fjk1 fjk2 . . . fjkd]⊤v0 , v0 = 1/

√N.

(2.55)

Note that in Eq.(2.55), we have used the fact that pjj = 0 (since fjjx = 0) andpjk = −pkj (since fjkx = −fkjx), cf. Eq.(2.51). Therefore, q ∈ Rd and Laff

D (·) =LD(·) + q : Rd → Rd.

Example 2.29. (Symmetric and Skew-Symmetric Decomposition of LD)

Consider the set of all (d× d) real symmetric matrices

sym(d,R) := X ∈ gl(d,R) | X⊤ = X,

and the set of all (d× d) real skew-symmetric matrices

so(d) := X ∈ gl(d,R) | X⊤ = −X.

From Eq.(2.50) for the matrix representation LH of LH , it is obvious that LH is askew-symmetric matrix since

LHlr =

∑

m=1

hmflmr = −∑

m=1

hmfrml = −LHrl ,

i.e. LH ∈ so(d). This corresponds to the fact LH(·) = −i[H, (·)] is a skew-adjointlinear operator. Moreover, in the view of the decomposition of LD, clearly we canalso decompose the corresponding LD ∈ gl(d,R) in Eq.(2.54) into the symmetric andskew-symmetric matrices. It is apparent that Ljj and Re(Ljk) are symmetric matricessince from Eq.(2.51) we have

Ljjlr = −1

2

∑m fjmlfjmr = Ljj

rl

Re(Ljklr ) = −1

4

∑m fjmlfkmr + fkmlfjmr = Re(Ljk

rl ).

However, Im(Ljk) is neither symmetric nor skew-symmetric. So we can further de-compose

Im(Ljk) := Sjk +Kjk,

where

Sjk =1

2Im(Ljk − L⊤

jk) ∈ so(d) , Kjk =1

2Im(Ljk + L⊤

jk) ∈ sym(d,R).


Finally, we have the following decomposition of LD ∈ gl(d,R) in accordance with thedecomposition of LD(her0(N)), i.e.

LD = LD1 + LD2 + LD3 ∈ gl(d,R), (2.56)

where the matrix representation LD1 of the self-adjoint LD1 (cf. Eq.(2.29)) is given by

LD1 =

d∑

j=1

ajjLjj + 2

d∑

j<k

Re(ajk)Re(Ljk) ∈ sym(d,R), (2.57)

and the matrix representations LD2 and LD3 of the skew-adjoint LD2 (cf. Eq.(2.30))and the self-adjoint LD3 (cf. Eq.(2.31)) are given by

LD2 = −2d∑

j<k

Im(ajk)Sjk ∈ so(d) , LD3 = −2d∑

j<k

Im(ajk)Kjk ∈ sym(d,R),

(2.58)respectively. We note that for the particular class of unital equations with the realGKS matrices, obviously LD = LD1 is symmetric, corresponding to the self-adjointLD1, cf. Lemma 2.23.

Remark 2.30. 1. (Trace of LD). In Proposition 2.21, we already proved thatthe Lindblad-Kossakowski generator LD has strictly negative trace. Since thetrace of a linear operator is independent of a particular matrix representation, itfollows immediately that Tr(LD) < 0. For the sake of completeness, an explicitcalculation of Tr(LD) by directly calculating the sum of the diagonal elementsof LD ∈ gl(d,R) in Eq.(2.54) is presented in Appendix C.3.

2. (Dimensional Aspect of LH +LD). In the view of Proposition 2.24, we can alsodeduce that the set of all admissible Lindblad-Kossakowski generators LH+LD ∈gl(d,R) is exactly described by the statement in Proposition 2.24 by simplyreplacing gl(her0(N)) with gl(d,R) since her0(N) and Rd, d = N2 − 1 being thedimension of her0(N), are isomorphic as real vector spaces.

Chapter 3

Controllability of the Lindblad-Kossakowski Master Equation

3.1 Bilinear Control Systems on Lie Groups

In what follows, we will use a Lie-theoretical approach to address control-theoreticissues for the Lindblad-Kossakowski master equation. Therefore, we recall some well-known results of bilinear control systems evolving on Lie groups and systems in-duced via a smooth Lie group action. This serves as our main machinery to derivecontrollability and accessibility results for closed quantum systems modelled by theLiouville-von Neumann master equation and for open quantum systems described bythe Lindblad-Kossakowski master equation, respectively.

3.1.1 Controllability and Accessibility on Lie Groups

We summarize some well-known control-theoretic ideas of bilinear control systems onmatrix Lie groups. Consider a bilinear or, more precisely, a right invariant controlsystem (Σ) on a connected matrix Lie group G with Lie algebra g,

(Σ) : X =

(A0 +

m∑

k=1

uk(t)Ak

)X, X(0) = X0 ∈ G, uk(t) ∈ R, (3.1)

where the Lie algebra elements A0 ∈ g and A1, . . . , Am ∈ g are called drift term andcontrol directions, respectively. The set of all admissible controls uk : R → R containsat least all piecewise constant functions. For the sake of simplicity, in what follows,we further assume that the set of admissible controls is equal to the set of all piecewiseconstant functions. For more general classes of admissible controls, one can consult[13, 33] for further information and details. Then, we define the following reachablesets of system (Σ), see e.g. [13, 33, 35],

56

3.1. Bilinear Control Systems on Lie Groups 57

• RΣ,T (X0) is defined as the set of all possible final points X(T ) of the solutionsof Eq.(3.1) at T ≥ 0, starting from the initial point X0. Thus, RΣ,T (X0) is theset of states reachable from X0 at time T .

• RΣ,≤T (X0) :=⋃

0≤t≤T RΣ,t(X0) is called the reachable set of (Σ) from X0 up totime T .

• RΣ(X0) :=⋃

t≥0 RΣ,t(X0) is called the reachable set of (Σ) from X0.

By right invariance of (Σ), we obtain

RΣ,T (X0) = RΣ,T (I)X0, RΣ,≤T (X0) = RΣ,≤T (I)X0, RΣ(X0) = RΣ(I)X0, (3.2)

for all X0 ∈ G. Therefore, we can study the reachable sets from any initial point byinspecting the reachable sets from the identity element of G.

Definition 3.1. The system (Σ) is called

(i) accessible, if the reachable set RΣ(I) has non-empty interior in G.

(ii) controllable, if the reachable set RΣ(I) coincides with the whole Lie group, i.e.RΣ(I) = G.

Note that the concept of controllability originally means that the system iscontrollable when any point X ∈ G can be reached from any initial point X0 ∈ G.That is, RΣ(X0) = G, the reachable set from any X0 is equal to G. This holdssimilarly for accessibility, the reachable set from any initial point RΣ(X0) has non-empty interior in G. For control systems on G where the right invariance propertyEq.(3.2) holds, the original concept of controllability and accessibility can be expressedvia Definition 3.1. Hence, it is enough to study controllability and accessibility in theoriginal sense via the reachable set RΣ(I).

Definition 3.2. Consider the system (Σ). We define

(i) the system Lie algebra sΣ as the smallest Lie subalgebra of g which containsA0, A1, . . . , Am, or for short sΣ := 〈A0, A1, . . . , Am〉LA.

(ii) the system group GΣ as the smallest subgroup of G which contains the reachableset RΣ(I).

The system Lie algebra sΣ is the smallest linear subspace of g spanned by A0, A1, . . . , Am

and all possible iterated commutators

[A0, Ai], [Ai, Aj ], [A0, [Ai, Aj ]], [Ak, [Ai, Aj]], . . . (3.3)

We remark that 〈exp(sΣ)〉G is the smallest subgroup of G containing all one-parametersubgroups eAt, A ∈ sΣ. Thus, by Yamabe’s theorem, 〈exp(sΣ)〉G is a Lie subgroup of

58 Chapter 3. Controllability of the Lindblad- Kossakowski Master Equation

G with the Lie algebra sΣ. Moreover, it is easy to see that the subgroup 〈exp(sΣ)〉Gcontains the reachable set RΣ(I) and one can show

〈exp(sΣ)〉G = 〈RΣ(I)〉G =: GΣ. (3.4)

In particular, the system group GΣ is a Lie subgroup of G. For further details, see[33, 35].

Next, we recall some results characterizing necessary and sufficient conditionsfor controllability and accessibility of bilinear control systems on Lie groups [33, 35].

Theorem 3.3. (Jurdjevic and Sussmann [35])

(a) The system (Σ) is accessible if and only if the system Lie algebra satisfies sΣ = g.

(b) If G is a connected Lie group, then the system (Σ) is controllable if and only ifit is accessible and the reachable set RΣ(I) is a group.

(c) If G is a compact connected Lie group, then controllability is equivalent to ac-cessibility.

The following notion of small-time local controllability was introduced in [13].

Definition 3.4. Small-time local controllability (STLC). The system (Σ) hasthe STLC property if, for every T > 0, a neighborhood of identity I in G is containedin the reachable set RΣ,T (I).

Note that R−1Σ,T (I) := X−1 | X ∈ RΣ,T (I) coincides with the set of all X ∈ G

that can be driven by admissible controls to the identity at time T ≥ 0. Thus, STLCimplies

• The identity I is also in the interior of R−1Σ,T (I).

• Every point X0 ∈ G is an interior point of RΣ,T (X0) and R−1Σ,T (X0).

Therefore, the STLC property means that for any initial point X0 ∈ G, we can alwaysfind a sufficiently small neighborhood V0 of X0 such that we can steer X0 to any pointin V0 in arbitrary time T > 0. Similarly, for any point Xf ∈ G, we can always finda sufficiently small neighborhood Vf of Xf such that we can steer any point in Vf toXf in arbitrary time T > 0. The following theorem provides a sufficient condition ofSTLC for systems on compact Lie groups.

Theorem 3.5. Small-time local controllability (STLC)[3]. Consider the sys-tem (Σ) where G is compact. Assume there exists a time T such that for every τ ≤ Tthere exists a piecewise constant control uτ (t) = [u1(t) u2(t) . . . um(t)]⊤ ∈ Rm steering

from I to I in time τ . Denote U τ := uτi N(τ)

i=1 , where uτi ∈ Rm is the constant value


assumed by the control uτ(t) at time interval ti−1 < t ≤ ti for i = 1, 2, . . . , N(τ), andt0 = 0, tN(τ) = τ . For each uτ

j ∈ U τ , define the matrix

Fj := A0 +m∑

k=1

ukjAk,

where ukj, k = 1, 2, . . . , m, are the k-th components of uτj ∈ Rm. Then, the system

has the STLC property if, for every τ , there exists an uτj ∈ Uτ such that

adnFjAk , n = 0, 1, . . . , dim(g) , k = 1, . . . , m,

span the whole Lie algebra g of G.

Definition 3.6. Fast Controllability (FC). The system (Σ) has the FC propertyif, for arbitrary small τ > 0, there exist controls uk : [0, τ ] → R such that RΣ,τ (I) = G,i.e. every X ∈ G can be reached from I in an arbitrarily short time.

FC just means that we can drive the system from any initial point to any finalpoint in G arbitrarily fast (almost in zero time), hence the name “fast controllability”.This can occur practically when admisible control signals are very large compared tothe drift term A0.

Theorem 3.7. Fast Controllability (FC)[35]. Consider the system (Σ) with thecontrols uk : R → R. Let cΣ be the control Lie algebra of (Σ), i.e. the Lie subalgebraof g generated by the control terms A1, . . . , Am. The system (Σ) has the FC propertyif cΣ = g.

We give the following counter-example showing that the converse of Theorem 3.7does not hold in general. However, if we assume that G is compact, then one canprove that the condition in Theorem 3.7 is also necessary for FC. In this case, we callK := 〈exp(cΣ)〉 the fast subgroup of G, the set of X ∈ G which can be approximatedarbitrarily fast from I.

Example 3.8. Consider a bilinear control system (Σ) in Eq.(3.1) on the Lie group

G :=

[X b0 1

] ∣∣∣ X ∈ SL(n,R) , b ∈ Rn

with the drift term A0 =

[0 c0 0

], for a non-zero c ∈ Rn, and the control directions

Akmk=1 such that

cΣ =

[A 00 0

] ∣∣∣ A ∈ sl(n,R)

( g.


For any τ > 0 arbitrarily small, there exists Y ∈ SL(n,R) such that any element inG can be decomposed as

[X b0 1

]=

[I b0 1

] [X 00 1

]=

[Y 00 1

] [I τc0 1

] [Y −1 00 1

] [X 00 1

],

with τY c = b. This is due to the fact that SL(n,R) acts transitively on Rn \ 0, see

[11, 12]. Note that

[I τc0 1

]= exp(τA0). The above decomposition shows that any

element in G can be reached arbitrarily fast from I, assuming there are no bounds onthe controls uk(t). Thus, the system is FC without having cΣ = g.

3.1.2 Induced Control Systems and Lie Group Actions

Let α denotes a Lie group action of G on a real analytic connected manifold M

α : G×M −→M , (X, p) 7−→ X · p. (3.5)

Let (Σ) be a right invariant control system Eq.(3.1) starting from the identity X0 = I.Via the Lie group action α, the one-parameter group etA(u) of (Σ) on G, with A(u) :=A0 +

∑mk=1 ukAk ∈ g, naturally induces a flow p0 7→ etA(u) · p0 on M . Therefore, α

induces a bilinear control system on M

(Σi) : p = D1α(I, p)

(A0 +

m∑

k=1

uk(t)Ak

), p(0) = p0, (3.6)

where D1α is the tangent map of α with respect to the first component. Each solutionX(t) ∈ G of the bilinear system (Σ) yields a solution p(t) ∈M of (Σi) and vice versavia

p(t) = X(t) · p0 , for all t ∈ R. (3.7)

We recall that controllability and accessibility properties for the induced bilinearcontrol systems on M are closely related to the corresponding property of (Σ). Let(Σ) and (Σi) be a bilinear control system on G and M , respectively, defined as before.The reachable set RΣ(I) is a subsemigroup of G and we call

SΣ := RΣ(I) ⊆ GΣ

the system semigroup which is obviously contained in the system group GΣ. Note thatSΣ and GΣ are connected sets. With respect to the induced system (Σi), we definethe reachable set from p0 and the orbit of p0 as

RΣi(p0) :=

⋃

t≥0

RΣi,t(p0) and OΣi(p0) :=

⋃

t∈R

RΣi,t(p0),


respectively, where RΣi,t(p0) is the set of all possible final points p(t) of the solutionsof Eq.(3.6) at t ∈ R. By the corresponding Lie group action α, we have

RΣi(p0) = SΣ · p0 and OΣi

(p0) = GΣ · p0. (3.8)

Controllability and accessibility of the induced system (Σi) then are defined withrepect to the underlying manifold M , similarly as the corresponding definitions on G.That is, the induced system is accessible when RΣi

(p0) has non-empty interior in Mfor all p0 ∈M , and it is controllable when RΣi

(p0) = M for all p0 ∈M .

We recall some results dealing with controllability and accessibility of the in-duced system on M by a Lie group action.

Theorem 3.9. [18] The induced bilinear system (Σi) is controllable on M providedthe following two conditions hold

• The induced system (Σi) is accessible on M .

• There exist constant controls u1, . . . , um such that the induced system (Σi) isweakly positively Poisson stable.

Theorem 3.10. [18] The induced system (Σi) is accessible on M if and only if thesystem group GΣ acts transitively on M .

Corollary 3.11. Let G be a compact connected Lie group. Then, the induced system(Σi) is controllable on M if and only if the system group GΣ acts transitively on M .

Proof. Any constant right-invariant vector field on a compact Lie group is weaklypositively Poisson stable [48], (or likewise, recurrent, see e.g. [33]). Hence, the inducedflow on M also has the same corresponding property. By Theorem 3.9, we have thatcontrollability of the induced system on M is equivalent to accessibility of the inducedsystem on M . By Theorem 3.10, the result follows.

Theorem 3.10 and Corollary 3.11 underline the important and interesting factthat characterizing accessibility of bilinear control systems on M and correspondingly,controllability in the particular compact case, boils down to the classification of Liegroups which act transitively on the underlying manifold M of interest. Indeed, Liegroup actions are classical topics in topology and the theory of transformation groups.For example, the complete classification of all closed connected subgroups of GL(N,R)acting transitively onM is well-known for the case M := RN \0 (see [11, 12, 18, 44]),for the real sphere M := S(RN) (see [14, 18, 51]), for the complex projective spaceM := CPN−1 ∼= S(CN)/U(1), and more generally, for complex Grassmann manifolds(see [18, 60]). Some of these classification results relevant for quantum control arelisted in the Appendix B.


3.2 Controllability of Quantum Systems

We consider the Lindblad-Kossakowski master equation

ρ = LH(ρ) + LD(ρ) , ρ0 = ρ(0), ρ ∈ P, (3.9)

with LH(ρ) = −i[H, ρ] and LD in the diagonal Lindblad form, cf. Eq.(2.21),

LD(ρ) =

NA∑

j=1

VjρV†j − 1

2V †

j Vjρ−1

2ρV †

j Vj , Vj ∈ CN×N , (3.10)

where NA ∈ N is arbitrary but finite, or LD in the GKS form, cf. Eq.(2.28),

LD(ρ) =1

2

N2−1∑

j,k=1

ajk ([Bj, ρBk] + [Bjρ,Bk]) , [ajk] ≥ 0, (3.11)

where BjN2−1j=1 is an orthonormal basis of her0(N). We assume that H is of the form

H = H0 +

m∑

k=1

uk(t)Hk , H0, Hk ∈ her0(N), (3.12)

where uk : R → R is the control signal. The terms H0 and Hk are called Hamiltoniandrift and control, respectively. Note that the Lindblad-Kossakowski master equation(3.9) constitutes a bilinear control system on her(N) which leaves the set of densityoperators P invariant.

3.2.1 Controllability of Closed Quantum Systems

We first focus on the Lindblad-Kossakowski master equation (3.9) with LD = 0. Theabsence of the dissipative part LD corresponds to (an ensemble of) closed quantumsystems having no interaction with the environment. The dynamical equation thenreduces to the so-called Liouville-von Neumann master equation

(Σi) : ρ(t) = −i[H0 +

m∑

k=1

uk(t)Hk, ρ(t)], ρ(0) = ρ0 ∈ P. (3.13)

Equation (3.13) “lifts” to a bilinear control system on the Lie group of special unitarymatrices SU(N) as

(Σ) : U(t) = −i(H0 +

m∑

k=1

uk(t)Hk

)U(t), U(0) = I ∈ SU(N), (3.14)

3.2. Controllability of Quantum Systems 63

in the sense that (Σ) induces the original Liouville-von Neumann equation (3.13) viathe Lie group action defined by the unitary conjugation

α(U, ρ) := UρU−1 = UρU † , U ∈ SU(N). (3.15)

Any solution of Eq.(3.13) can be expressed as

ρ(t) = U(t)ρ0U(t)†,

where U(t) is a solution of Eq.(3.14). Thus, the evolution of (an ensemble of) closedquantum systems is described by a unitary conjugation of its density operator. Sim-ilarly, for the Lie group action of SU(N) on the unit complex sphere S(CN) ∈ CN

defined by

α(U, ψ) := Uψ , ψ ∈ S(CN), U ∈ SU(N),

the lifted Liouville-von Neumann equation (3.14) induces the following control systemon S(CN),

(Σi) : ψ(t) = −i(H0 +

m∑

k=1

uk(t)Hk

)ψ(t) , ψ(0) = ψ0 ∈ S(CN), (3.16)

well-known as the Schrodinger equation for a closed (single) quantum system. Anysolution of Eq.(3.16) can be expressed as

ψ(t) = U(t)ψ0,

where U(t) is a solution of Eq.(3.14). These different equations for closed quantumsystems give rise to the following different notions of controllability:

Definition 3.12. A closed quantum system is called

(a) pure state controllable, when the reachable set RΣi(ψ0) of (3.16) is equal to the

entire sphere S(CN) for any initial pure state ψ0 ∈ S(CN), i.e. the system (3.16)is controllable on the complex sphere S(CN).

(b) projective state controllable, when the reachable set RΣi(ρ0) of (3.13) is equal

to the unitary orbit O(ρ0) := Uρ0U† | U ∈ SU(N), for any ρ0 = ψ0ψ

†0, ψ0 ∈

S(CN). Note that such O(ρ0) is a complex projective space CPN−1 ∼= P1,N(C),cf. Eq.(3.17).

(c) pure state like controllable, when the reachable set RΣi(ρ0) of (3.13) is equal to

the unitary orbit O(ρ0) := Uρ0U† | U ∈ SU(N), where ρ0 has two non-zero

distinct eigenvalues with algebraic multiplicity 1 and N − 1, respectively. Suchρ0 is called pure state like ensemble.


(d) density operator controllable, when the reachable set RΣi(ρ0) of (3.13) is equal

to the unitary orbit O(ρ0) := Uρ0U† | U ∈ SU(N) for each initial density

operator ρ0 ∈ P, i.e. the system (3.13) is controllable on O(ρ0) for each ρ0.

(e) operator controllable or fully controllable, when the reachable set RΣ(I) of (3.14)is equal to the entire group SU(N), i.e. the system (3.14) is controllable on theLie group SU(N).

The characterization of different notions of controllability and their relationshipsin between has been addressed, for example, in [1, 53]. Here, we pursue a slightlydifferent route by considering Volklein’s [60] characterization of Lie groups which acttransitively on complex Grassmann manifolds. This leads to more straightforwardproofs which simplify and close a gap in [1].

As a first step, we consider the case where the Liouville-von Neumann equation(3.13) is assumed to evolve on the complex Grassmannian, which is defined as the setof all orthogonal projections of rank k, i.e.

Pk,N(C) := P ∈ CN×N | P † = P, P 2 = P, Tr(P ) = k. (3.17)

Clearly, Pk,N(C) can be identified with the set of all k-dimensional complex subspacesof CN , i.e. with

Gk,N(C) := V ∈ CN | V linear subspace, dim V = k,

via the diffeomorphism Φ : Pk,N(C) → Gk,N(C), P 7→ Image(P ), see [28]. Theset Gk,N(C) is called Grassmann manifold. Due to the diffeomorphism Φ and byCorollary 3.11, controllability of the Liouville-von Neumann equation on Pk,N(C) isequivalent to the transitivity of the system group action G of (3.14) on the Grassmannmanifold Gk,N(C), see [18]. For the classification of Lie groups which act transitivelyon Grassmann manifolds, we refer to [60]. This classification leads to the followingnecessary and sufficient controllability criterion for the Liouville equation on Pk,N(C)[18].

Theorem 3.13. Let sΣ be the system Lie algebra generated by iH0, iH1, . . . iHm ∈su(N). The Liouville equation (3.13) is controllable on the complex GrassmannianPk,N(C) if and only if

(a) sΣ is equal to su(N) or conjugate to sp(N/2), for N even and k = 1 or N − 1.

(b) sΣ is equal to su(N), for N odd or 1 < k < N − 1.

Proof. Consult the classification of Lie groups which act transitively on Grassmannmanifolds by Volklein [60] as summarized in the Appendix B. The result directlyfollows since all iHj are skew-Hermitian. Hence, the Lie algebras being involved areonly su(N) and its subalgebras.


We apply Theorem 3.13 to characterize controllability of the Liouville equationon the unitary orbit O(ρ0) and thus, clarify the results in [1, 53] for the differentnotions of controllability for closed quantum systems as described in Definition 3.12.

Corollary 3.14. (Main Result 1 : Controllability of Closed QuantumSystems). Let ρ0 ∈ P be a fixed density operator.

(a) If N is even and ρ0 has only two distinct eigenvalues λ1 and λ2 with alge-braic multiplicity 1 and N − 1, respectively, then the Liouville equation (3.13)is controllable on the unitary orbit O(ρ0) if and only if sΣ is equal to su(N) orconjugate to sp(N/2).

(b) If N is odd or ρ0 has an eigenvalue configuration other than in (a), then theLiouville equation (3.13) is controllable on the unitary orbit O(ρ0) if and onlyif sΣ is equal to su(N).

Proof. First, consider the case where ρ0 has two distinct eigenvalues λ1 and λ2 withalgebraic multiplicity k and N − k, respectively. The stabilizer of ρ0 is obviously ofthe form

Stab(ρ0) := U ∈ U(N) | Uρ0U† = ρ0 = U(k) × U(N − k).

Hence, O(ρ0) is diffeomorphic to the Grassmann manifold Gk,N(C) and the complexGrassmannian Pk,N(C) as well, i.e.

O(ρ0) ∼=U(N)

Stab(ρ0)=

U(N)

U(k) × U(N − k)∼= Gk,N(C) ∼= Pk,N(C).

Note that Pk,N(C) can be written as a unitary orbit

Pk,N(C) = UXU † | U ∈ U(N), X = diag(Ik, 0) = O(X),

where X has the same stabilizer as ρ0. Hence, the system group acts transitively onO(ρ0) if and only if it also acts transitively on Pk,N(C). By Theorem 3.13, this provesstatement (a) and statement (b) when N is odd.

For the rest of statement (b), consider the case where ρ0 has more than twodistinct eigenvalues λ1, λ2, . . . , λr, r > 2, with algebraic multiplicity n1, n2, . . . , nr,respectively. The stabilizer of ρ0 then has the form

Stab(ρ0) = U(n1) × U(n2) × . . .× U(nr),

such that the orbit O(ρ0) is diffeomorphic to the flag manifold, i.e.

O(ρ0) ∼=U(N)

U(n1) × U(n2) × . . .× U(nr).


Suppose N > 3. Choose k = n1 + n2 such that 1 < k < N − 1 and consider the map

π : O(ρ0) −→ Pk,N(C) = O(X) , Uρ0U† 7→ UXU †,

with U ∈ SU(N) and X := diag(Ik, 0). One can easily check that π is well-definedand satisfies the relation π(U ·ρ) = U ·π(ρ), where U ·ρ is defined by U ·ρ := UρU † forU ∈ SU(N). Then, transitivity on O(ρ0) necessarily implies transitivity on Pk,N(C),1 < k < N − 1, which is, by Theorem 3.13(b), equivalent to sΣ = su(N). ForN = 3 (with three distinct eigenvalues), the same argument can be repeated fork = n1 + n2 = 2 and again Theorem 3.13(b) yields sΣ = su(N) since N is odd. Thiscompletes the proof.

Corollary 3.15. Let sΣ be the system Lie algebra generated by iH0, iH1, . . . iHm ∈su(N).

(a) The Schrodinger equation (3.16) is pure state controllable if and only if sΣ isequal to su(N) or conjugate to sp(N/2) for N even.

(b) The Liouville equation (3.13) is projective state controllable if and only if sΣ isequal to su(N) or conjugate to sp(N/2) for N even.

(c) The Liouville equation (3.13) is pure state like controllable if and only if sΣ isequal to su(N) or conjugate to sp(N/2) for N even.

(d) Pure state, projective state and pure state like controllability of closed quantumsystems are equivalent.

Proof. For (a), apply Volklein’s result [60] for Lie groups which act transitively onthe complex sphere S(CN). To prove statement (b), apply Corollary 3.14(a) for ρ0

having two distinct eigenvalues λ1 = 1 and λ2 = 0 with algebraic multiplicity 1 andN−1, respectively. For statement (c), apply again Corollary 3.14(a) for ρ0 having twodistinct eigenvalues with algebraic multiplicity 1 and N − 1 respectively. So finally,(d) is clear.

Note that, the notion of density operator controllability was introduced in quan-tum control literature to mean controllability of the Liouville equation on O(ρ0) foreach ρ0 ∈ P. Therefore, Corollary 3.14 immediately implies the following.

Corollary 3.16. Let sΣ be the system Lie algebra generated by iH0, iH1, . . . iHm. TheLiouville equation (3.13) is density operator controllable if and only if sΣ = su(N).

Proof. The “if” statement is clear. Now, suppose sΣ = sp(N/2) for N is even. Then,by Corollary 3.14, the system is controllable on O(ρ0) for ρ0 with rank one, but notfor ρ0 with more than two distinct eigenvalues. This concludes the statement.


Therefore, density operator controllability is completely equivalent to controllabilityof the lifted Liouville equation (3.14) on SU(N),

Theorem 3.17. Let sΣ be the system Lie algebra generated by iH0, iH1, . . . iHm. Thelifted Liouville equation (3.14) is controllable on SU(N) (hence operator controllableor fully controllable) if and only if sΣ = su(N).

Therefore, we obtain the following relations between the different notions of control-lability of closed quantum systems.

density operator controllable ⇐⇒ operator controllable=⇒ (pure state, projective state, pure state like) controllable.

Note that when sΣ = sp(N/2), for N even, the system is pure state controllable, butthis is not enough to have density operator controllability.

Examples

We give a simple example from two-level closed quantum systems dealing with thenotions of small-time local controllability (STLC) and fast controllability (FC), seee.g. [2, 3]. We consider the Pauli matrices

Sx =

[0 −i

−i 0

], Sy =

[0 −11 0

], Sz =

[−i 00 i

](3.18)

with the commutation relations

[Sx, Sy] = 2Sz , [Sy, Sz] = 2Sx , [Sz, Sx] = 2Sy. (3.19)

Example 3.18. Fast Controllability (FC). Consider the following two levelsystem

U(t) =(Sz + u1(t)Sx + u2(t)Sy

)U(t) , U(0) = I. (3.20)

It is easy to see that the system is fast controllable, i.e. by Theorem 3.7, the controlLie algebra cΣ generated by Sx and Sy is equal to su(2) such that the fast Lie-subgroupK := 〈exp(cΣ)〉 = SU(2) contains the whole possible unitary Lie group SU(2). Hence,any U ∈ SU(2) can be reached arbitrarily fast from I. Correspondingly for theLiouville equation

ρ(t) =[Sz + u1(t)Sx + u2(t)Sy , ρ(t)

], ρ(0) = ρ0,

any ρ0 can be steered to any target state ρf ∈ O(ρ0) arbitrarily fast.

Example 3.19. Small-Time Local Controllability (STLC). Consider the fol-lowing two level system

U(t) =(Sz + u2(t)Sy

)U(t) , U(0) = I. (3.21)


This system is not FC since cΣ = span〈Sy〉 6= su(2) and SU(2) is compact. However,it is fully controllable since sΣ = su(2). The fast subgroup K = 〈exp(cΣ)〉 ( SU(2)containing the set of unitary matrices that can be reached arbitrarily fast from identityis SO(2). The proof of the following corollary using Theorem 3.5 can be seen in [2],

Corollary 3.20. The system (3.21) has STLC property, i.e. the identity I iscontained in the interior of Rτ (I) and R−1

τ (I) for any small time τ > 0.

Proof. The system Lie algebra sΣ generated by Sz and Sy is su(2). Now choose aconstant control u2(t) = u. For τ > 0 sufficiently small we have a solution

U(τ) = exp ((Sz + Syu)τ) = exp

([−i −uu i

]τ

).

The eigenvalues λ1,2 of (Sz +Syu)τ can be explicitly calculated to be ±i√

1 + u2τ suchthat we can write the solution as

U(τ) = WDW † = W

[exp(i

√1 + u2τ) 0

0 exp(−i√

1 + u2τ)

]W †,

for W a unitary matrix. Thus we can choose some control u 6= 0 such that λ1,2 =±i2πn to obtain U(τ) = I. Hence, we have the control u that steers the system fromI to I. Next, we apply Theorem 3.5 by setting the matrix F = Sz + Syu for a singleconstant control u. We calculate adn

FSy,

ad0FSy = Sy

ad1FSy = [Sz + Syu, Sy] = −2Sx

ad2FSy = [Sz + Syu,−2Sx] = −4Sy + 4Szu,

which span the Lie algebra g = su(2) since, as shown above, we can choose u differentfrom zero. Hence by Theorem 3.5, the system has the STLC property.

Similarly for the Liouville equation

ρ(t) =[Sz + u2(t)Sy , ρ(t)

], ρ(0) = ρ0,

we can reach any point in a small neighborhood of ρ0 within O(ρ0) in arbitrary smalltime. The set of ρf ∈ O(ρ0) that can be reached arbitrarily fast from ρ0 is theorthogonal orbit OO(ρ0) := Oρ0O

⊤ | O ∈ SO(2).

3.2.2 Controllability Issues of Open Quantum Systems

Under the presence of dissipation LD 6= 0, we show that open quantum systemsare never controllable. Statements dealing with non-controllability results of theLindblad-Kossakowski master equation can be found in [7, 8, 53, 56].


Theorem 3.21. The Lindblad-Kossakowski master equation (3.9) with LD 6= 0 isnever controllable on P.

Proof. First we consider the unital case. Indeed, Lemma 2.20 shows that the norm||ρ||2 = Tr(ρ2) is monotonically decreasing. Hence, non-controllability is concluded.

For the non-unital case, the lack of controllability can be checked as follows [18].Consider the inner product 〈x, y〉 := x†y on CN and the induced norm ||x|| =

√〈x, x〉.

Now evaluate the norm ddt||ρ||2 = d

dt(Tr(ρ2)) on the extreme points of P, i.e. on the

set of projective (pure) states ρ = ψψ†,

ddt

(Tr(ρ2)) = 2∑

j Tr(Vjψψ

†V †j ψψ

† − ψψ†V †j Vjψψ

†)

= 2∑

j

(〈ψ, Vjψ〉2 − ||ψ||2||Vjψ||2

)≤ 0,

where the inequality follows from the Cauchy-Schwarz inequality. Suppose equalityholds for all pure states. Then, controllability fails since the set of all pure stateswould be invariant under the flow and thus, states with ||ρ|| < 1 were not reachable.Suppose strict inequality holds for a pure state ρ0 := ψ0ψ

†0. Then, again controllability

does not hold since ρ0 /∈ R(ρ) for all ρ ∈ P \ ρ0.

For general non-unital systems, the norm Tr(ρ2) can increase for some ρ ∈ P.However, since

d

dt(||ρ||2) =

d

dt(Tr(ρ2)) = 2〈ρ,LD(ρ)〉

according to Eq.(2.27), the norm change is only due to the dissipative part LD andcan not be altered at all by the controls uk(t). For any ρ(t) ∈ P at any instance oftime t, the rate change of the norm depends only on the current state ρ(t) and LD,independent of the control. This failure of ”local controllability” for open quantumsystems reflects the lack of analogous small time local controllability (STLC). Namely,for any initial condition ρ0, the neighborhood of ρ0 in P is not contained in thereachable set R≤τ (ρ0), for τ > 0 sufficiently small. In other words, for any initialpoint ρ0 ∈ P, there does not exist a sufficiently small neighborhood V0 of ρ0 such thatthe controls can steer ρ0 to any point in V0 for some time T , 0 ≤ T ≤ τ .

h-Controllability and wh-Controllability

In the general situation of open quantum systems when controllability does not hold,weaker notions of controllability are desirable. Such notions like h-controllability andwh-controllability were introduced in [21]. First, we shall describe some facts on theLindblad-Kossakowski master equation in the vector of coherence representation andin what follows, we will formulate the h-controllability and wh-controllability results.We consider the following unital Lindblad-Kossakowski master equation on the Bloch


ball Bd,v = LHv + LDv , v(0) = v0, v ∈ Bd

=(LH0

+

m∑

k=1

uk(t)LHk

)v +

d∑

j,k=1

ajkLjkv,(3.22)

with H = H0 +∑m

k=1 uk(t)Hk, and H0, Hk ∈ her0(N). Note that LHk∈ so(d) and

Ljk ∈ gl(d,R) appear as concrete matrix representations of LHk= −i[Hk, (·)] ∈

gl(her0(N)) and LD ∈ gl(her0(N)) whose explicit forms are given in Eq.(2.50) andEq.(2.51), respectively. Recall that d is the dimension of her0(N), i.e. d = N2 − 1.Open quantum systems in terms of vector of coherence representation can be viewedas a bilinear control system on Rd which leaves Φ(P) ⊂ Bd invariant, where Φ is themap in Eq.(2.44).

Recall the adjoint representation of su(N),

ad : su(N) −→ gl(su(N)) , X 7→ adX ,

which is a Lie-algebra homomorphism. For each X ∈ su(N), adX is a linear operatoracting on su(N), i.e. adXY := XY −Y X = [X, Y ] ∈ su(N). Let her0(N) = isu(N) ∼=su(N), seen as a real vector space. Note that her0(N) is not a Lie-algebra. Now, we

introduce another representation ad of su(N) which acts on her0(N),

ad : su(N) −→ gl(her0(N)) , X 7→ adX := adX .

Thus, for eachX ∈ su(N), adX is the linear operator adX regarded as a linear operatoracting on her0(N).

Lemma 3.22. Consider her0(N) ∼= Rd with the inner product 〈X, Y 〉 = Tr(X†Y ),

X, Y ∈ her0(N). Let ksu(N) ⊂ gl(d,R) be a matrix representation ofadX | X ∈

su(N)

with respect to any orthonormal basis Bjdj=1 of her0(N). Then, we have :

(a) ksu(N)∼= adsu(N)

∼= su(N) ∼= adsu(N) as Lie algebras.(b) ksu(N) ⊆ so(d) where equality holds only for N = 2, or equivalently, for d = 3.(c) ksu(N) is irreducible, i.e. the only ksu(N)-invariant subspaces are 0 and Rd.

Proof. (a). We show only adsu(N)∼= su(N). Note that adX adY − adY adX = ad[X,Y ]

for all X, Y ∈ su(N), hence ad is a Lie algebra homomorphism which is, by definition,

surjective on its image. We show that ad is injective. For X ∈ su(N), adX = 0 if andonly if adX = 0, since her0(N) = isu(N). Now, adX = 0 implies that X must be inthe center of su(N), which is trivial. The claim follows.

(b). Let Ω := [ωrs] ∈ ksu(N) be a matrix representation of adX for some

X ∈ su(N). With respect to above inner product, we have ωrs = 〈Br, adXBs〉 =

−〈Bs, adXBr〉 = −ωsr. Hence Ω is skew-symmetric. By dimension counting, we have


dim(ksu(N)) = dim(su(N)) = N2 − 1 = d ≤ d(d − 1)/2 = dim(so(d)), where equalityholds only for N = 2, or equivalently, d = 3.

(c). Recall that su(N) is simple, i.e. the only ideals are 0 and su(N) so that its

adjoint representation is irreducible. From (a), we conclude that adsu(N) is irreducibleand thus ksu(N) is also irreducible on Rd.

By Lemma 3.22, we can view ksu(N) as a Lie subalgebra of so(d) with the corre-sponding compact Lie subgroup

Ksu(N) := 〈exp(ksu(N))〉G ⊆ SO(d).

When N = 2, or equivalently, d = 3, we have ksu(2) = so(3) ∼= su(2) and Ksu(2) =SO(3). Now, with respect to the bilinear control system in Eq.(3.22), we define thefollowing two Lie subalgebras of ksu(N),

kD := 〈LH0, LH1

, . . . , LHm〉LA ⊆ ksu(N) ⊆ so(d)

kC := 〈LH1, . . . , LHm

〉LA ⊆ ksu(N) ⊆ so(d).(3.23)

Definition 3.23. The unital Lindblad-Kossakowski master equation (3.22) is called

(a) Hamiltonian controllable (h-controllable), when the Ksu(N)-orbit of any initialpoint v0 is contained in the closure of its reachable set, i.e.

Ksu(N) · v0 ⊆ R(v0) for all v0 ∈ Φ(P) ⊂ Bd.

(b) weakly Hamiltonian controllable (wh-controllable), when the Ksu(N)-orbit of anyinitial point v0 is contained in the closure of the semigroup R+ · R(v0), i.e.

Ksu(N) · v0 ⊆ R+ · R(v0) for all v0 ∈ Φ(P) ⊂ Bd.

Note that in general, the notions of h-controllability and wh-controllability areactually independent from a particular representation, see [21]. Further, if the unitalequation (3.22) is not accessible, the closure R(v0) in Definition 3.23 is taken withrespect to the orbit of Eq.(3.22), not to the ambient space Rd. The notion of wh-controllability has not been studied so far in the literature and was first introducedin [21] with the purpose to provide a partial answer for the problem of finding thebest approximation (or at least a bound) to some ρ∗ ∈ O(ρ) on the reachable setR(ρ). The notions of h-controllability and wh-controllability are clearly weaker thanthe usual controllability property in the sense that the system might not be control-lable. In Example 3.44, Section 3.3.3, we even show that wh-controllability does notnecessarily imply accessibility. Moreover, a quantum system which is density oper-ator controllable in the absence of dissipation, i.e. when LD = 0, is not necessarilywh-controllable when the dissipation term LD 6= 0 is included, cf. Example 3.25.This surprising fact remains true even if the corresponding open quantum systemwith LD 6= 0 is indeed accessible, see also Example 3.44. Sufficient conditions forh-controllability and wh-controllability are the following.


Theorem 3.24. The unital Lindblad-Kossakowski master equation (3.22) with thecontrols uk : R → R is

(a) h-controllable, if sC := 〈iH1, iH2 . . . , iHm〉LA = su(N).

(b) wh-controllable, if sD := 〈iH0, iH1, . . . , iHm〉LA = su(N) and LD = γId, γ < 0.

Proof. (a). This is indeed an application of Lemma 6.4 of [35]. For the sake ofcompleteness we briefly repeat the proof here. Write the reachable set of Eq.(3.22) asR(v0) = S · v0, where S is the corresponding semigroup. Now, choose j ∈ 1, . . . , mand the constant controls

uk(t) :=

n , k = j0 , k 6= j

.

The solution of Eq.(3.22) at time t/n > 0 starting from v0 using uk(t) is given by

exp((LH0

+ LD + nLHj)(t/n)

)· v0.

By letting n → +∞, this shows that exp(tLHj) ∈ S for each j ∈ 1, . . . , m and

t > 0. Similarly for n → −∞, we have exp(tLHj) ∈ S for each j ∈ 1, . . . , m and

t < 0. Note that LHjis the matrix representation of −adiHj

so that the assumptionsC = su(N) is equivalent to kC := 〈LH1

, . . . , LHm〉LA = ksu(N). The corresponding Lie

group of kC is therefore Ksu(N). Since exp(tLHj) ∈ S, for t ∈ R, j ∈ 1, . . . , m, and

S is a semigroup, we obtain Ksu(N) ⊆ S and h-controllability follows.

(b). When LD = 0, the condition sD = su(N) coincides with the densityoperator controllability on the unitary orbit O(ρ0) = Uρ0U

† | U ∈ SU(N) for closedquantum systems. Hence there exists controls u∗k(t) which steer ρ0 to an arbitrary finalpoint ρf ∈ O(ρ0) in some finite time T ∗, see [19, 37]. Without loss of generality, weassume that u∗k(t) is piecewise constant with only one switch at time τ , i.e.

u∗k(t) :=

ak ∈ R , 0 ≤ t < τbk ∈ R , τ ≤ t ≤ T ∗ .

In general, to steer ρ0 to an arbitrary final point ρf , the controls might need toswitch more than once. But nevertheless, the following arguments can be extendedto any finite number of switchings. Let X1 := LH0

+∑m

k=1 akLHkand X2 := LH0

+∑mk=1 bkLHk

. Note that the density operator controllability on the unitary orbit iscompletely equivalent to the controllability of closed quantum systems in the vectorof coherence representation on the orbitKsu(N) ·v0. Hence, for arbitrary vf ∈ Ksu(N) ·v0

which corresponds to ρf ∈ O(ρ0), we have

vf = exp(X2(T∗ − τ)) exp(X1τ)v0.


By applying the same controls u∗k(t) up to time T ∗ to Eq.(3.22) when LD = γId, γ < 0,we terminate at a final point v∗f , where

v∗f = exp((X2 + γId)(T∗ − τ)) exp((X1 + γId)τ)v0

= exp(γT ∗) · vf ∈ R(v0).

This implies that for any vf ∈ Ksu(N) · v0, we also have vf ∈ exp(−γT ∗)R(v0) ⊂R+ · R(v0), and wh-controllability holds.

Example 3.25. (wh-controllability and density operator controllability).We give a particular example for a two-level open quantum system which is densityoperator controllable when there were no dissipation, but not wh-controllable afterthe dissipation is taken into account. This fact remains true even when the openquantum system is accessible, i.e. the system in this example is also accessible, seeExample 3.42 in Section 3.3.3.

Consider a two-level system in the vector of coherence representation where thedrift Hamiltonian, control Hamiltonian and dissipation are given by

LH0=

0 −d 0d 0 00 0 0

, LH1

=

0 0 10 0 0−1 0 0

, LD =

−a 0 00 0 00 0 −a

,

respectively. Note that in this case, the corresponding GKS matrix is given by A =diag(0, a, 0), a > 0. The equation for the vector of coherence is then given by

vx

vy

vz

=

−a −d u(t)d 0 0

−u(t) 0 −a

vx

vy

vz

. (3.24)

Note that if there were no dissipation, the system is density operator controllable,i.e. kD = 〈LH0

, LH1〉LA = ksu(2) = so(3) ∼= su(2). Denote by 〈v, w〉 := v⊤w the inner

product of R3. Now consider the following wedge (i.e. closed convex cone) in vx-vy

plane,

S :=

[vx vy 0]⊤∣∣ (vy ≥ − tan θ · vx) ∧ (vy ≥ tan θ · vx)

, θ ∈ [0,

π

2],

and the “ice-cream” cone W obtained by fully rotating S with respect to vy-axis. Weclaim that there exists θ ∈ [0, π

2] such that W is invariant under the flow of Eq.(3.24)

if and only if a ≥ 2d. This claim will particularly show that the system in Eq.(3.24)can not be wh-controllable, i.e. there does not exist λ ∈ R+ such that

Ksu(2) · v0 = SO(3) · v0 ⊆ λ · R(v0)

for all v0 ∈ R3. Note that the control u(t) rotates v with respect to vy-axis andtherefore leaves W invariant. So we may consider only the drift and dissipative part


LH0+LD. Now, we shall use again the linear invariance theorem for wedges, Theorem

2.9, for W . Note that W is a generating wedge and its dual W ∗ is given by fullyrotating

S∗ :=

[vx vy 0]⊤∣∣ (vy ≥ − tan(

π

2− θ)vx) ∧ (vy ≥ tan(

π

2− θ)vx)

,

with respect to vy-axis. Since W is a pointed wedge, E1(W ) is exactly the set of allexposed points in W , which is given by

E1(W ) =[vx vy vz]

⊤ = k[cos θ cos φ sin θ cos θ sin φ]⊤∣∣ k ≥ 0, φ ∈ [0, 2π)

.

For any point v ∈ E1(W ) with v = k[cos θ cosφ sin θ cos θ sinφ]⊤, its opposite wedgeon W is given by

opW (v) := v⊥ ∩W ∗ =r[− sin θ cosφ cos θ − sin θ sin φ]⊤

∣∣ r ≥ 0.

By Theorem 2.9, the flow of Eq.(3.24) leaves the cone W invariant if and only if(LH0

+LD)(v) ∈ T sv (W ) := opW (v)∗, for all v ∈ E1(W ), or equivalently, if and only if

the following condition

⟨

−a −d 0d 0 00 0 −a

k cos θ cosφk sin θ

k cos θ sinφ

,

−r sin θ cos φr cos θ

−r sin θ sinφ

⟩

≥ 0

holds, for all k, r ≥ 0 and φ ∈ [0, 2π), which finally yields

a cos θ sin θ ≥ −d cosφ , for all φ ∈ [0, 2π).

Since cosφ ∈ [−1, 1] and cos θ sin θ ∈ [0, 12], we conclude the claim that there exists

θ ∈ [0, π2] such that W is invariant if and only if a ≥ 2d.

We can also inspect the eigenvalues of LH0+ LD, which are given by s1,2 =

−a±√

a2−4d2

2and s3 = −a. The above condition precisely says that the system can not

be wh-controllable when there does not exist oscillations with respect to vx-vy axis,i.e. the condition a ≥ 2d is fulfilled when the eigenvalues are both real and negativesuch that no oscillations occur.

3.2.3 Reachable Sets of Two-Level Systems

Since the Lindblad-Kossakowski master equation with LD 6= 0 does not preserve theeigenvalues of the density operators, the reachable sets can not be conveniently char-acterized via a unitary orbit. Hence, the study of reachable sets for open quantumsystems becomes more complicated. In what follows, we illustrate some examples ofwell-studied physical processes in open quantum systems for the simplest two-levelsystems : depolarizing channel, amplitude damping and phase flip e.g. see [52]. We


give some numerical simulations to catch some insight on how the failure of control-lability can be immediately recognized and how the reachable sets might look like.This will have some connection to the claim made by C. Altafini [6, 7, 8] dealing withthe global structure of the reachable sets.

Consider the simplest two-level open quantum systems,

ρ = −i[H, ρ] + LD(ρ) , ρ ∈ P, (3.25)

with −iH = 12(dSz + u(t)Sy), where Sz and Sy are as in Eq.(3.18). The control u(t)

takes value in R. Practically, this means that the control u(t) is allowed to be verylarge compared to the parameter d representing the strength of the drift Hamiltonian.When there were no dissipation (LD = 0), as has been discussed in Example 3.18and 3.19, the system in Eq.(3.25) is density operator controllable and moreover, it issmall-time local controllable (STLC). However, it is not fast controllable (FC).

We choose the Pauli matrices as a particular orthonormal basis of her0(2)

B1 :=i√2Sx , B2 :=

i√2Sy , B3 :=

i√2Sz, (3.26)

where Sx, Sy and Sz are as in Eq.(3.18). With respect to this orthonormal basis, thematrix representation for the adjoint action of the drift Hamiltonian −i[H0, (·)] :=

[d2Sz, (·)] and the control Hamiltonian −i[u(t)H1, (·)] := [u(t)

2Sy, (·)] can be calculated

using formula in Eq.(2.50), with

LH0=

0 −d 0d 0 00 0 0

∈ so(3) , LH1

=

0 0 10 0 0−1 0 0

∈ so(3),

respectively. In the vector of coherence representation, we consider the evolution ofthe three dimensional vector v = [vx vy vz]

⊤ using the notation in Eq.(2.47),

v = LHv + LaffD v =

0 −d u(t)d 0 0

−u(t) 0 0

v + LDv + q,

where LD and q will be calculated explicitely by Eq.(2.54) and Eq.(2.55) for differentprocesses later on. Suppose we associate the vector of coherence v with the Euclideancoordinate axis (x, y, z). Then we can visualize the effect of the drift Hamiltonian LH0

in the equation of motion is to (slowly) rotate v around z-axis, while the effect of thecontrol Hamiltonian u(t)LH1

is to (quickly) rotate v with respect to y-axis. Here weloosely use the phrases “slowly” and “quickly” with repect to the fact that u(t) can bemade relatively large compared to the drift d. Note that in all numerical simulations,we rescale the vector of coherence such that the maximum size of ||v|| equals to one.Hence, the Bloch ball here is a 3-D ball with radius one centered at the origin.


Example 3.26. (Depolarizing Channel). The dissipation process for the depo-larizing channel is modelled by the real diagonal GKS matrix

A :=a

2I3 ≥ 0 , a > 0,

such that LD =3∑

j=1

a

2Ljj, where Ljj can be calculated explicitely via Eq.(2.51),

L11 = diag(0,−1,−1) , L22 = diag(−1, 0,−1) , L33 = diag(−1,−1, 0).

Since the GKS matrix is real, the depolarizing channel is a unital process, cf. Lemma2.16, with the translational part q = 0. The equation for the vector of coherence isthen given by

vx

vy

vz

=

−a −d u(t)d −a 0

−u(t) 0 −a

vx

vy

vz

, (3.27)

where a accounts for the rate of dissipation. Correspondingly, the dissipation termLD in the GKS form can be written as the sum of double-commutators

LD(ρ) = −3∑

j=1

a

4[Bj , [Bj, ρ]] , a > 0.

Similarly for the Lindblad diagonal form Eq.(2.21), we have

LD(ρ) =

3∑

k=1

VkρV†k − 1

2V †

k Vkρ−1

2ρV †

k Vk , Vk = V †k =

√a

2Bk.

We fix the different values for the drift d > 0 and the rate of dissipation a > 0. Thenwe give some simple simulations of the unforced evolution of the system followed byan illustration of possible (unbounded or fast) control actions u(t) in the Bloch ball.

As seen from the figures, when the dissipation rate a is relatively larger thanthe drift d, (Fig. 3.1a), the dissipation dominates the evolution such that the vectorof coherence is attracted rapidly toward the origin. In contrast, when the dissipationrate is relatively smaller than the drift (Fig.3.1c), the effect of the drift Hamiltonianas slow rotation of v with respect to z-axis can be clearly seen before the systemasymptotically reachs the origin. For all cases, due to unitality, the system eventuallyevolves to the completely mixed state (the origin). This situation holds no matter wehave “unbounded” controls since the effect of control here is to give rotations of v withrespect to y-axis. Hence, this example gives some quick insight why controllabilityfails for open quantum systems.


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

X

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

Z

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

Z

−1

−0.5

0

0.5

1

−1−0.5

00.5

1−1

−0.5

0

0.5

1

XY

Z

(a) a = 3d

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

X

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

Z

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

Z

−1

−0.5

0

0.5

1

−1−0.5

00.5

1−1

−0.5

0

0.5

1

X

Y

Z

(b) a = d

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

X

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

Z

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Y

Z

−1−0.5

00.5

1

−10

1−1

−0.5

0

0.5

1

XY

Z

(c) a = 0.1d

Figure 3.1: Depolarizing Channel

Example 3.27. (Amplitude Damping). The amplitude damping process ischaracterized by the (complex) GKS matrix

A :=a

2

1 −i 0i 1 00 0 0

≥ 0 , a > 0.

By Lemma 2.16, the condition for unital process for two-level systems is equivalent tothe real GKS matrix. Thus, the amplitude damping is a non-unital process. Observethat the real part of the off-diagonal elements of A is zero and since the structureconstant gjkl of su(2) is always zero, we have Im(Ljk) = 0, by Eq.(2.51). Thus, for


the amplitude damping we have simply

LD =

2∑

j=1

a

2Ljj =

−a2

0 00 −a

20

0 0 −a

.

The translational part q can be explicitely calculated

q = − 2√N

3∑

j<k

Im(ajk)

fjk1

fjk2

fjk3

1√

N= − 2√

2(−a

2)

00√2

1√

2=

00a

1√

2.

The dynamical equation for the vector of coherence is then given by

vx

vy

vz

=

−a2

−d u(t)d −a

20

−u(t) 0 −a

vx

vy

vz

+

00a

1√

2. (3.28)

Note that the GKS matrix A can be diagonalized

D = diag(0, 0, a) = W †AW , W =

1√2i 0 − 1√

2i

1√2

0 1√2

0 1 0

,

such that the dissipation term LD in the Lindblad diagonal form Eq.(2.21) reads

LD(ρ) = V3ρV†3 − 1

2V †

3 V3ρ−1

2ρV †

3 V3,

where V3 is given by

V3 =√λ3

∑3j=1wj3Bj =

√a(− 1√

2iB1 + 1√

2B2 + (0)B3

)=

√a

2(Sx + iSy)

V †3 =

√a

2(Sx − iSy).

The operators V3 and V †3 are well-known as the atomic ladder operators characterizing

the spontaneous emission process in two-level atomic systems.

It is clear from the figures that the effect of the dissipation rate a and thedrift d in this case is similar to the depolarizing channel. However, the amplitudedamping is a non-unital process such that the quantum state evolves asymptoticallytoward a pure state (in this case v = [0 0 1]⊤), hence purification happens. Here, wecan conclude as well that the control, eventhough unbounded, can not speed up thepurification process, i.e. the purification is only due to the dissipation rate a. Hence,controllability also fails in this case.


−1 −0.5 0 0.5 1−1

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(a) a = 3d

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Z

(b) a = d

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Z

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0

1

−1−0.500.51−1

−0.5

0

0.5

1

XY

Z

(c) a = 0.1d

Figure 3.2: Amplitude Damping

Example 3.28. (Phase Flip). The phase flip process, also known as the phasedamping, is characterized by the real GKS matrix

A := diag(0, 0, a) ≥ 0 , a > 0.

In this case we have the unital process

LD = aL33 =

−a 0 00 −a 00 0 0

, q = 0,


such that the equation for the vector of coherence is given byvx

vy

vz

=

−a −d u(t)d −a 0

−u(t) 0 0

vx

vy

vz

. (3.29)

Correspondingly in the GKS form, the dissipative part LD is given by

LD(ρ) = −a2[B3, [B3, ρ]],

while in the Lindblad diagonal form Eq.(2.21), LD reads

LD(ρ) = V ρV † − 1

2V †V ρ− 1

2ρV †V , V =

√aB3.

−1 −0.5 0 0.5 1−1

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0

0.5

1

Y

X

−1 −0.5 0 0.5 1−1

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Z

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0

0.5

1

Y

Z

−1

0

1

−1

0

1−1

−0.5

0

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1

XY

Z

(a) a = 3d

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Z

−1

0

1

−1

0

1−1

−0.5

0

0.5

1

XY

Z

(b) a = 0.1d

Figure 3.3: Phase Flip

For the phase flip, the dissipation rate a affects only the components vx and vy

of the vector of coherence v. Oscillations occur in vx and vy coordinates while vz canbe altered only by the control. Nevertheless, controllability also fails since once thesystem evolves leaving a pure state, it never comes back to a pure state.

Example 3.29. (The Reachable Set of a Depolarizing Channel).

The figures in Examples 3.26-3.28 should give a fairly rough idea how the reach-able sets might look like in the simplest case. In the following Figure 3.4, we providean image of a slice of the reachable set for a depolarizing channel. The image was pro-duced using the software Subdiv based on the Subdivision Algorithm developed by


L. Grune [25] to compute domains of attractions of a dynamical system, which is re-lated to the corresponding reachable sets. However, the algorithm is computationallytoo expensive when applied to higher dimensional quantum systems.

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−0.2

0

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0.4

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0.8

1

Y

X

Figure 3.4: Slice of reachable set of Depolarizing Channel (a = d)

Analytical studies of the global structure of reachable sets of the Lindblad-Kossakowski master equation appear scarcely in the quantum control literature. Werecall the following Claim 3.30 dealing with a characterization for the reachable setsof the unital Lindblad-Kossakowski master equation. However, we shall see that thestatement in the claim is false especially for open quantum systems with dimensionN > 2. Even for the special case of two-level systems N = 2, the claim is still incorrectunless we have a stronger assumption. Therefore, we shall correct this mistake forunital two-level systems N = 2 in Theorem 3.32.

Claim 3.30. (C. Altafini [6, 7, 8]). Consider the unital (N -level) Lindblad-Kossakowski master equation in the vector of coherence representation Eq.(3.22)which satisfies the following assumptions,

1. The system is accessible, i.e. R(v0) has non-empty interior for all v0.

2. When there were no dissipation (LD = 0), the system is density operator con-trollable, i.e. sD := 〈iH0, iH1, . . . , iHm〉LA = su(N).

Then, the reachable set R≤t(v0) is an annulus of inner and outer radius ||v(t)|| and

||v0||, respectively. The closure of the reachable set R(v0) is the whole ball of radius||v0||.


Remark 3.31. 1. The failure of the claim follows immediately even for the sim-plest case of two-level systems N = 2 by the counter-examples provided inExample 3.26 and 3.28. Both systems, depolarizing channel and phase flip, areunital and accessible, see Example 3.42 in Section 3.3.3. Moreover, when therewere no dissipation we have for d 6= 0,

sD = 〈iH0, iH1〉LA = 〈d2Sz,

1

2Sy〉LA = su(2),

i.e. the system is density operator controllable and even it is STLC. But never-theless, by observing the figures in the Example 3.26 and 3.28, the reachable setsdo not look like an annulus. This is further clarified by Figure 3.4 in Example3.29 depicting a slice of the approximated reachable set. Hence, the reachableset is not an annulus even for two-level systems.

2. In relation with the weaker notions of controllability for open quantum systems,we can also analyze why the reachable set is not an annulus. Observe that thedepolarizing channel in Example 3.26 is wh-controllable by Theorem 3.24(b).Following [35], when LD = 0, the condition sC := 〈iH1〉LA 6= su(2) implies thatthere exists a point vf ∈ Ksu(2) · v0 = SO(3) · v0 which can only be reached fromv0 after some time T ∗, where T ∗ can not be made arbitrarily small. Now, whenLD = −aI3, a > 0, at time T ∗ we have the norm ||v∗f || = exp(−aT ∗)||vf || <||vf ||. This shows that v∗f can not be arbitrarily close to the point vf which isobviously in the annulus with outer radius ||v0||. Therefore, reachable sets ofwh-controllable systems are in general not an annulus.

3. For general N -level systems, we recall Lemma 2.25 and Proposition 2.27 showingthat for N > 2 there exists a set of points with non-empty interior in the Blochball BN2−1 that do not even correspond to physically valid quantum states.Hence in this case, the reachable sets can not be an annulus with respect to theEuclidean norm.

However for unital two-level systems, the reachable set can be still expectedto be an annulus provided that we replace the STLC or the density operator con-trollability (when there were no dissipation) by the stronger assumption of FC (fastcontrollability). This is exactly the case where h-controllability comes into play.

Theorem 3.32. (Annulus reachable set for two-level systems). Considera unital two-level Lindblad-Kossakowski master equation in the vector of coherencerepresentation Eq.(3.22) which is h-controllable, e.g. sC := 〈iH1, . . . , iHm〉LA = su(2).Then the reachable set R≤t(v0) is an annulus of inner and outer radius ||v(t)|| and||v0||, respectively, where v(t) is any solution of Eq.(3.22) at time t. The closure ofthe reachable set R(v0) is the whole ball of radius ||v0||.

3.3. Accessibility of the Lindblad-Kossakowski Master Equations 83

Proof. h-controllability implies that any v0 can be steered arbitrarily close and fastto any element vf on the orbit Ksu(2) · v0 = SO(3) · v0. Since SO(3) acts transitivelyon the real sphere in R3, any vector vf ∈ R3 with ||vf || = ||v0|| can be reachedarbitrarily close and arbitrarily fast. Now as time elapses t > 0, Lemma 2.20 showsthat a unital system monotonically decreases the norm ||v(t)|| ≤ ||v0||, independentof the applied control. Then, repeating the argument of h-controllability on the orbitKsu(2) · v(t) = SO(3) · v(t) for t > 0, we have the desired annulus result.

Quite naturally, one is tempted to generalize Theorem 3.32 for the general uni-tal N -level systems when h-controllability holds, i.e. to say that the closure of thereachable set is an annulus in the sense that

R≤t(ρ0) = ρ ∈ P | ||ρ(t)|| ≤ ρ ≤ ||ρ0||, (3.30)

where ||ρ|| :=√

Tr(ρ2). However, we shall show shortly that this is not the case.Indeed, suppose we have h-controllable systems for N -level systems, for N > 2, orequivalently d > 3. Note that the Euclidean norm ||v|| on the vector of coherencecorresponds to the trace norm ||ρ||. Consider an initial condition ρ0 with ||ρ0|| < 1.By h-controllability, ρ0 can be steered arbitrarily close and fast to any ρf ∈ O(ρ0),which has the same norm as ρ0. Now suppose another target state ρ1, which also hasthe same norm as ρ0, but need not lie on the unitary orbit of ρ0. In this case, eventhough h-controllability holds, it is clear that ρ1 can not be reached arbitrarily fastsince ρ1 /∈ O(ρ0). Now, since the unital system monotonically decreases the norm ||ρ||,the reachable set can never be an annulus for higher dimensional quantum systems.

3.3 Accessibility of the Lindblad-Kossakowski Mas-

ter Equations

According to Theorem 3.21, the Lindblad-Kossakowski master equation is never con-trollable on P. This issue immediately raises the question under what circumstancesthe Lindblad-Kossakowski master equation is at least accessible, i.e. when do thereachable sets R(ρ0) have at least non-empty interior in P? Since the evolutionof open quantum systems can no longer be represented as unitary conjugation, theLindblad-Kossakowski master equation on P can no longer be lifted to the specialunitary group SU(N) as before. Instead, it is appropriate to lift the master equationto a bilinear control system on the general Lie group GL(d,R) and the semidirectproduct GL(d,R) ⋊ Rd for the unital and non-unital systems, respectively.

3.3.1 The Unital Case

We consider the unital N -level Lindblad-Kossakowski master equation in the vectorof coherence representation as a bilinear control system evolving on Rd, d = N2 − 1,


which leaves Φ(P) ⊂ Bd invariant,

(Σi) :

v = LHv + LDv , v(0) = v0, v ∈ Rd

=(LH0

+m∑

k=1

uk(t)LHk

)v +

d∑

jk=1

ajkLjkv

=:(A0 +

m∑

k=1

uk(t)Ak

)v,

(3.31)

where A0 := LH0+ LD is the overall drift term composed from the Hamiltonian

drift and the dissipation term, and Ak := LHkare the Hamiltonian controls. Hence,

A0 ∈ gl(d,R) and Ak ∈ so(d). Now we can consider the bilinear control system (Σi)on the manifold M := Rd as an induced system of the corresponding “lifted” bilinearcontrol system evolving on the Lie group G = GL(d,R),

(Σ) : X =(A0 +

m∑

k=1

uk(t)Ak

)X , X(0) = I, (3.32)

where A0 and Ak are as in Eq.(3.31). It is clear that the prescribed Lie-group actionα : G× Rd → Rd is simply given by the usual matrix-vector multiplication

α(X, v) := Xv , X ∈ GL(d,R), v ∈ Rd.

The system Lie-algebra sΣ of the Lindblad-Kossakowski master equation is the Liealgebra generated by A0, A1, . . ., Am, i.e.

sΣ := 〈A0, A1, . . . , Am〉LA,

and the system group is then given by GΣ := 〈exp(sΣ)〉G.Since there exists a diffeomorphism

Φ : P → Φ(P) ⊂ Rd

from the set of density operators to the set of corresponding vectors of coherence, thequestion of accessibility of the unital Lindblad-Kossakowski master equation originallydefined on P (i.e. whether the reachable set R(ρ0) has non-empty interior in P for allρ0 ∈ P \ IN/N) is completely the same as asking for accessibility of the Lindblad-Kossakowski master equation in the vector of coherence representation on Φ(P) (i.e.whether the reachable set RΣi

(v0) of the system (Σi) in Eq.(3.31) has non-emptyinterior in Φ(P) for all v0 ∈ Φ(P) \ 0). We remark here that the point IN/N ∈ P,and correspondingly the origin 0 ∈ Φ(P), is excluded because it is a common fixedpoint which is left invariant by the flow of the unital Lindblad-Kossakowski masterequation. Hence, the reachable set R(ρ0) for ρ0 = IN/N and correspondingly RΣi

(v0)for v0 = 0 can not have non-empty interior in P and Φ(P), respectively.


Lemma 3.33. The unital Lindblad-Kossakowski master equation in the vector ofcoherence representation Eq.(3.31) is accessible on Φ(P) \ 0 if and only if it isaccessible on Rd \ 0.

Proof. First note that Φ(P) has non-empty interior in Rd. The “if” statement isobvious since Φ(P) ⊂ Rd. Suppose Eq.(3.31) is accessible on a point v0 ∈ Φ(P) \ 0,i.e. the reachable set RΣi

(v0) = SΣ · v0 has non-empty interior in Φ(P) \ 0, (orequivalently, also in Rd \ 0). This implies that, for any point v = av0 ∈ Rd \ 0where a ∈ R is non-zero, the reachable set RΣi

(v) = aRΣi(v0) = a(SΣ · v0) has non-

empty interior in Rd \ 0. Now observe that IN/N is an interior point in P andso is the origin in Φ(P). Thus, we can choose any such point v0 ∈ Φ(P) \ 0 inan open neighborhood of 0 and repeat the scaling argument above to obtain that forall v ∈ Rd \ 0 the reachable set RΣi

(v) has non-empty interior in Rd \ 0, i.e.accessibility on Rd \ 0 holds.

Due to Lemma 3.33 and Theorem 3.10, it is clear that the accessibility of theunital Lindblad-Kossakowski master equation is equivalent to classifying all subgroupsof G = GL(d,R) which act transitively on M := Rd \ 0, and checking whetherthe system group GΣ of the lifted system (Σ) in Eq.(3.32) coincides to one of thosetransitive subgroups. Indeed, the classification of all connected Lie subgroups ofGL(d,R) acting transitively on M = Rd \ 0 is a well-studied classical topic intopology where results can be found in e.g. [11, 12, 60]. However, the classification listsare incomplete therein. The complete list was provided in e.g. [44], and is reproducedin the Appendix B. By exploiting the classification of Lie groups which act transitivelyon M = Rd \ 0, we state the accessibility result of the unital Lindblad-Kossakowskimaster equation in term of its system Lie algebra sΣ. Note that in the sequel, we willoften use the short-cut term “transitive Lie groups” to mean the Lie groups which acttransitively on Rd \ 0 via usual matrix-vector multiplication. Similarly, “transitiveLie algebras” means the Lie algebras of the corresponding transitive Lie groups.

Theorem 3.34. (Main Result 2a : Accessibility of Open Quantum Systems,Unital Case). The unital N-level Lindblad-Kossakowski master equation

ρ = −i[H0 +

m∑

k=1

uk(t)Hk, ρ]

+ LD(ρ), ρ(0) = ρ0 ∈ P \ IN/N

is accessible if and only if the system Lie algebra sΣ = 〈A0, A1, . . . , Am〉LA ⊆ gl(d,R),d = N2 − 1, of the corresponding vector of coherence equation (3.31) is conjugate toone of the following types

(a) For N even: gl(d,R), so(d) ⊕ R.

(b) For N odd:


– so(d) ⊕ R, su(d/2) ⊕ eiαR, su(d/2) ⊕ C,

– sp(d/4) ⊕ eiαR, sp(d/4) ⊕ C, sp(d/4) ⊕ H,

– gl(d,R), gl(d/2,C), sl(d/2,C) ⊕ eiβR,

– sl(d/4,H) ⊕ eiβR, sl(d/4,H) ⊕ C, sl(d/4,H) ⊕ H,

– sp(d/2,R) ⊕ R, sp(d/4,C) ⊕ eiβR, sp(d/4,C) ⊕ C,

– spin(7) ⊕ R, if N = 3, or equivalently, d = 8,

where α ∈ (−π2, π

2) and β ∈ [−π

2, π

2].

Proof. From Proposition 2.21, we have the strict inequality Tr(LD) < 0 for all LD 6= 0.Correspondingly, cf. Remark 2.30 and Appendix C.3, the matrix representation LD

has stricly negative trace, i.e. Tr(LD) < 0. Now, since

sΣ = 〈A0, A1, . . . , Am〉LA = 〈LD + LH0, LH1

, . . . , LHm〉LA,

with LHj∈ so(d), for all j = 0, 1, . . . , m, it is clear that there always exist elements

of sΣ with non-zero trace. Therefore, we exclude all transitive Lie groups in theclassification list whose the corresponding Lie algebras are subalgebras of sl(d,R).Thus, the system Lie algebra sΣ can never be of the form (or conjugate) to thefollowing Lie algebras,

• sl(d,R), sl(d/2,C), sl(d/4,H)

• sl(d/4,H) ⊕ su(2) ∼= sl(d/4,H) ⊕ sp(1)

• sp(d/2,R), sp(d/4,C).

Additionally, the following special transitive Lie algebras with their specific matrixrepresentations,

• g2 ⊕ R ⊂ gl(7,R),

• spin(9) ⊕ R ⊂ gl(16,R),

• spin(9, 1,R) ⊂ gl(16,R),

• spin(9, 1,R) ⊕ R ⊂ gl(16,R),

can never be (or conjugate to) the system Lie algebra sΣ since the sizes of their matrixrepresentations never match with the dimension of N -level quantum systems in thefollowing sense. Quantum states of N -level systems with N = 2, 3, 4, 5, . . ., have therespective dimension d = N2 − 1 = 3, 8, 15, 24, . . ., in the vector of coherence repre-sentation and thus, d = 7, 16 does not appear as the dimension of N -level quantumstates.


If N is even, (and hence d = N2 − 1 is odd), most of the Lie algebras on the listare excluded due to the fact that the sizes of their matrix representations d are even.The only Lie algebras left are gl(d,R) and so(d)⊕R. However, for N odd (d is even),many more Lie algebras are possible and those are listed in Theorem 3.34(b).

Moreover, for a particular class of unital master equations, provided that anadditional assumption is satisfied, we can improve the above accessibility conditionby excluding more transitive Lie algebras. We start with the following lemma.

Lemma 3.35. Let k ⊆ so(d) be a Lie subalgebra which acts irreducibly on Rd,i.e. there are no nontrivial k-invariant subspaces of Rd. Then, the set C(k) :=

A ∈

GL(d,R) | AkA−1 ⊆ so(d)

is given by R+ ·O(d), where O(d) denotes the orthogonalgroup.

Proof. Suppose A ∈ C(k). Then for any B ∈ k we have

0 = (ABA−1)⊤ + ABA−1 = −A−⊤BA⊤ + ABA−1.

Equivalently, we have [A⊤A,B] = 0 for all B ∈ k, which implies that A⊤A must be inthe centralizer of k within gl(d,R). Since there are no nontrivial k-invariant subspacesof Rd, following [12], this implies that every non-zero element of the centralizer of k

is invertible, i.e. it is a division algebra. It is known that every finite dimensionaldivision algebra over R is isomorphic (as fields) to either R, C or H.

If the centralizer of k is isomorphic to R, then A⊤A = λId, λ > 0. This impliesA = ±

√λK, with K⊤K = Id, or equivalently, A = γG, with γ > 0, G ∈ O(d), and we

are done. Suppose the centralizer of k is isomorphic to C. Any matrix representationof w1 + xi ∈ C in Rd×d is given by a map of the form

1 7→ Id , i 7→ X ∈ Rd×d,

with X2 = −Id. This implies that X can not be symmetric since X⊤ = X yieldsthe contradiction X2 = X⊤X = −Id. Therefore, since A⊤A is symmetric, it can bewritten as A⊤A = λId + xX, with λ > 0, x = 0, and the result follows as before.Similarly, suppose the centralizer of k is isomorphic to H. Any matrix representationof quaternions q = w1 + xi + yj + zk ∈ H in Rd×d is given by a map of the form

1 7→ Id , i 7→ X , j 7→ Y , k 7→ Z , X, Y, Z ∈ Rd×d,

where X2 = Y 2 = Z2 = −Id and XY = Z, Y Z = X, ZX = Y , cf. [12]. This showsthatX, Y and Z can not be symmetric. Moreover, any representation of non-zero purequaternions q = xX+yY +zZ ∈ Rd×d satisfies q2 = −kId, where k = x2 +y2+z2 > 0.This shows that any matrix representation of pure quaternions can not be symmetric.Hence, we can again write A⊤A = λId + xX + yY + zZ, with λ > 0 and x, y, z = 0.The proof is complete.


Theorem 3.36. (Main Result 2b : Accessibility of Open Quantum Systems,Unital Case). Consider the unital N-level Lindblad-Kossakowski master equation

ρ = −i[H0 +

m∑

k=1

uk(t)Hk, ρ]

+ LD(ρ), ρ(0) = ρ0 ∈ P \ IN/N.

Assume that the GKS matrix is real and moreover, the system Lie algebra sΣ =〈A0, A1, . . . , Am〉LA ⊆ gl(d,R), d = N2 − 1, of the corresponding vector of coherenceequation (3.31) contains a subalgebra k ⊆ so(d) which acts irreducibly on Rd. Thenthe system is accessible if and only if sΣ is conjugate to one of the following types

(a) For N > 2 even: gl(d,R)

(b) For N odd:

– gl(d,R), gl(d/2,C), sl(d/2,C) ⊕ eiβR,

– sl(d/4,H) ⊕ eiβR, sl(d/4,H) ⊕ C, sl(d/4,H) ⊕ H,

– sp(d/2,R) ⊕ R, sp(d/4,C) ⊕ eiβR, sp(d/4,C) ⊕ C,

where α ∈ (−π2, π

2) and β ∈ [−π

2, π

2].

(c) For N = 2: gl(3,R), so(3) ⊕ R.

Proof. For N > 2, we will exclude the following Lie algebras from Theorem 3.34 :

• so(d) ⊕ R

• su(d/2) ⊕ eiαR, su(d/2) ⊕ C,

• sp(d/4) ⊕ eiαR, sp(d/4) ⊕ C, sp(d/4) ⊕ H,

• spin(7) ⊕ R, for d = 8.

First, we show that all of the Lie algebras above can be regarded as a Lie subalgebraof the form k0 ⊕ R, with k0 ⊆ so(d). The first case so(d) ⊕ R is trivial. Considersu(d/2) ⊕ C. Any element A ∈ su(d/2) ⊕ C can be expressed as

A = X + iY + (a+ bi)Id/2,

where X ∈ so(d/2), Y ∈ sym(d/2,R) and a, b ∈ R. The representation of A inso(d) ⊕ R ⊂ gl(d,R) is given by

A =

[X + aId/2 −Y − bId/2

Y + bId/2 X + aId/2

]=

[X −Y − bId/2

Y + bId/2 X

]+ aId

= A0 + aId ∈ so(d) ⊕ R,


with A0 ∈ so(d). The same reasoning applies to su(d/2) ⊕ eiαR. Now recall that thecompact symplectic Lie algebra sp(d/4) is defined by

sp(d/4) = sp(d/4,C) ∩ u(d/2) ⊂ su(d/2).

Hence, sp(d/4)⊕eiαR and sp(d/4)⊕C can be regarded as Lie subalgebras of so(d)⊕R.Further, note that any quaternion q = w+ xi + yj + zk ∈ H considered as a real d× dmatrix takes the form

w −x −y −zx w z −yy −z w xz y −x w

⊗ Id/4 ∈ gl(d,R),

see [11]. Then, the representation of A ∈ sp(d/4)⊕H in so(d)⊕R ⊂ gl(d,R) is givenby

A =

X +

[0 −xx 0

]⊗ Id/4 −Y +

[−y −zz −y

]⊗ Id/4

Y +

[y −zz y

]⊗ Id/4 X +

[0 x

−x 0

]⊗ Id/4

+ wId

= A0 + wId ∈ so(d) ⊕ R,

for some X ∈ so(d/2) and Y ∈ sym(d/2,R). One can easily verify that A0 ∈ so(d).For the special case of spin(7)⊕R with d = 8, we recall that spin(7) is represented asa Lie subalgebra of gl(8,R) which is isomorphic to so(7). A Lie algebra isomorphismφ : spin(n) → so(n) can be found in e.g. [9], Chapter 5.

Suppose the system Lie algebra sΣ is conjugate to any of the listed Lie subalgebraof so(d) ⊕ R, i.e. sΣ = A−1(k0 ⊕ R)A for some A ∈ GL(d,R) and k0 ⊆ so(d). Byassumption that sΣ contains k ⊆ so(d), we have k ⊆ A−1k0A ⊆ sΣ, or equivalently,AkA−1 ⊆ k0 ⊆ so(d). Then, Lemma 3.35 implies that A must be a real positivemultiple of an orthogonal matrix. Consequently, we obtain

sΣ = A−1(k0 ⊕ R)A ⊆ so(d) ⊕ R.

Now, when the GKS matrix is real, Lemma 2.23 shows that LD = LD1 is self-adjointand corespondingly, its matrix representation LD is symmetric. Since A0 = LH0

+LD ∈sΣ with LH0

∈ so(d), it follows from sΣ ⊆ so(d) ⊕ R that LD has to be of the formLD = γId with γ < 0. See also Example 3.44 to show that LD = γId is indeed possiblefor arbitrary N -level systems. Recall that

〈LH0, LH1

, . . . , LHm〉LA ⊆ ksu(N) ⊆ so(d),

where equality in the last inclusion never holds for N > 2 due to Lemma 3.22. Then,for LD = γId, we obtain

sΣ = 〈LD + LH0, LH1

, . . . , LHm〉LA ⊆ ksu(N) ⊕ R ( k0 ⊕ R.


Note that for N > 2, the last inclusion above is proper, i.e. equality never holds, sinceby simple dimension counting one has

dim(ksu(N)) = dim(su(N)) = N2 − 1 = d < d(d− 1)/2 = dim(so(d))

dim(ksu(N)) = dim(su(N)) = N2 − 1 = d < d2

4− 1 = dim(su(d/2))

dim(ksu(N)) = dim(su(N)) = N2 − 1 = d < d4(2(d

4) + 1) = dim(sp(d/4))

dim(ksu(3)) = dim(su(3)) = 8 < 21 = dim(so(7)) = dim(spin(7)),

for d ≥ 8 (or equivalently, N ≥ 3). Obviously, this is a contradiction to the initialassumption that sΣ = A−1(k0 ⊕R)A, which must have the same dimension as k0 ⊕R.It follows that the above listed Lie subalgebras have to be excluded.

Now for the exceptional case N = 2 (or equivalently d = 3). Suppose LD = γI3(see e.g. the depolarizing channel in Example 3.26), then we have


, . . . , LHm〉LA ⊆ ksu(2) ⊕ R = so(3) ⊕ R,

since ksu(2) = so(3). Hence so(3) ⊕ R is possible for two-level systems.

The assumption in Theorem 3.36, which requires that sΣ contains a subalgebrak ⊆ so(d) acting irreducibly on Rd, can be fulfilled for example, when

sC := 〈iH1, . . . , iHm〉LA = su(N).

Indeed, this implies kC := 〈LH1, . . . , LHm

〉LA = ksu(N) ⊆ sΣ, where ksu(N) is irreducibleaccording to Lemma 3.22, as required. Other subalgebras k which act irreducibly onRd are e.g. the embeddings of su(d/2) and sp(d/4) in so(d).

The following corollary shows that for n-coupled spin- 12

systems (e.g. n-qubitsystems in quantum computing applications [38, 39, 40, 57, 58]), the characterizationof accessibility becomes surprisingly simple.

Corollary 3.37. (n-coupled spin-12

systems). Consider the unital Lindblad-Kossakowski master equation of n-coupled spin-1

2systems.

(i) For n ≥ 1, the spin system is accessible if and only if the system Lie algebra sΣ

is equal to gl(22n − 1,R) or conjugate to so(22n − 1) ⊕ R.

(ii) For n > 1, if additionally the GKS matrix is real and sΣ contains an irreduciblesubalgebra of so(22n − 1), then the spin system is accessible if and only if sΣ isequal to gl(22n − 1,R).

Proof. This remarkable result is of course due to the fact that n-coupled spin- 12

systemsare N = 2n-level systems, and thus N is always even. Thus, (i) and (ii) follows fromTheorem 3.34 and 3.36, respectively, for N even.


We briefly remark that Theorem 3.34 and 3.36 correct the accessibility resultspreviously stated in [6, 8] by C. Altafini, where the listed classification of transitiveLie groups on Rd \ 0 is incomplete. Moreover in [6, 8], it was incorrectly statedthat the system Lie algebra sΣ being equal to gl(d,R) is necessary and sufficient foraccessibility of arbitrary unital N -level systems. Our theorems also slightly revise theaccessibility results which appeared in [47].

3.3.2 The Non-Unital Case

In this subsection we consider the non-unital N -level Lindblad-Kossakowski masterequation in the vector of coherence representation Eq.(2.47) as a bilinear controlsystem on Rd,

(Σi) :

v = LHv + LaffD v = LHv + LDv + q , v(0) = v0, v ∈ Rd

=(LH0

+

m∑

k=1

uk(t)LHk

)v +

d∑

jk=1

ajkLjkv + q

=:(A0 +

m∑

k=1

uk(t)Ak

)v + q,

(3.33)

where A0 := LH0+LD ∈ gl(d,R), Ak := LHk

∈ so(d) and q ∈ Rd non-zero. The system(Σi) is a bilinear control system on the manifold M = Rd, but now the origin of Rd

is no longer a common fixed point due to the constant drift q. Moreover, the system(Σi) can be considered as an induced system of a bilinear control system evolving onsome Lie group. Precisely, we lift the system (Σi) to a bilinear control system on thesemidirect product G′ := GL(d,R) ⋊ Rd.

Before describing the lift and the corresponding group action, we recapitulatesome facts dealing with the semidirect product Lie group (and also the correspondingLie algebra). Let the Lie group G := GL(d,R) act linearly on a vector space Rd andlet G′ be the semidirect product of G and Rd which is denoted by

G′ = G⋊ Rd.

By the semidirect product, we mean that the group operation is defined by

X ′1X

′2 := (X1, x1) · (X2, x2) = (X1X2, X1x2 + x1), (3.34)

for all (X1, x1), (X2, x2) ∈ G′. Likewise, the Lie algebra g′ of G′ is written as

g′ = g ⋊ Rd,

where the Lie bracket in g′ is defined as

[(A, a), (B, b)] := ([A,B], Ab− Ba), (3.35)


for all (A, a), (B, b) ∈ g′. In a matrix representation, elements of G′ and g′ areconveniently expressed by

X ′ =

[X x0 1

]∈ G′ = G⋊ Rd , X ∈ G, x ∈ Rd,

A′ =

[A a0 0

]∈ g′ = g ⋊ Rd , A ∈ g, a ∈ Rd,

(3.36)

respectively, such that the group operation and the Lie bracket can be done as usualby matrix multiplication. We shall denote π and τ be the canonical projections of G′

onto G and Rd, respectively, i.e.

π : G′ → G , (X, x) 7→ Xτ : G′ → Rd , (X, x) 7→ x

and their corresponding differentials

dπ : g′ → g , (A, a) 7→ Adτ : g′ → Rd , (A, a) 7→ a.

Note that π is a Lie group homomorphism and consequently dπ is a Lie algebrahomomorphism satisfying

dπ([A′1, A

′2]) = [dπ(A′

1), dπ(A′2)] , for all A′

1, A′2 ∈ g′. (3.37)

For any A′ = (A, a) ∈ g′, the exponential map of A′ is

exp(A′) = exp(

[A a0 0

]) =

[exp(A) (I + A

2!+ A2

3!+ . . .)a

0 1

]∈ G′, (3.38)

and hence,π(exp(A′)) = exp(dπ(A′)). (3.39)

Now consider the following bilinear control system on G′ = GL(d,R)⋊Rd as thecorresponding lifted version of the non-unital Lindblad-Kossakowski master equation(3.33)

(Σ′) : X ′ =(A′

0 +

m∑

k=1

uk(t)A′k

)X ′ , X ′(0) = I, X ′ ∈ G′, (3.40)

where A′0 = (A0, q) ∈ g′ and A′

k = (Ak, 0) ∈ g′. Note that A0, Ak and q are as inEq.(3.33). The non-unital Lindblad-Kossakowski master equation (3.33) then can beconsidered as an induced system of (Σ′) on M = Rd with respect to the Lie groupaction α : G′ × Rd → Rd defined by

α(X ′, p) = α((X, x), p) = Xp+ x, (3.41)


for all X ′ = (X, x) ∈ G′. Indeed, since

D1α(X ′, p)(A′) = D1α((X, x), p)(A, a) = Ap + a,

one has, according to Eq.(3.6),

p = D1α(I, p)(A′

0 +

m∑

k=1

uk(t)A′k

)=(A0 +

m∑

k=1

uk(t)Ak

)p+ q,

which is exactly the induced system (Σi) in Eq.(3.33). Alternatively, by using theexplicit matrix representation Eq.(3.36), we have for the system (Σ′),

[X x0 0

]=

([A0 q0 0

]+

m∑

k=1

uk(t)

[Ak 00 0

])[X x0 1

]

=

[(A0 +

∑mk=1 uk(t)Ak)X (A0 +

∑mk=1 uk(t)Ak) x+ q

0 0

].

Thus, the system (Σi) can be viewed as a projection of the system (Σ′) onto the secondfactor and likewise, the projection of (Σ′) onto the first factor yields the bilinear controlsystem on GL(d,R),

(Σ) : X =

(A0 +

m∑

k=1

uk(t)Ak

)X , X(0) = I, X ∈ GL(d,R). (3.42)

We write formally Σi = τ(Σ′) and Σ = π(Σ′).

As far as the question of accessibility of the non-unital Lindblad-Kossakowskimaster equation is concerned, Theorem 3.10 again plays a crucial role. Consider thelifted system (Σ′) in Eq.(3.40) and the corresponding induced system (Σi) in Eq.(3.33).Then the accessibility of the non-unital Lindblad-Kossakowski master equation isequivalent to determining when the system group GΣ′ of (Σ′) acts transitively onM = Rd with respect to the group action α given by Eq.(3.41). Recall that thesystem Lie algebra sΣ′ of (Σ′) can be expressed as

sΣ′ = 〈A′0, A

′1, . . . , A

′m〉LA = 〈(A0, q), (A1, 0), . . . , (Am, 0)〉LA

⊆[〈A0, A1, . . . , Am〉LA W

0 0

]=

[sΣ W0 0

],

where W ⊆ Rd is a linear subspace and sΣ is the system Lie algebra of the correspond-ing projected system (Σ) in Eq.(3.42). Therefore, we have dπ(sΣ′) = sΣ. Similarly forthe system (Lie) group GΣ′ of (Σ′), by using the exponential map Eq.(3.38), we have

GΣ′ =⟨exp(sΣ′)

⟩G⊆⟨

exp(

[sΣ W0 0

])

⟩

G

=

[〈exp(sΣ)〉G W1

0 1

]=

[GΣ W1

0 1

],


where W1 is a subgroup of Rd, and GΣ is the system (Lie) group of the correspondingprojected system (Σ) in Eq.(3.42). In this case we obtain

π(GΣ′) = GΣ.

We will see that the accessibility property of the non-unital Lindblad-Kossakowskimaster equation can be characterized via the system Lie algebra sΣ of the projectedsystem (Σ) on G. We start with the following auxiliary result.

Lemma 3.38. Let G be a Lie subgroup of the semidirect product GL(d,R) ⋊ Rd,with the corresponding Lie group action on Rd as defined in Eq.(3.41). Then, G actstransitively on Rd if the following two conditions hold,

(a) The group π(G) acts transitively on Rd \ 0,

(b) There exists at least one element B′ in the Lie algebra g′ of G, which correspondsto a “pure translation”, i.e. B′ = (0, b), b 6= 0.

Proof. Suppose there exists (0, b) ∈ g′ which implies T := (Id, b) ∈ G. Take an elementX ′ = (X, x) ∈ G. Since G is a group we have

X ′T (X ′)−1 =

[X x0 1

]·[Id b0 1

]·[X−1 −X−1x

0 1

]=

[Id Xb0 1

]∈ G.

As π(G) acts transitively on Rd \ 0, we have π(G)b = Rd \ 0. Then, as X ′ runsover G (and X = π(X ′) runs over π(G)), we obtain

[Id w0 1

]∈ G , for all w ∈ Rd \ 0,

i.e. we have all translations in G. This basically shows transitivity of G on Rd. Indeed,for two arbitrary points z1, z2 ∈ Rd, choose X ′ = (Id, z2−z1) ∈ G such that z2 = X ′ ·z1,which implies transitivity of G on Rd.

Concerning the non-unital Lindblad-Kossakowski master equation, Lemma 3.38immediately provides the following sufficient condition for accessibility.

Theorem 3.39. (Main Result 3a : Accessibility of Open Quantum Systems,Non-Unital Case) Let (Σi) be the non-unital N-level Lindblad-Kossakowski masterequation in Eq.(3.33), and the corresponding lifted system (Σ′) in Eq.(3.40). Then,(Σi) is accessible if the following two conditions hold.

(a) The system Lie algebra sΣ = dπ(sΣ′) of the projected system (Σ) in Eq.(3.42) isconjugate to one of the transitive Lie algebras corresponding to the unital caseas listed in Theorem 3.34.


(b) There exists at least a “pure translation” in the Lie algebra sΣ′, i.e. B′ = (0, b) ∈sΣ′, with b 6= 0.

While the transitive action of GΣ on Rd \ 0 has been provided in the classifi-cation list, it is however not so straightforward to check whether a pure translation(0, b), b 6= 0 is really contained in sΣ′. Surprisingly, as will be shown in the followingTheorem 3.41, the transitivity of π(GΣ′) = GΣ on Rd \ 0 even implies the exis-tence of all pure translation in sΣ′, provided an additional assumption is satisfied.Therefore, we have an alternative sufficient condition for accessibility of the generalN -level non-unital Lindblad-Kossakowski master equation. Theorem 3.41 is basicallyan adaptation of the results by Jurdjevic, Sallet and Kupka [10, 36]. We start withthe following definition of a common fixed point.

Definition 3.40. Let (Σi) be the non-unital Lindblad-Kossakowski master equationin Eq.(3.33). Let

F :=Fu(v) :=

(A0 +

m∑

k=1

ukAk

)v + q

∣∣ uk ∈ R,

be the set of vector fields on Rd representing the bilinear control system (Σi) on Rd.The system (Σi) is said to have no common fixed point in Rd if, for any v ∈ Rd, therealways exists Fu(v) ∈ F such that Fu(v) 6= 0.

Theorem 3.41. (Main Result 3b : Accessibility of Open Quantum Systems,Non-Unital Case). Let (Σi) be the non-unital N-level Lindblad-Kossakowski masterequation in Eq.(3.33), and the corresponding lifted system (Σ′) in Eq.(3.40). Assumethat (Σi) has no common fixed point in Rd. Then, (Σi) is accessible if the system Liealgebra sΣ = dπ(sΣ′) of the projected system (Σ) in Eq.(3.42) is conjugate to one ofthe transitive Lie algebras corresponding to the unital case as listed in Theorem 3.34.

Proof. Suppose sΣ is as desired, which means that the corresponding system groupGΣ of the projected system (Σ) acts transitively on Rd \ 0. Such GΣ acts irreduciblyon Rd; i.e. the only GΣ-invariant subspaces of Rd are 0 and Rd; and hence the sameis true for the Lie algebra sΣ. Now consider the restricted projection dπ on the systemLie algebra sΣ′ of the system (Σ′) which is a linear operator

dπ′ := dπ|sΣ′

: sΣ′ → sΣ.

Since dπ(sΣ′) = sΣ, the restricted projection dπ′ is onto. Note that the kernel of dπ′

can be considered as a linear subspace of Rd,

ker(dπ′) = (0, a) ∈ sΣ′ | a ∈ Rd ⊆ Rd,

and moreover, ker(dπ′) is an ideal of sΣ′ , i.e. we have

adsΣ′

(ker(dπ′)) = ker(dπ′),


since [(B, b), (0, a)] = (0, Ba) ∈ ker(dπ′), for any (B, b) ∈ sΣ′ . So ker(dπ′) is invariantunder adjoint action of sΣ′ . Note that ker(dπ′) viewed as a linear subspace of Rd isalso invariant under linear action of sΣ. Since sΣ acts irreducibly on Rd, it followsthat the only sΣ-invariant subspaces are Rd or 0. Hence ker(dπ′) is either Rd or0.

Suppose ker(dπ′) = Rd. This clearly means that all translations (Id, a) | a ∈Rd are contained in the system group GΣ′ . Hence, we have transitivity of GΣ′ on Rd

and correspondingly, the accessibility result for the non-unital Lindblad-Kossakowskimaster equation follows by Theorem 3.10. So finally we shall disprove that ker(dπ′) =0.

To show that ker(dπ′) = 0 is impossible under the prescribed assumptions, werefer to [10, 36]. For the sake of completeness, we present here a more detailed andelaborated version of the original proof. We also clarify some subtleties and correctminor mistakes. Assume ker(dπ′) = 0. Then the restricted projection dπ′ : sΣ′ → sΣ

is a Lie algebra isomorphism. Now recall that any Lie algebra sΣ on the list of Theorem3.34 is of the form

sΣ = s0Σ ⊕ cΣ,

where s0Σ is semi-simple, acts irreducibly on the underlying Euclidean space Rd, and

cΣ is the center of sΣ, see [11, 12]. By the Lie algebra isomorphism dπ′, we also have

sΣ′ = s0Σ′ ⊕ cΣ′

where s0Σ′ is also semisimple and cΣ′ is the center of sΣ′ . We recall Levi’s Theorem

or the Levi decomposition e.g. [32] stating that every finite dimensional Lie algebracan be written as semidirect product of a semisimple Lie algebra and its radical(largest solvable ideal). The corresponding semisimple part is called Levi factor orLevi subalgebra.

In the following, we will show that(A0+

∑mk=1 ukAk

)z0+q = 0 for some z0 ∈ Rd,

i.e. there does exist a common fixed point z0. First, note that dπ′(cΣ′) = cΣ and hencedπ′(s0

Σ′) = s0Σ. Consider the semidirect product

g0 := s0Σ ⋊ Rd = s0

Σ′ ⊕ (0 ⋊ Rd).

By direct verification, we have 0 ⋊ Rd is an abelian ideal of g0. Since any abelianideal is trivially solvable we obtain 0 ⋊ Rd ⊆ rad(g0), where “rad” means radical.Conversely, we have dπ(rad(g0)) ⊆ rad(s0

Σ) = 0, since s0Σ is semisimple, i.e. its

radical is always zero. So dπ(rad(g0)) = 0 implies rad(g0) ⊆ 0 ⋊ Rd, and we getthe equality

0 ⋊ Rd = rad(g0).

Thus, s0Σ is indeed a Levi factor (or Levi subalgebra) of g0 and so is s0

Σ′, due tothe Lie algebra isomorphism dπ′. Now apply Levi-Malcev-Harish-Chandra Theorem


[32], stating that Levi factors are unique up to conjugation by an element in exp(Z),Z ∈ rad(g0), i.e.

s0Σ′ = exp(adZ)(s0

Σ ⋊ 0) , Z = (0, z0) ∈ 0 ⋊ Rd.

Next, we use the following notation. For any A′ = (A, a) ∈ gl(d,R) ⋊ Rd and z ∈ Rd,we write

A′ · (z) := D1α(I, z)(A′) = D1α((Id, 0), z)(A, a) = Az + a, (3.43)

where the action α is given by Eq.(3.41). Now, let E = z ∈ Rd | s0Σ′ · (z) = 0. By

conjugacy of Levi factors, for any A′ = (A, a) ∈ s0Σ′, we obtain

A′ · (z0) = exp(Z)(A, 0) exp(Z−1) · (z0)

=

[Id z00 1

] [A 00 0

] [Id −z00 1

]· (z0)

=

[A −Az00 0

]· (z0) = Az0 −Az0 = 0.

Thus, z0 ∈ E. Suppose there exists z1 6= z0 such that z1 ∈ E. Then, for anyA′ = (A, a) ∈ s0

Σ′, we have

0 = A′ · (z1) −A′ · (z0) = Az1 + a− Az0 − a = A(z1 − z0),

which impliess0Σ(z1 − z0) = 0.

It follows that there exists a one dimensional linear subspace 〈z1 − z0〉 of Rd which isinvariant under s0

Σ. This is impossible since s0Σ acts irreducibly on Rd. Hence we only

have E = z0. Note that cΣ′ ⊆ (cΣ, w) | w ∈ Rd. For any B′ = (B, b) ∈ s0Σ′ and

C ′ = (C,w) ∈ cΣ′ , we have

[B′, C ′] = [(B, b), (C,w)] = 0,

which implies Bw = Cb, as [B,C] = 0. Since B′ = (B, b) ∈ s0Σ′ , we have also

B′ · (z0) = Bz0 + b = 0,

which yields Bz0 = −b. Hence, by [B,C] = 0, we obtain

B(w + Cz0) = Bw +BCz0 = Bw + CBz0 = Cb+ C(−b) = 0,

for all B ∈ s0Σ and C ∈ cΣ. Thus, s0

Σ(w + Cz0) = 0, which implies that w = −Cz0,C ∈ cΣ, such that cΣ′ = (A,−Az0) | A ∈ cΣ. This shows

sΣ′ · (z0) = (s0Σ′ ⊕ cΣ′) · (z0) = 0,


and in particular, for A′ :=(A0 +

∑mk=1 ukAk , q

)∈ sΣ′ , we obtain

A′ · (z0) =(A0 +

m∑

k=1

ukAk

)z0 + q = 0,

which is a contradiction to the assumption that (Σi) has no common fixed points. Itfollows that ker(dπ′) 6= 0 and the proof is therefore complete.

Theorem 3.41 shows that when the system Lie algebra sΣ of the projected system(Σ) is conjugate to one of the Lie algebras on the list of Theorem 3.34 (unital case)and provided the non-existence of a common fixed point, it is verified that not onlya pure translation is contained in the system Lie algebra sΣ′ of (Σ′). But indeed, allpure translations are automatically contained in the system Lie algebra sΣ′ such thatsΣ′ = sΣ ⋊ Rd, and we have transitivity of the system group GΣ′ on Rd. On the otherhand, we note that there are solvable Lie subalgebras of gl(d,R) ⋊ Rd without anypure translation which do act transitively on Rd, see e.g.[17].

3.3.3 Some Examples

We provide some basic examples of accessibility of two-level systems and two coupledspin-1/2 systems with cross-relaxation which frequently appear in nuclear magneticresonance (NMR) applications. We also show that open quantum systems satisfyingthe sufficient condition for wh-controllability in Theorem 3.24(b) do exist for arbitraryN -level systems.

Example 3.42. (Accessibility of Two-Level Systems)

Depolarizing Channel

Consider again the depolarizing channel for two-level open quantum systems as inExample 3.26 with the GKS matrix A := a

2I3, a ≥ 0, which yields

LD =

−a 0 00 −a 00 0 −a

, LH0

=

0 −d 0d 0 00 0 0

, LH1

=

0 0 10 0 0−1 0 0

.

The system Lie algebra sΣ can be explicitely calculated to be


〉LA = so(3) ⊕ R,

such that the depolarizing channel is accessible according to Theorem 3.37.


Amplitude Damping

We consider the non-unital amplitude damping process in Example 3.27 characterizedby

LD =

−a2

0 00 −a

20

0 0 −a

, q =

00a

1√

2.

The matrix representations of the drift Hamiltonian LH0and control Hamiltonian LH1

are the same as in the depolarizing channel process. The corresponding lifted system(Σ′) evolves on the semidirect product G′ = GL(3,R) ⋊ R3. The system Lie algebraof the projected system sΣ = dπ(sΣ′) can be calculated explicitely,

sΣ = dπ(sΣ′) = dπ

(⟨[LD + LH0

q0 0

],

[LH1

00 0

]⟩

LA

)

= 〈LD + LH0, LH1

〉LA = gl(3,R),

which is one of the transitive Lie-algebras listed in Theorem 3.34 for unital case. Notethat, for all v ∈ R3, there exists u1 ∈ R such that (LD + LH0

)v + u1LH1v + q 6= 0.

Therefore, the corresponding non-unital process does not have a common fixed pointin R3 and hence, according to Theorem 3.41, the process is accessible with the systemLie algebra sΣ′ = gl(3,R) ⋊ R3.

Phase Flip

We recall the phase flip or phase damping process in Example 3.28 where

LD =

−a 0 00 −a 00 0 0

.

The matrix representation for the drift Hamiltonian LH0and control Hamiltonian

LH1are the same as in the depolarizing channel and the amplitude damping process.

Then, the unital phase flip process is accessible since the system Lie algebra is


〉LA = gl(3,R).

System in Example 3.25

We refer to the unital system in Example 3.25 where

LD + LH0=

−a −d 0d 0 00 0 −a

, LH1

=

0 0 10 0 0−1 0 0

.

This system is accessible since sΣ = 〈LD + LH0, LH1

〉LA = gl(3,R).


Example 3.43. (Two-Coupled Spin-1/2 Systems)

We recall the dynamics of two (weakly) coupled spin-1/2 systems with relax-ations frequently encountered in applications of nuclear magnetic resonance (NMR)spectroscopy [38, 39, 40, 57, 58]. The underlying Hilbert space is the tensor prod-uct C2 ⊗ C2. Two-coupled spin-1/2 systems then can be seen as two copies ofsingle two-level systems and considered as 22 = 4-level systems. Correspondingly,the density operators are represented by 4 × 4 complex matrices. In terms of theLindblad-Kossakowski master equation, an orthonormal basis Bjd

j=1 of her0(4),where d = N2 − 1 = 42 − 1 = 15, can be chosen as appropriate tensor productsof Pauli matrices. In this case we denote the Pauli matrices by

σx =1

2

[0 11 0

], σy =

1

2

[0 −ii 0

], σz =

1

2

[1 00 −1

]

and we use the orthonormal basis Bj15j=1 for her0(4) of the form

Iα = σα ⊗ I2 , α = x, y, zSα = I2 ⊗ σα , α = x, y, z

2IαSβ = 2σα ⊗ σβ , α, β = x, y, z.

The Lindblad-Kossakowski master equation captures various NMR relaxation mech-anisms via the GKS matrix

A = π

2kDD kIDD/CSA kS

DD/CSA

kIDD/CSA 2kI

CSA 0

kSDD/CSA 0 2kS

CSA

0

0 0

∈ her(15). (3.44)

We arrange the GKS matrix A = [ajk] in such a way that ajk, for j, k = 1, 2, 3, willcorrespond to the the following three basis elements

B1 = 2IzSz , B2 = Iz , B3 = Sz

in the Lindblad-Kossakowski master equation. Here, kDD represents a parameterwhich accounts for the dipole-dipole relaxation. Similarly, kI

CSA and kSCSA are rep-

resenting relaxations due to chemical shift anisotropy of spin I and S respectively,while kI

DD/CSA and kSDD/CSA are parameters representing cross-relaxations of spin I

and S, respectively. Note that to construct the Lindblad-Kossakowski master equation(2.28), we do not need to use other basis elements Bj15

j=4 due to the fact that ajk = 0for j, k > 3. With this particular type of relaxation, the Lindblad-Kossakowski masterequation (2.28) can be expressed as a double commutator form

ρ = −i[H, ρ] − πkDD[2IzSz, [2IzSz, ρ]] − πkICSA[Iz, [Iz, ρ]]

−πkSCSA[Sz, [Sz, ρ]] − πkI

DD/CSA[2IzSz, [Iz, ρ]]

−πkSDD/CSA[2IzSz, [Sz, ρ]],

(3.45)


with the Hamiltonian H = H0 +∑4

k=1 uk(t)Hk, where H0 = 2πJIzSz is the driftHamiltonian and

H1 = 2πIx , H2 = 2πSx , H3 = 2πIy, , H4 = 2πSy

are the control Hamiltonians. The real GKS matrix A indicates that the Lindblad-Kossakowski master equation is unital. Complete positivity then can be ensured bypositive semidefiniteness of A, thus imposing some inequality restrictions on the valuesof relaxation parameters, i.e.

kDD, kICSA, k

SCSA ≥ 0 , 4kDDk

ICSA ≥ (kI

DD/CSA)2 , 4kDDkSCSA ≥ (kS

DD/CSA)2

kSCSA

(4kDDk

ICSA − (kI

DD/CSA)2)≥ kI

CSA(kSDD/CSA)2.

Equation for the Vector of Coherence and Accessibility

We transform the Lindblad-Kossakowski master equation (3.45) to the vector of co-herence representation. Denote 〈Bi〉 := vi = Tr(Biρ). The 15 components of thevector of coherence v then can be arranged as

v =[〈Iz〉〈Ix〉〈Iy〉〈IySz〉〈IxSz〉〈IzSz〉〈IzSx〉〈IzSy〉

〈Sy〉〈Sx〉〈Sz〉〈IxSy〉〈IySy〉〈IxSx〉〈IySx〉]⊤.

(3.46)

Since the GKS matrix A is real, we have a unital equation for the vector of coherence

v =

(LH0

+

4∑

k=1

ukLHk

)v + LDv. (3.47)

Explicitely, we obtain the following equation

102

Chapte

r3.

Contro

llability

ofth

eLin

dbla

d-K

ossa

kow

skiM

aste

rEquatio

n

v = Gv

ddt

〈Iz〉〈Ix〉〈Iy〉

〈IySz〉〈IxSz〉〈IzSz〉〈IzSx〉〈IzSy〉〈Sy〉〈Sx〉〈Sz〉〈IxSy〉〈IySy〉〈IxSx〉〈IySx〉

= π

−u3 u1

u3 −ka −J −kc

−u1 −ka −kc JJ −kc −ka −u1 u2 −u4

−kc −J −ka u3 u2 −u4

u1 −u3 −u4 u2

u4 −k′a −J −k′c −u3 u1

−u2 −k′a −k′c J −u3 u1

J −k′c −k′a −u2

−k′c −J −k′a u4

u2 −u4

−u2 u3 −kb

−u2 −u1 −kb

u4 u3 −kb

u4 −u1 −kb

〈Iz〉〈Ix〉〈Iy〉

〈IySz〉〈IxSz〉〈IzSz〉〈IzSx〉〈IzSy〉〈Sy〉〈Sx〉〈Sz〉〈IxSy〉〈IySy〉〈IxSx〉〈IySx〉

whereka = kDD + kI

CSA , k′a = kDD + kSCSA

kb = kICSA + kS

CSA

kc = kIDD/CSA , k′c = kS

DD/CSA

Note that ka, k′a, kb on the diagonal part are representing self-relaxations of the corresponding elements of v and kc, k

′c

on the off-diagonal part are representing cross-relaxations.


For many choices of nonzero relaxation parameters ka, k′a, kb, kc, k

′c, the numerical

experiments to compute the system Lie algebra sΣ of the two-coupled spin-1/2 systemsalways end up with a Lie algebra of dimension 225. Hence, we expect that for a genericchoice of parameters, two-coupled spin-1/2 systems are accessible with the system Liealgebra gl(15,R). Indeed in Chapter 4, we will show that the accessibility of theLindblad-Kossakowski master equation is generic with the system Lie algebra equalto gl(d,R).

Example 3.44. (Generalized Depolarizing Channel, LD = γId).

In this example, we show the existence of the so-called generalized depolarizingchannel process for any arbitrary N -level open quantum systems, i.e. the existence ofa unital N -level Lindblad-Kossakowski master equation where the dissipative part isrepresented by a negative multiple of identity matrix LD = γId, γ < 0, for d = N2−1.

Indeed, such a system is exactly the case when the GKS matrix A is diagonalwith the same entries, i.e. ajj = a > 0 and ajk = 0, for j 6= k. We shall show thatthis holds for arbitrary N -level systems. When such a GKS matrix A represents theunital Lindblad-Kossakowski master equation, the matrix representation LD in thevector of coherence equation is simply

LD =

d∑

j=1

ajjLjj = a

d∑

j=1

Ljj,

where Ljj is given by Eq.(2.51). We check the main diagonal and off-diagonal elementsof LD,

1. Off-diagonal elements of LD : For any l 6= r, we obtain

d∑

j=1

Ljjlr = −

d∑

j=1

1

4

d∑

m=1

fjml(fjmr − igjmr) + fjml(fjmr + igjmr)

= −1

2

d∑

j,m=1

fjmlfjmr = −1

2

d∑

j,m=1

fljmfrjm = 0,

due to Lemma C.1 in Appendix C.3 and by renaming the indices. Thus, anyoff-diagonal elements of LD is zero.

2. Main diagonal elements of LD : For any l, we have∑d

j=1Ljjll = −1

2

∑dj,m=1 f

2ljm.

Using the same method as in the proof of Lemma C.1, we have

−d=N2−1∑

j,m

f 2ljm =

d∑

j

Tr([Bl, Bj ]2)

=∑

j

Tr(BlBjBlBj −BlBjBjBl − BjBlBlBj +BjBlBjBl)

=∑

j

Tr(2BlBjBlBj)︸︷︷︸A

−∑

j

Tr(2BjBlBlBj)︸︷︷︸B

.


We use again Lemma 2.3 to calculate part A and B. Note that any orthonormalbasis BjN2−1

j=1 of her0(N) plus BN2 = IN/√N forms an orthonormal basis of

CN×N when complex span is considered.

Part (A): Tr(2Bl

∑N2−1j BjBlBj + 2Bl

IN√NBl

IN√N

)= Tr(2BlIN Tr(Bl)︸︷︷︸

=0

)

Tr(2Bl

∑N2−1j BjBlBj

)+ 2/N Tr(B2

l )︸︷︷︸=1

= 0

∑N2−1j 2Tr(BlBjBlBj) = −2/N.

Part(B): Tr(∑N2−1

j 2BjBlBlBj + 2 IN√NBlBl

IN√N

)= Tr(2IN Tr(BlBl)︸︷︷︸

=1

)

Tr(∑N2−1

j 2BjBlBlBj

)+ 2/N Tr(BlBl)︸︷︷︸

=1

= 2N

∑N2−1j Tr(2BjBlBlBj) = 2N − 2/N.

Hence for any diagonal entry of LD, we simply have

a

d∑

j=1

Ljjll =

a

2(−2/N − (2N − 2/N)) = −aN.

This calculation shows that, for arbitrary unital N -level open quantum systems whenthe GKS matrix is a positive multiple of identity, i.e. A = aId, with a > 0 andd = N2 − 1, the matrix representation LD yields a negative multiple of identitymatrix,

LD = γId , γ = −aN.We call such a Lindblad-Kossakowski master equation with LD = γId the generalizeddepolarizing channel for arbitrary unital N -level systems. Note that in relation toTheorem 3.24(b), the generalized depolarizing channel gives an example of a wh-controllable system provided that

sD := 〈iH0, iH1, . . . , iHm〉LA = su(N),

or in the vector of coherence representation,

kD := 〈LH0, LH1

, . . . , LHm〉LA = ksu(N).

However, eventhough the generalized depolarizing channel is wh-controllable, theprocess is never accessible for N ≥ 3. This is due to the fact that the system Liealgebra is at most of the form sΣ = ksu(N)⊕R, where the dimension of ksu(N) is strictlysmaller than the dimension of so(d), su(d/2) and sp(d/4). This is of course verydifferent to the usual depolarizing channel process for the exceptional case of twolevel systems N = 2 in Example 3.42, where the system is wh-controllable and alsoaccessible with the system Lie algebra sΣ = ksu(2) ⊕ R = so(3) ⊕ R.


Example 3.45. (Accessibility of System with LD = γId + Skew)

In this example, we construct a unital system which is accessible with the systemLie algebra of the form sΣ = so(d) ⊕ R. Let the GKS matrix A satisfy the followingconstruction

• Take the diagonal elements of the GKS matrix A to be equal, i.e. ajj = a, a > 0,for all j and set the real parts of the off-diagonal elements Re(ajk) = 0, for allj 6= k.

• Choose Im(ajk), j 6= k, such that∑

j<k Im(ajk)[Bj, Bk] = 0, i.e. the processis desired to be unital despite that the GKS matrix A is no longer real. Thisis always possible for N > 2 by finding Bj and Bk such that [Bj, Bk] = 0 andsetting Im(ajk) 6= 0, whenever [Bj , Bk] = 0, see Example 2.18.

• Beside choosing Im(ajk), j 6= k, such that the process is still unital, it is alsorequired to choose Im(ajk), j 6= k, in such a way that A ≥ 0, i.e. it does notviolate complete positivity. Note that this is always possible as long as we allowIm(ajk) to be not too large compared to ajj.

By the above construction of the GKS matrix we have

LD = −1

2

N2−1∑

j

ajj[Bj , [Bj , ρ]]︸︷︷︸self-adjoint

+1

2

N2−1∑

j<k

Im(ajk) (2iBjρBk − 2iBkρBj)︸︷︷︸skew-adjoint

.

Note that in the vector of coherence representation we obtain

LD =∑

j

ajjLjj +Q = γId +Q , γ = −aN, Q ∈ so(d),

where the self-adjoint part is exactly represented by a negative multiple of identityas in Example 3.44, and the skew-adjoint part is represented by a skew-symmetricmatrix Q ∈ so(d). Then we have the system Lie algebra is of the form


, . . . , LHm〉LA

= 〈γId +Q+ LH0, LH1

, . . . , LHm〉LA

= 〈Q+ LH0, LH1

, . . . , LHm〉LA ⊕ R.

Note that LH0, LH1

, . . . , LHm∈ ksu(N), where ksu(N) is considered as a Lie subalgebra

of so(d). Now we estimate what can happen with the system Lie algebra sΣ. Theskew symmetric matrix Q ∈ so(d) can be chosen in infinitely many ways (from theconstruction) and more importantly, Q is not necessarily contained in the Lie subal-gebra ksu(N). This is true since Q is linearly independent from ksu(N), see Lemma 2.4.As a result, when the Lie bracket of the form

[Q+ LH0, LHj

] , j = 1, 2, . . . , m,


and the corresponding higher bracket are computed, they generate new elements inso(d) which are not necessarily contained in the subalgebra ksu(N). Hence in thiscase, the skew-symmetric part Q plays role in enlarging the system Lie algebra out ofksu(N). Indeed, numerical experiments for low dimensional systems N = 3 and N = 4show that the system Lie algebras are eventually reaching so(8) ⊕ R and so(15)⊕ R,respectively, for many choices of parameters. This shows the possibility of accessiblesystem with the system Lie algebra of the form so(d) ⊕ R.

Chapter 4

Genericity of Accessibility and anAlgorithm for DeterminingAccessibility

4.1 Genericity Problem

Concerning the accessibility question of a bilinear control system on Rd \ 0, onehas to calculate the system Lie algebra generated by matrices representing the driftand control terms, according to Theorem 3.10. In relation with this fact, we recalltwo important results dealing with the generation of semisimple and reductive Liealgebras. Lemma 4.1, which is due to Kuranishi, and Lemma 4.2 can be found in[11, 32].

Lemma 4.1. (Kuranishi). Let g0 be a real semisimple Lie algebra. Then thereexist two elements A,B ∈ g0 such that 〈A,B〉LA = g0.

Lemma 4.2. Let g be a reductive Lie algebra which can be decomposed as g = g0⊕c,where g0 is a semisimple Lie algebra and c is the center of g with dim(c) ≤ k. If g0 isgenerated by k elements then g is also generated by k elements.

Now consider a bilinear control system on Rd \ 0,

(Σi) x =

(A0 +

m∑

k=1

uk(t)Ak

)x , A0, Ak ∈ gl(d,R).

The following accessibility result on Rd \ 0 is well-known.

Theorem 4.3. Consider the system (Σi) with m = 1, i.e. with only one singlecontrol. Then, there exists a pair (A0, A1) ∈ gl(d,R) × gl(d,R) such that the system(Σi) is accessible.

107

108 Chapter 4. Genericity of Accessibility and an Algorithm for Determining Accessibility

Proof. The system (Σi) is accessible if and only if the system Lie algebra sΣ =〈A0, A1〉LA is conjugate to one of the transitive Lie algebras on Rd \ 0 (see thelist in Appendix B). Note that all of the transitive Lie algebras on Rd \ 0 are of theform g = g0 ⊕ c where g0 is semisimple and c is the center of g with dim(c) ≤ 2. ByLemma 4.1, for a semisimple Lie algebra g0, there exist two elements A,B ∈ g0 suchthat they generate g0. The result follows from Lemma 4.2.

Theorem 4.3 reveals an important fact that no matter how large the dimensiond is, the system (Σi) is accessible using a single control, provided that a proper choiceof (A0, A1) is given. Suppose now we have a bilinear control system (Σi) on Rd, whereA0, . . . , Ak are restricted to be elements in sl(d,R), which is a semisimple Lie algebra.The following theorem is due to Jurdjevic and Kupka, see [33], Chapter 6.4. The proofof Theorem 4.4 mainly relies on the fact that sl(d,R) is a semisimple Lie algebra.

Theorem 4.4. (Genericity of Accessibility [33]). The set of (A0, A1) ∈sl(d,R) × sl(d,R) such that

〈A0, A1〉LA = sl(d,R)

is open and dense in sl(d,R)× sl(d,R). Therefore, almost all bilinear control systems(Σi) with m = 1 are accessible i.e., the system (Σi) is generically accessible even ifonly a single control is available.

A similar situation arises when we consider open quantum systems modelled bythe Lindblad-Kossakowski master equation,

(Σi) v =

(A0 +

m∑

k=1

uk(t)Ak

)v + q.

For the unital case where q = 0, we have precisely a bilinear control system on Rd

which additionally leaves Φ(P) ⊆ Bd invariant. Therefore, motivated by the resultof Theorem 4.4, it is meaningful to ask whether the unital Lindblad-Kossakowskimaster equation is generically accessible. The same question can be posed as wellfor the non-unital case where q 6= 0. However, note that in the context of theLindblad-Kossakowski master equation, the matrix representation A0 is restrictedto be within some subset of gl(d,R), while the matrices A1, . . . , Am are restricted tobe in ksu(N) ⊆ so(d). Hence, to obtain a similar genericity result for the Lindblad-Kossakowski master equation, the proof of Theorem 4.4, which basically uses thesemi-simplicity of sl(d,R) and the full flexibility to choose the matrices A0, A1, . . . , Am

within sl(d,R), has to be modified.

4.2. Genericity Results for the Lindblad-Kossakowski Master Equation 109

4.2 Genericity Results for the Lindblad-Kossakowski

Master Equation

The main goal is to prove that accessibility of the Lindblad-Kossakowski master equa-tion is a generic property. Similar statements dealing with the genericity of accessi-bility also appeared in a series of papers by C.Altafini [6, 7, 8]. However, a rigorousproof is missing and only a sketchy proof referring to the work of Jurdjevic [33] isavailable. Note that the proof in [33] has to be considerably refined due to the factthat some of the matrix representations in the Lindblad-Kossakowski master equa-tion are restricted to a proper subalgebra of gl(d,R). To start with, we state severalauxiliary results which will be used later on to prove the genericity results for theLindblad-Kossakowski master equation.

Proposition 4.5. The set of (Ω, D) ∈ ksu(N) × gl(d,R) such that the correspondinggenerated Lie algebra

s := 〈Ω, D〉LA

equals to gl(d,R) is open and dense in ksu(N) × gl(d,R).

Before we proceed to the proof, we consider the following easy fact.

Lemma 4.6. Let (Ω, D) ∈ ksu(N) × gl(d,R) and L := D − Tr(D)dId ∈ sl(d,R). Then,

〈Ω, L〉LA = sl(d,R) is equivalent to 〈Ω, D〉LA = gl(d,R).

Proof. Note that 〈Ω, D〉LA = 〈Ω, L〉LA ⊕ R. The result follows since gl(d,R) =sl(d,R) ⊕ R.

Therefore, the problem in Proposition 4.5 is equivalent to showing that the set ofall (Ω, L) ∈ ksu(N)×sl(d,R) such that the corresponding Lie algebra 〈Ω, L〉LA equals tosl(d,R) is open and dense in ksu(N)×sl(d,R). In this way, we can adapt or modify theproof in [33] using the semi-simplicity of sl(d,R) plus an additional idea coming froma standard controllability result in linear system theory. In fact, as is well-known, theset

A := (A, b) ∈ gl(n,R) × Rn | span〈b, Ab, . . . , An−1b〉 = Rnis open and dense in gl(n,R)×Rn. The idea is as follows. Define Φ : gl(n,R)×Rn → R,with

Φ(A, b) := det([b Ab . . . An−1b]).

Then, due to the fact that Φ is real analytic (in this case it is a polynomial function)and not identically zero, the set

N := (A, b) ∈ gl(n,R) × Rn | Φ(A, b) = 0is closed and nowhere dense. Therefore, the set

A = gl(n,R) × Rn \ N


is open and dense in gl(n,R) × Rn. Moreover, it is non-empty by inspection of anexample of a controllable system. In our case for the Lindblad-Kossakowski masterequation, we will use this idea by replacing A and b with adL and Ω, respectively.

In order to adapt the proof in [33, 34], we clarify some basic ideas dealing withstrongly regular elements in a real semisimple Lie-algebra. Let g be a real semisimpleLie algebra and gC = g⊕ ig be the complexification of g. Consider the correspondingadjoint representations

ad : g → End(g) , adC : gC → End(gC).

Now consider L ∈ g and let

Sp(L) :=a ∈ C

∣∣ a 6= 0, ker(adC

L − aI) 6= 0,

where I is the identity operator. Define

EC

a (L) := ker(adC

L − aI) , a ∈ Sp(L)

as the a-eigenspace of adC

L. An element L ∈ g is called strongly regular if it satisfiesthe following conditions

(i) All nonzero eigenvalues of adC

L are simple, i.e. for each a ∈ Sp(L) the corre-sponding eigenspace EC

a (L) is a one-dimensional complex subspace of gC.

(ii) The generalized kernel

EC

0 (L) :=⋃

n≥1

ker((adC

L)n)

does not contain any non-trivial ideal of gC.

It is known that the set of all strongly regular elements is generic in g, i.e. it is openand dense in g. Then we have the following important properties (see e.g. [34]).

1. With respect to a strongly regular element L ∈ g, the complex Lie algebra gC

decomposes as a direct sum

gC = EC

0 (L) ⊕⊕

a∈Sp(L)

EC

a (L).

2. For every a ∈ Sp(L), the set [EC

a (L), EC

−a(L)] is a one-dimensional vector spacecontained in EC

0 (L). The sum of all [EC

a (L), EC

−a(L)], a ∈ Sp(L), equals EC

0 (L)as a vector space, i.e.

∑

a∈Sp(L)

[EC

a (L), EC

−a(L)] = EC

0 (L).


3. For any a, b ∈ Sp(L) such that a+ b ∈ Sp(L),

[EC

a (L), EC

b (L)] = EC

a+b(L).

4. It turns out that EC

0 (L) = ker(adC

L) and moreover, EC

0 (L) is a Cartan subalgebraof gC (see Definition A.14 and Proposition A.15 in Appendix A).

5. With respect to a strongly regular element L ∈ g, the real Lie algebra g decom-poses as a direct sum

g = E0(L) ⊕⊕

a∈Sp(L), Im(a)≥0

Ea(L),

where E0(L) := EC

0 (L)∩g and Ea(L) := (EC

a (L)⊕EC

a (L))∩g is the corresponding“real eigenspace” of EC

a (L). Note that Ea(L) = Ea(L), where a denotes thecomplex conjugate of a. Thus, any A ∈ g has a unique decomposition

A = A0 +∑


Aa,

where A0 ∈ E0(L) and Aa ∈ Ea(L) for a ∈ Sp(L), Im(a) ≥ 0.

Lemma 4.7. Let L ∈ g be a strongly regular element. Then

E0(L) ⊆∑


[Ea(L), E−a(L)].

Proof. Consider a ∈ Sp(L), where a is complex. Then −a, a,−a ∈ Sp(L) and eachEC

a (L), EC

−a(L), EC

a (L) and EC

−a(L) is a one-dimensional complex vector space. Weclaim that

([EC

a (L), EC

−a(L)] + [EC

a (L), EC

−a(L)])∩ g ⊆ [Ea(L), E−a(L)]. (4.1)

We recall that each [EC

a (L), EC

−a(L)] and [EC

a (L), EC

−a(L)] is a one-dimensional complexvector space in EC

0 (L), cf. Property 2 above. We occasionally use the notations 〈 · 〉C

and 〈 · 〉R to denote the complex and real span, respectively. Suppose

EC

a (L) = 〈X + iY 〉C , X, Y ∈ g

EC−a(L) = 〈V + iW 〉C , V,W ∈ g.

(4.2)

Then, it is clear that ECa (L) = 〈X − iY 〉C and EC

−a(L) = 〈V − iW 〉C. Calculating theleft hand side of Eq.(4.1), we obtain

([EC

a (L), EC−a(L)] + [EC

a (L), EC−a(L)]

)∩ g = 〈A+ iB,A− iB〉C ∩ g

= 〈A,B〉C ∩ g = 〈A,B〉R,(4.3)


with A := [X, V ] − [Y,W ] ∈ g and B := [X,W ] + [Y, V ] ∈ g. On the other hand, wehave

[Ea(L), E−a(L)] =[(EC

a (L) ⊕EC

a (L)) ∩ g , (EC

−a(L) ⊕ EC

−a(L)) ∩ g]

= [〈X, Y 〉R, 〈V,W 〉R]

= 〈[X, V ], [X,W ], [Y, V ], [Y,W ]〉R.

(4.4)

Now comparing Eq.(4.3) and Eq(4.4) we obviously have

〈A,B〉R ⊆ 〈[X, V ], [X,W ], [Y, V ], [Y,W ]〉R,

and Eq.(4.1) follows. Now, summing up the left and right hand sides of Eq.(4.1) forall a ∈ Sp(L), Im(a) ≥ 0 (including the real eigenvalues) we obtain

∑

a

[Ea(L), E−a(L)] ⊇∑

a

(g ∩

([EC

a (L), EC

−a(L)] + [EC

a (L), EC

−a(L)]))

= g ∩∑

a

([EC

a (L), EC

−a(L)] + [EC

a (L), EC

−a(L)])

= g ∩( ∑

a∈Sp(L)

[EC

a (L), EC

−a(L)])

= g ∩EC

0 (L) =: E0(L).

Note that interchanging the summation and intersection in the second line of abovecalculation is allowed since, for arbitrary C,D ∈ g, one has

(〈C〉C + 〈D〉C

)∩ g = 〈C,D〉C ∩ g = 〈C,D〉R = 〈C〉R + 〈D〉R

=(〈C〉C ∩ g

)+(〈D〉C ∩ g

).

Hence, we conclude the desired inclusion.

Proposition 4.8. Let L be a strongly regular element in a real semisimple Liealgebra g. Choose X ∈ g such that X =

∑aXa, where Xa ∈ Ea(L) is non-zero for

all a ∈ Sp(L) and Im(a) ≥ 0. Then we have 〈X,L〉LA = g

Proof. The inclusion 〈X,L〉LA ⊆ g is clear. Now, each Ea(L) is invariant under adL.Then we have

span〈adLX, . . . , adkLX〉 =

⊕


Ea(L) ⊂ g, (4.5)

for some k ∈ N, since all Ea(L) are irreducible subspaces of adL and Xa 6= 0 byassumption. By Lemma 4.7, summing [Ea(L), E−a(L)] for all a ∈ Sp(L), Im(a) ≥ 0,we eventually generate E0(L). Therefore we obtain

〈X,L〉LA ⊇ E0(L) ⊕⊕


Ea(L) = g,

and the result follows.


Lemma 4.9. Let Ω ∈ ksu(N) ⊆ so(d) be non-zero. Then there exists θ ∈ SO(d) suchthat (θΩθ⊤)ij 6= 0 for all 1 ≤ i 6= j ≤ d.

Proof. Fix i, j and i 6= j. The map θ → (θΩθ⊤)ij is real analytic and not identicallyzero (this can be seen by an appropriate Givens rotations) such that the set Nij :=θ ∈ SO(d) | (θΩθ⊤)ij = 0 is closed and nowhere dense in SO(d). Then, the set

⋃

1≤i6=j≤d

Nij =: N

is also closed and nowhere dense in SO(d). This implies that the set

SO(d) \ N = θ ∈ SO(d) | (θΩθ⊤)ij 6= 0, for all 1 ≤ i 6= j ≤ d

is open and dense in SO(d), and particularly, non-empty.

Now we are in the position to prove Proposition 4.5.

Proof. (Proposition 4.5). For Ω ∈ ksu(N) and L ∈ sl(d,R), we define the following realvector space

V (Ω, L) := span〈adLΩ, ad2LΩ, . . . , adk

LΩ〉 , k := d2 − d.

We show that the following set

A :=(Ω, L) ∈ ksu(N) × sl(d,R) | dim(V ) = d2 − d

is open and dense in ksu(N) × sl(d,R). First, construct the following matrix

M(Ω, L) :=[Lω|L2ω| . . . |Lkω

]∈ Rd2×(d2−d) , k = d2 − d,

whereL := Id ⊗ L− L⊤ ⊗ Id , ω := Vec(Ω).

Define the map

Φ : Rd2×(d2−d) −→ RD , D :=

(d2

d2 − d

),

where(Φi(X)

)i=1,...,D

consists of the determinants of all possible (d2 − d) × (d2 − d)

square matrices constructed by deleting out d rows of the matrix X ∈ Rd2×(d2−d). Ifthe map (Ω, L) → Φ

(M(Ω, L)

)is real analytic and not identically zero, then it follows

that the setN :=

(Ω, L) ∈ ksu(N) × sl(d,R) | Φ

(M(Ω, L)

)= 0

is closed and nowhere dense in ksu(N) × sl(d,R), and thus, the set

A = (ksu(N) × sl(d,R)) \ N


is open and dense in ksu(N) × sl(d,R). Indeed, Φ is real analytic, i.e. it is a polynomialfunction. To show that it is not identically zero, it is enough to find a pair (Ω, L) suchthat Φ(M(Ω, L)) 6= 0, which is done as follows.

Without loss of generality, cf. Lemma 4.9, we can choose Ω ∈ ksu(N) with all off-diagonal elements being non-zero. We choose L ∈ sl(d,R) diagonal where all possibledifferences of the diagonal elements are distinct. It is easy to check that such L isa strongly regular element in sl(d,R). With respect to the strongly regular L, wedecompose

sl(d,R) = E0(L) ⊕⊕


Ea(L)

whereE0(L) = Diag(λ1, . . . , λd) | λj ∈ R, λ1 + . . .+ λd = 0

andEa(L) = span〈Eij , 1 ≤ i 6= j ≤ d〉,

with Eij := eie⊤j . Then, for all such Ω with nonzero off-diagonal elements, by Eq.(4.5)

in the proof of Proposition 4.8, we obtain

V (Ω, L) = span〈adLΩ, ad2LΩ, . . . , adk

LΩ〉 = span〈Eij , 1 ≤ i 6= j ≤ d〉,

with dim(V (Ω, L)

)= d2 − d such that Φ(M(Ω, L)) 6= 0. Thus, we are done to

show that Φ is not identically zero and therefore, the set A is open and dense inksu(N) × sl(d,R).

Now, we define the following set

Ar :=(Ω, L) ∈ A | L strongly regular in sl(d,R)

.

Since the set of all strongly regular element L ∈ sl(d,R) is open and dense in sl(d,R),cf. [33, 34], obviously the set Ar is also open and dense in ksu(N) × sl(d,R). Moreover,due to the fact that dim

(V (Ω, L)

)= d2 − d for every (Ω, L) ∈ Ar, we have the

corresponding real vector space

V (Ω, L) =⊕


Ea(L).

By applying Lemma 4.7, we obtain

〈Ω, L〉LA ⊇ E0(L) ⊕⊕


Ea(L) = sl(d,R)

for all (Ω, L) ∈ Ar. Therefore, the set of pair (Ω, L) ∈ ksu(N) × sl(d,R) such that

〈Ω, L〉LA = sl(d,R)

is open and dense. By Lemma 4.6, we conclude Proposition 4.5.


We are in the position to state the genericity results for accessibility of theLindblad-Kossakowski master equation, both for the unital and non-unital case. Re-garding the term “genericity” in our results, note that we call a set generic if eitherit is open and dense or it contains an open and dense subset. Recall again that, forthe unital Lindblad-Kossakowski master equation, accessibility is meant with respectto Rd \ 0 while for the non-unital case, it is meant with respect to Rd.

Theorem 4.10. (Main Result 4 : Genericity of Accessibility of Open Quan-tum Systems, Unital Case). The unital Lindblad-Kossakowski master equation

(Σi) v =(A0 +

m∑

k=1

uk(t)Ak

)v , v ∈ Rd,

where A0 := LD + LH0∈ Lu ⊂ gl(d,R), Ak := LHk

∈ ksu(N), is generically accessible.More precisely, let m ∈ N be arbitrary. Then, the set of all (A0, A1, . . . , Am) such that(Σi) is accessible is open and dense in Lu × ksu(N) × . . .× ksu(N).

Proof. Obviously, it is enough to prove the above statement for m = 1, since acces-sibility for m = 1 implies accessibility for arbitrary m ∈ N. By applying Proposition4.5, the set of all (A0, A1) ∈ gl(d,R) × ksu(N) such that

〈A0, A1〉LA = gl(d,R)

is open and dense in gl(d,R) × ksu(N). However, the drift term A0 = LD + LH0is

restricted to the set Lu (and can not be the whole gl(d,R)) in order to represent a validLindblad-Kossakowski generator L that satisfies complete positivity requirement. Werecall Proposition 2.24 (and correspondingly Remark 2.30), the set Lu of all admissibleunital Lindblad-Kossakowski generators in its matrix representation is a closed convexcone of gl(d,R) with non-empty interior. Now, the intersection between an open anddense subset of gl(d,R) and Lu yields an open and dense subset in Lu. Therefore, theset of pairs (A0, A1) ∈ Lu × ksu(N) such that


is open and dense in Lu × ksu(N). This implies that the system (Σi) is genericallyaccessible.

For the non-unital case, we first consider the set Ln ⊂ gl(d,R) of all admissibleLindblad-Kossakowski generators in its matrix representation. Any element A0 ∈ Ln

can be written as A0 = LD + LH0, where LD is linearly independent from LH0

, with

LD =d∑

j,k=1

ajkLjk , [ajk] ≥ 0,


and Ljk ∈ gl(d,R) are the fixed matrices defined in Eq.(2.51). We also recall Eq.(2.55)for the translation part of the non-unital Lindblad-Kossakowski master equation

q = − 2

N

d∑

j<k

Im(ajk)[fjk1 fjk2 . . . fjkd]⊤ ∈ Rd,

where fjkl are the structure constants of su(N). Since the GKS matrix A = [ajk]is uniquely defined by a given A0 ∈ Ln, the translation q can be regarded as q(A0),where one can check that the dependence on A0 is linear. Before we state the resultfor the non-unital case, we need the following results.

Lemma 4.11. Let Ln ⊂ gl(d,R) be the set of admissible non-unital Lindblad-Kossakowski generators in its matrix representation. Define the set

Lin := GL(d,R) ∩ Ln,

andB :=

(A0, A1) ∈ Lin × ksu(N) | A1A

−10 q(A0) 6= 0

,

where q : Ln → Rd represents the translation part of the non-unital Lindblad-Kossakowskimaster equation. Then, the set B is open and dense in Ln × ksu(N).

Proof. The set Lin is open and dense in Ln, since Ln has non-empty interior in gl(d,R)(cf. Proposition 2.24 and correspondingly Remark 2.30) and GL(d,R) is open anddense in gl(d,R). Therefore, it is enough to show that B is open and dense inLin × ksu(N) as follows.

The function f(A0, A1) := A1A−10 q(A0) is real analytic since q depends linearly

on A0. We show that f is not identically zero by choosing a pair (A0, A1) ∈ Lin×ksu(N)

such that f(A0, A1) 6= 0. We choose A0 ∈ Lin where q(A0) 6= 0. Note that this ispossible since we are in the non-unital case. Then, we fix a non-zero vector x :=A−1

0 q(A0) ∈ Rd and define the function

g : ksu(N) → Rd , g(A) := Ax.

We claim that ker(g) is not the whole ksu(N). Indeed, suppose ker(g) = ksu(N). Thenksu(N)x = 0 and consequently, there exists a one-dimensional linear subspace Rx whichis invariant under ksu(N). This is a contradiction since ksu(N) is irreducible accordingto Lemma 3.22. Therefore, ker(g)⊥ is non-trivial. Now, we can choose a non-zeroA1 ∈ ker(g)⊥. This particular choice of A0 and A1 yields f(A0, A1) = A1A

−10 q(A0) =

A1x = g(A1) 6= 0 and we are done.

Lemma 4.12. Let Ln ⊂ gl(d,R) be the set of admissible non-unital Lindblad-Kossakowski generators in its matrix representation. Define the set

A :=

(A0, A1) ∈ Ln × ksu(N)

∣∣∣∣∣for all v ∈ Rd, there exists u ∈ R

such that (A0 + uA1)v + q(A0) 6= 0

,


where q : Ln → Rd represents the translation part of the non-unital Lindblad-Kossakowskimaster equation. Then, the set A contains an open and dense subset of Ln × ksu(N).

Proof. It follows that the set A can be equivalently expressed as

A =(A0, A1) ∈ Ln × ksu(N)

∣∣ for all v ∈ Rd : A0v + q(A0) 6= 0 or A1v 6= 0.

We show that A ⊃ B, where the set B is as defined in Lemma 4.11 and we are done.Take any element (A0, A1) ∈ B and define a non-zero vector x := −A−1

0 q(A0) ∈ Rd.Any v ∈ Rd such that v 6= x can be written as

v = −A−10 q(A0) + ∆ , ∆ 6= 0.

For any v 6= x, one has

A0v + q(A0) = A0(−A−10 q(A0) + ∆) + q(A0) = A0∆ 6= 0,

since A0 is invertible. For v = x,

A1v = −A1A−10 q(A0) 6= 0,

since (A0, A1) ∈ B. Therefore, for all v ∈ Rd, either A0v + q(A0) 6= 0 or A1v 6= 0 issatisfied and hence (A0, A1) ∈ A. Then B ⊂ A and the result follows.

Theorem 4.13. (Main Result 5 : Genericity of Accessibility of OpenQuantum Systems, Non-Unital Case). The non-unital Lindblad-Kossakowskimaster equation

(Σi) v =(A0 +

m∑

k=1

uk(t)Ak

)v + q(A0) , v ∈ Rd,

where A0 := LD + LH0∈ Ln ⊂ gl(d,R), Ak := LHk

∈ ksu(N), q(A0) 6= 0, isgenerically accessible. More precisely, let m ∈ N be arbitrary. Then, the set ofall (A0, A1, . . . , Am) such that (Σi) is accessible contains an open and dense subset ofLn × ksu(N) × . . .× ksu(N).

Proof. Again, it is enough to prove the statement only for m = 1. For the non-unitalcase, a sufficient condition for accessibility (cf. Theorem 3.41) is


and the non-existence of a common fixed point, i.e. for all v ∈ Rd, there exists u ∈ R

such that

(A0 + uA1) v + q(A0) 6= 0.


Repeating the proof of Theorem 4.10, the set of all (A0, A1) ∈ gl(d,R) × ksu(N) suchthat


is open and dense in gl(d,R) × ksu(N). By Proposition 2.24 (and correspondinglyRemark 2.30), the set Ln of all admissible non-unital Lindblad-Kossakowski generatorsin its matrix representation is a closed convex cone of gl(d,R) with non-empty interior.Hence, we obtain the set

D :=(A0, A1) ∈ Ln × ksu(N) | 〈A0, A1〉LA = gl(d,R)

is open and dense in Ln × ksu(N).

Now for the non-existence of a common fixed point. The set A in Lemma 4.12exactly defines all pairs (A0, A1) ∈ Ln × ksu(N) such that there exists no common fixedpoint. Since A contains an open and dense subset of Ln×ksu(N), the intersection A∩D

also contains an open and dense subset of Ln × ksu(N). Therefore, we have verifiedthat the system (Σi) is generically accessible.

4.3 Algorithm for Determining Accessibility

In Section 3.3, we have exploited the list of transitive Lie algebras for determiningaccessibility of the unital Lindblad-Kossakowski master equation. We emphazise againhere that the term “transitive Lie algebra” is meant to be the Lie algebra of thecorresponding Lie group which acts transitively on Rd \ 0. For the case of N -levelsystems where N is even (including the case of n-coupled spin 1/2 systems), theaccessibility characterization is very concise in the sense that only two possible Liealgebras are left. However, for the case when N is odd, many more Lie algebras arepossible. Thus, one faces the following practical task : given the fixed set of matricesA0, A1, . . . , Am ∈ gl(d,R) in the unital Lindblad-Kossakowski master equation, decidewhether the system Lie algebra sΣ is conjugate to one of the transitive Lie algebrasin Theorem 3.34, i.e. decide whether the system is accessible or not, and determinethe corresponding type of transitive Lie algebra according when the system is indeedaccessible.

In principle, it is possible to give an algorithm such that, with the finite numberof iterations, a conclusive answer on accessibility can be obtained, i.e. either theunital Lindblad-Kossakowski master equation is not accessible, or it is accessible andsΣ is conjugate or equal to a certain type of Lie algebras listed in Table 4.1. Wedescribe such algorithm adapted from [12] with some minor modifications. For thecase where N is even or particularly for n-coupled spin 1/2 systems, the steps ofthe algorithm can be reduced significantly due to our theoretical results. We alsogive some further clarification and explanation on parts of the algorithm [12] whichcontain some subtleties.

4.3. Algorithm for Determining Accessibility 119

We start by recalling some facts on transitive Lie algebras g ⊆ gl(d,R), cf.[11, 12]. Note that transitivity holds up to conjugation, i.e. any g′ = AgA−1, forA ∈ GL(d,R), is transitive whenever g is transitive. Hence, the list of transitive Liealgebras only displays one representative for each equivalence classes. Any transitiveLie algebra is represented by g = g0 ⊕ c where g0 is semisimple, acts irreducibly onRd and c is the center of g,

c := X ∈ g | [X, Y ] = 0, for all Y ∈ g.We have z0 and z, the centralizer of g0 and g in gl(d,R), respectively,

z0 := X ∈ gl(d,R) | [X, Y ] = 0, for all Y ∈ g0z := X ∈ gl(d,R) | [X, Y ] = 0, for all Y ∈ g.

Since we know that g0 acts irreducibly on Rd, the centralizer z0 must be isomorphic(as a field) to either R, C or H [12]. It is not too difficult to see that c ⊆ z and z ⊆ z0

such that c ⊆ z0. Since the center c is abelian, it has the real dimension ǫ ≤ 2. If z0

is abelian, we have z0 ⊆ z such that z0 = z ∼= R or z0 = z ∼= C. In the Table 4.1, wereproduce the possible transitive Lie algebras for the Lindblad-Kossakowski masterequation according to Theorem 3.34.

Remark 4.14. All Lie algebras appearing on the list of Table 4.1 are regarded withrespect to their representations as subalgebras of gl(d,R). For the corresponding formof the representations, we refer to [11, 12].

Algorithm 4.15. (Preliminary Test). Suppose we are given a set of fixed ma-trices A0, A1, . . . , Am ∈ gl(d,R), for some fixed d, representing the unital Lindblad-Kossakowski master equation.

1. Find a basis b1, b2, . . . , bD ∈ gl(d,R) of the system Lie algebra sΣ generated byA0, A1, . . . , Am.

2. (Dimension test). Check the dimension index (d,D) if it matches one (or pos-sibly more) of the dimension indices in Table 4.1. If there does not exist such(d,D) in the table, sΣ can not be transitive and the system is not accessible. Ifthere do exist such (d,D), go to step 3.

3. (Necessary open orbit test). Take any non-zero v ∈ Rd and compute a basis of

lv := X ∈ sΣ | Xv = 0.If D − dim(lv) 6= d, then sΣ is not transitive. Otherwise, go to Step 4.

4. (Necessary centralizer test). Determine a basis of the centralizer of sΣ,

z′ =X ∈ Rd×d | [X, bi] = 0 , i = 1, . . . , D

,

by solving the corresponding set of linear equations. If z′ is isomorphic (as afield) to either R, C or H, go to step 5. Otherwise sΣ is not transitive.


Type g = g0 ⊕ c D := dim(g) z0 z

1. so(d) ⊕ R (1/2)d(d− 1) + 1 R R

2. su(d/2) ⊕ eiαR, su(d/2) ⊕ C (d/2)2 − 1 + ǫ, (ǫ = 1, 2) C C

A 3. sp(d/4) ⊕ eiαR, sp(d/4) ⊕ C 2(d/4)2 + (d/4) + ǫ, (ǫ = 1, 2) H ⊆ H

4. sp(d/4) ⊕ H∼= sp(d/4) ⊕ su(2) ⊕ R 2(d/4)2 + (d/4) + 4 R R

5. spin(7) ⊕ R∼= so(7) ⊕ R (d = 8) 22 R R

1. gl(d,R) = sl(d,R) ⊕ R d2 R R

2. sl(d/2,C) ⊕ eiβR, 2((d/2)2 − 1) + ǫ, (ǫ = 1, 2) C C

gl(d/2,C) = sl(d/2,C) ⊕ C

B 3. sl(d/4,H) ⊕ eiβR, 4(d/4)2 − 1 + ǫ, (ǫ = 1, 2) H ⊆ H

sl(d/4,H) ⊕ C

4. sp(d/2,R) ⊕ R 2(d/2)2 + (d/2) + 1 R R

5. sp(d/4,C) ⊕ eiβR, 4(d/4)2 + 2(d/4) + ǫ, C C

sp(d/4,C) ⊕ C (ǫ = 1, 2)C sl(d/4,H) ⊕ H

∼= sl(d/4,H) ⊕ su(2) ⊕ R 4(d/4)2 + 3 R R

Table 4.1: Transitive Lie algebras for the Lindblad-Kossakowski master equation.Note that α ∈ (−π

2, π

2), β ∈ [−π

2, π

2] and D is dimension of g considered as real vector

space.


5. Based on the matching dimension index (d,D) in the Table 4.1, go to the corre-sponding Section 4.3.1, 4.3.2 or/and 4.3.3 according to whether the dimensionindex is of Type A, B or/and C with respect to Table 4.1 and execute thecorresponding subalgorithms there.

A basic algorithm constructing a basis of a Lie algebra, for example, is given byAlgorithm D.1 in Appendix D. Likewise, we also refer to Algorithm D.3 in AppendixD for deciding whether a particular real subspace is isomorphic (as a field) to eitherR, C or H. For the necessary open orbit test, note that D − dim(lv) is equal to thedimension of the subspace Xv | X ∈ sΣ, which coincides as well to the dimensionof the orbit Gv. So, if D − dim(lv) 6= d, the orbit Gv is not open and sΣ can not betransitive. The centralizer check is meant to fulfill the necessary condition that thecentralizer z of any transitive algebra g must be isomorphic (as a field) to either R, C

or H.

Remark 4.16. For the specific case where the size N of the quantum system at handis even, only the algorithms for Type A and B1 are relevant, due to our accessibilityresults in Theorem 3.34 and 3.36. Particularly, this applies to all n-coupled spin 1/2systems.

4.3.1 Type A

We recall the important fact that any transitive Lie algebra of Type A (see Table4.1) decomposes as g = g0 ⊕ c where g0 is compact semisimple and the center c of g

has the dimension ǫ ≤ 2. According to the Cartan criterion for compact semisimpleLie algebras (cf. Theorem A.9, Appendix A), g0 has a negative definite Killing form.Now, since the Killing form is Ad-invariant (see Eq.(A.5a)), we can use this Cartancriterion to decide further whether sΣ obtained in Algorithm 4.15 is conjugate to aLie algebra of Type A with the same dimension index.

Algorithm 4.17. (Type A). Suppose a basis b1, b2, . . . , bD ∈ gl(d,R) of sΣ isalready obtained.

1. With respect to the above basis, calculate the matrix representation of eachadbi

, i = 1, . . . , D.

2. Calculate the corresponding Killing form κ represented by a real symmetricmatrix K of size D whose elements are

Kij := κ(bi, bj) := Tr(adbiadbj

).

Diagonalize this Killing form such that the diagonalized version K has elementsKij = 0 for i 6= j.


3. Let ǫ′ be the number of zero appearing on the diagonal of K. The algebra sΣ

is transitive and conjugate to a Lie algebra of Type A with the correspondingdimension index if and only if ǫ′ = 1 or 2, and the rest diagonal elements of Kare all strictly negative.

An algorithm with a finite number of steps, which converts a basis bi to ci,i = 1, . . . , D such that the corresponding Killing form Kij = κ(ci, cj) is diagonal withrespect to this basis, is summarized in Algorithm D.2, Appendix D. Note that we usethe fact that the so-called signature of a real symmetric bilinear form (e.g. Killingform) is invariant under any choice of basis. In Algorithm 4.17, the dimension of thecenter of sΣ is identified by ǫ′. Using the fact from the necessary open orbit test inAlgorithm 4.15, one can conclude with Algorithm 4.17 that sΣ is transitive of TypeA, see [12].

4.3.2 Type B

One of the important features of the transitive Lie algebras of Type B is that theyare decomposed into g = g0 ⊕ c, where g0 is actually simple (not only semisimple) butnon-compact.

Type B.1 : gl(d,R)

This is the easiest case since gl(d,R) is the most we can get. Therefore, a dimensioncheck suffices and this has been done in the preliminary test of Algorithm 4.15. More-over, due to the genericity result in the previous section, one can expect that gl(d,R)is indeed generated for almost all cases.

Algorithm 4.18. (Type B.1). From an obtained basis b1, b2, . . . , bD ∈ gl(d,R)of sΣ, the system Lie algebra sΣ is transitive and equal to gl(d,R) if and only ifD = d2.

Type B.2 : sl(d/2,C) ⊕ eiβR and sl(d/2,C)⊕ C

When the dimension index test sends sΣ to Type B.2 in Table 4.1, then transitivity canbe decided immediately by the centralizer test, which has been done in the preliminarytest of Algorithm 4.15. When the centralizer of sΣ is isomorphic (as a field) to C,it implies that sΣ is a subalgebra of Agl(d/2,C)A−1 (or equivalently, A−1sΣA is asubalgebra of gl(d/2,C)), for A ∈ GL(d,R). Then sΣ must contain Asl(d/2,C)A−1

since, for d/2 > 2, any subalgebra of gl(d/2,C) having dimension equal or greaterthan that of sl(d/2,C) must contain sl(d/2,C) [12]. By the result from the dimensionindex test, it follows that sΣ is transitive of Type B.2.


Algorithm 4.19. (Type B.2). The algebra sΣ is transitive and of Type B.2 if andonly if the centralizer z′ of sΣ is isomorphic (as a field) to C.

Type B.3 : sl(d/4,H) ⊕ eiβR and sl(d/4,H) ⊕ C

When the dimension index indicates Type B.3 in Table 4.1, then transitivity can bedecided by the centralizer test as well. However in this case, the centralizer z0 ofg0 = sl(d/4,H) is isomorphic to H, which is not abelian. So the centralizer z of g

might not be equal to z0∼= H, see Table 4.1. Thus, one has to check if the centralizer

z′0 of the complex non-compact semisimple part of sΣ is isomorphic to H. But first,we have to determine the corresponding semisimple part of sΣ before being able tocompute its centralizer. Now, note that for any transitive Lie algebra g = g0 ⊕ c, wehave

[g, g] = [g0, g0] = g0,

since g0 is semisimple and c is abelian. Hence, to determine the corresponding semisim-ple part of sΣ, we compute the derived algebra s0

Σ := [sΣ, sΣ], which is also includedin Algorithm 4.20. If the centralizer z′0 of s0

Σ is isomorphic (as a field) to H, it impliesthat s0

Σ is a subalgebra of Asl(d/4,H)A−1. Following the result from the complex case,for d/4 > 2, any subalgebra of gl(d/4,H) having dimension equal or greater than thatof sl(d/4,H) must contain sl(d/4,H) [12]. By the result from the dimension indextest, it follows that sΣ must be transitive of Type B.3.

Algorithm 4.20. (Type B.3). Given a basis b1, b2, . . . , bD ∈ gl(d,R) of sΣ.

1. Select a set of linearly independent matrices c1, c2, . . . , cD0obtained from the

derived algebraspan

⟨[bi, bj ] , 1 ≤ i < j ≤ D

⟩.

2. If D −D0 6= 1 or 2, then sΣ can not be transitive of Type B.3. If D −D0 = 1or 2, then compute a basis of the centralizer of [sΣ, sΣ],

z′0 = X ∈ Rd×d | [X, ci] = 0 , i = 1, . . . , D0.

Then, sΣ is transitive of Type B.3 if and only if z′0∼= H.

Note that D −D0 must be the same as the dimension of the center c of g (i.e. 1 or2), otherwise sΣ can not be transitive of Type B.3.

Type B.4 : sp(d/2,R) ⊕ R

When the dimension index of sΣ is of Type B.4, we have to check whether sΣ isconjugate to the Lie algebra of a Lie group which leaves some nondegenerate skew-symmetric bilinear form on Rd invariant. Before we proceed to the algorithm, we


recall the noncompact real symplectic Lie algebra

sp(d/2,R) :=X ∈ Rd×d | X⊤Jd + JdX = 0

,

where

Jd :=

[0 Id/2

−Id/2 0

].

The corresponding noncompact real symplectic Lie group is

Sp(d/2,R) = M ∈ Rd×d | M⊤JdM = Jd.

It is easy to show, for A ∈ GL(d,R), that

A−1sp(d/2,R)A =X ∈ Rd×d | X⊤J + JX = 0

=: sp(d/2,R) eJ ,

where now J := A⊤JdA ∈ GL(d,R). Therefore, the following algorithm decideswhether the system Lie algebra sΣ is conjugate to sp(d/2,R) ⊕ R.

Algorithm 4.21. (Type B.4) Given a basis b1, b2, . . . , bD ∈ gl(d,R) of sΣ.

1. Select a linearly independent set b′1, b′2, . . . , b

′D−1 from the “shifted” basis bi −

1/(d)Tr(bi)Id, for i = 1, . . . , D.

2. Solve the following set of linear equations for the unknown Ω ∈ Rd×d, for i =1, . . . , D − 1,

(i). Ω⊤ + Ω = 0,(ii). (b′i)

⊤Ω + Ω(b′i) = 0.

3. If the linear equations in Step 2 have only the trivial solution, then sΣ can notbe transitive of Type B.4. If there exist non-zero solutions, then sΣ is transitiveof Type B.4 if and only if any non-zero solution Ω is regular.

Type B.5 : sp(d/4,C) ⊕ eiβR and sp(d/4,C)⊕ C

Type B.5 involves a Lie group which leaves some nondegenerate skew-symmetric bilin-ear form on Cd/2 invariant. Similar to the real case, we recall the noncompact complexsymplectic Lie algebra

sp(d/4,C) :=X ∈ Cd/2×d/2 | X⊤Jd + JdX = 0

,

with the corresponding Lie group

Sp(d/4,C) = M ∈ Cd/2×d/2 | M⊤JdM = Jd.


It is clear that sp(d/4,C) = sp(d/4,R)C := sp(d/4,R) ⊕ i sp(d/4,R). Further, notethat sp(d/4,C) is regarded as a subalgebra of gl(d,R) by the following representation,

sp(d/4,C) :=

[X Y−Y X

] ∣∣∣ X, Y ∈ sp(d/4,R)

⊂ gl(d,R).

We observe carefully that sp(d/4,C) is not a subalgebra of sp(d/2,R). Instead, wehave

sp(d/4,C) ⊂ sp(d/2,R)Ω :=X ∈ Rd×d | X⊤Ω + ΩX = 0

,

where

Ω =

[Jd/2 Jd/2

Jd/2 −Jd/2

].

Note that with respect to sp(d/4,C), there exists the complex structure Jd, withJ2

d = −Id, which commutes with sp(d/4,C). This is due to the fact that the centralizerof sp(d/4,C) is isomorphic (as a field) to C. In relation with Ω, it is easy to see thatthe complex structure Jd satisfies J⊤

d Ω = ΩJd. Now, for A ∈ GL(d,R), we have

A−1sp(d/4,C)A ⊂ sp(d/2,R)eΩ :=X ∈ Rd×d | X⊤Ω + ΩX = 0

, (4.6)

where now Ω = A⊤ΩA. Correspondingly, the complex structure in A−1sp(d/4,C)A

reads J = A−1JdA, with J2 = −Id, which must also satisfy

J⊤Ω = A⊤J⊤d ΩA = A⊤ΩJdA = ΩJ . (4.7)

Therefore, according to Eq.(4.6) and Eq.(4.7), the following algorithm decides whethersΣ is transitive of Type B.5.

Algorithm 4.22. (Type B.5). Given a basis b1, b2, . . . , bD ∈ gl(d,R) of sΣ.

1. Select a set of linearly independent matrices c1, c2, . . . , cD0obtained from the

derived algebraspan

⟨[bi, bj ] , 1 ≤ i < j ≤ D

⟩.

If D −D0 6= 1 or 2, then sΣ can not be transitive of Type B.5. If D −D0 = 1or 2, then go to Step 2.

2. Recall the centralizer z′ of sΣ (obtained from Algorithm 4.15). If z′ is notisomorphic (as a field) to C, then sΣ can not be transitive of Type B.5. If

z′ = span〈Id, J〉 ∼= C, for some complex structure J2 = −Id, continue to Step 3.

3. Solve the following set of linear equations for the unknown Ω ∈ Rd×d

(i). Ω⊤ + Ω = 0,

(ii). (ci)⊤Ω + Ωci = 0,

(iii). J⊤Ω = ΩJ ,

for i = 1, . . . , D0.


4. If the linear equations in Step 3 have only the trivial solution, then sΣ can notbe transitive of Type B.5. If there exist non-zero solutions, then sΣ is transitiveof Type B.5 if and only if any non-zero solution Ω is regular.

Note that when a real non-zero matrix Ω solves the linear equations in Step 3 ofAlgorithm 4.22, we obtain a complex nondegenerate skew-symmetric bilinear form[12]

α(x, y) := 〈x, Ωy〉 − i〈Jx, Ωy〉,with 〈x, y〉 := x⊤y, for x, y ∈ Rd, which is left invariant by the corresponding Liegroup of sΣ. Therefore, sΣ is transitive of Type B.5.

4.3.3 Type C

The transitive Lie algebra conjugate to Type C is the most complicated one sinceg = g0 ⊕ c has the semisimple part g0 = sl(d/4,H) ⊕ su(2) which is actually neithersimple nor compact. The algorithm below is some necessary check which has to becontinued in the next step provided that the test is successful.

Algorithm 4.23. (Type C - Step 1). Given a basis b1, b2, . . . , bD ∈ gl(d,R) ofthe system Lie algebra sΣ.

1. (Centralizer test). If the subspace z′ computed in the preliminary test of Al-gorithm 4.15 is not isomorphic (as a field) to R, then sΣ can not be tran-sitive of Type C. If z′ ∼= R, select a linearly independent set b′1, b

′2, . . . , b

′D0

,D0 = D − 1, which forms a basis of the Lie algebra s0

Σ from the “shifted” basisbi − 1/(d)Tr(bi)Id, for i = 1, . . . , D.

2. With repect to the basis of s0Σ, calculate the matrix representation of each adb′i

,i = 1, . . . , D0.

3. Calculate the corresponding Killing form κ on s0Σ represented by a real symmetric

matrix K ∈ gl(D0,R) whose elements are

Kij := κ(b′i, b′j) := Tr(adb′i

adb′j) , i, j = 1, . . . , D0.

Diagonalize this Killing form such that the diagonalized version K ∈ gl(D0,R)

has elements Kij = 0 for i 6= j.

4. If there exist some zero diagonal entries of K, or if the number of negativediagonal entries of K is not equal to 2(d/4)2 + d/4 + 3, then sΣ is not transitiveof Type C. Otherwise, continue to Algorithm 4.24 (Type C - Step 2).


An algorithm with a finite number of steps, which converts a basis b′i to ci,i = 1, . . . , D0 such that the corresponding Killing form Kij = κ(ci, cj) is diagonalwith respect to this basis, is summarized in Algorithm D.2, Appendix D. The lastnecessary check in Algorithm 4.23 uses the fact that a semisimple Lie algebra has anon-degenerate Killing form. Furthermore, it also checks if s0

Σ has the same Cartanindex as g0 = sl(d/4,H) ⊕ su(2). Note that the number of negative diagonal entriesin the diagonalized Killing form determines the Cartan index of a semisimple Liealgebra and is equivalent to the dimension of the maximal compact subalgebra of asemisimple Lie algebra [59]. The maximal compact subalgebra of sl(d/4,H) ⊕ su(2)is sp(d/4) ⊕ su(2) (See [11], or table of classification in [27]) and therefore it hasdimension 2(d/4)2 + d/4 + 3.

Now, note that for any semisimple Lie algebra g0 = g1 ⊕ g2, where g1 and g2

are simple, the adjoint representation adg0also decomposes as the direct sum of two

irreducible representations with respect to g1 and g2. Due to irreducibility of eachsummand, the centralizer of adg0

must be either of the form [12]

R ⊕ C (for g1 non-complex, g2 complex)C ⊕ R (for g1 complex, g2 non-complex)R ⊕ R (for g1 non-complex, g2 non-complex)C ⊕ C (for g1 complex, g2 complex).

Since sl(d/4,H) and su(2) are both non-complex, the centralizer of adg0for g0 =

sl(d/4,H) ⊕ su(2) is of the form R ⊕ R which has real dimension 2. Hence, thefollowing algorithm checks if the centralizer of ads0

Σhas real dimension 2.

Algorithm 4.24. (Type C - Step 2). From the matrix representation of adb′i,

i = 1, . . . , D0, already obtained in Algorithm 4.23, compute a basis for the followinglinear subspace

z(ads0Σ) := X ∈ RD0×D0 | X(adb′i

) − (adb′i)X = 0 , i = 1, . . . , D0.

If the dimension of z(ads0Σ

) is not equal to 2, then sΣ can not be transitive of Type C.Else, continue to Algorithm 4.25 (Type C - Step 3).

Up to Algorithm 4.24 (Type C - Step 2), we only know that the centralizer ofads0

Σ

has real dimension 2. This still only means that s0Σ might be a non-complex

semisimple Lie algebra which consists of two non-complex simple ideals (as it wasintended to be the Type C), or s0

Σ might be just a complex simple Lie algebra, i.e.z(ads0

Σ

) ∼= C, which has also real dimension 2. To exclude the possibility of the latter

case, we use the following simple argument. Suppose s0Σ = g1 ⊕ g2 where g1, g2 are

non-complex simple such that the centralizer z(ads0Σ

) ∼= R⊕R has explicitely the form

z(ads0Σ) =

[aID01

00 bID02

] ∣∣∣ a, b ∈ R

,


with D01 := dim(g1) and D02 := dim(g2). Obviously any matrix Z ∈ z(ads0Σ

) witha 6= b 6= 0 commutes with the set of matrices

M =

[A 00 B

] ∣∣∣ A ∈ gl(D01,R) , B ∈ gl(D02,R)

.

In the case of Type C, we have g1 = sl(d/4,H) and g2 = su(2) such that D01 = D0−3and D02 = 3. Then, it is necessary to check if the set of matrices M which commuteswith such Z has the dimension (D0 − 3)2 + 9.

Algorithm 4.25. (Type C - Step 3). Take any nonzero Z ∈ z(ads0Σ) with Z 6= ID0

.Compute a basis of the following linear subspace

M = M ∈ RD0×D0 | MZ − ZM = 0.

Then sΣ can not be transitive of Type C if dim(M) 6= (D0 − 3)2 + 9. Else, continueto the final step in Algorithm 4.26 (Type C - Step 4).

Note that there is a small caveat in this step which does not matter in our case.Suppose s0

Σ is complex and simple such that the centralizer z(ads0Σ

) ∼= C has the form

z(ads0Σ) =

[aID0/2 bID0/2

−bID0/2 aID0/2

] ∣∣∣ a, b ∈ R

.

Any matrix Z ∈ z(ads0Σ

) with a 6= b 6= 0 commutes with the set of matrices

N =

[A B−B A

] ∣∣∣ A,B ∈ gl(D0/2,R)

.

Note that N ∼= gl(D0/2,C) has the real dimension D20/2. Now, in order to see that the

necessary dimension comparison test in Algorithm 4.25 really excludes the possibilitythat s0

Σ is complex and simple, we note that

dim(M) = (D0 − 3)2 + 9 = D20/2 = dim(N )

can only be satisfied for D0 = 6. Take D0 = 6, from Table 4.1, we have

dim(sl(d/4,H) ⊕ su(2)) = D0 = 4(d/4)2 + 2,

so that this coincidence can only happen for d = 4. But N -level quantum systemshave the underlying sizes d = N2 − 1, i.e. d = 3, 8, 15, . . .. Hence, for those values ofd, we have dim(M) 6= dim(N ), and s0

Σ can not be complex and simple.

Up to this stage, we know that s0Σ = g1 ⊕ g2 where

g1 : simple, non-complex,dim(g1) = D0 − 3 = 4(d/4)2 − 1 = d2/4 − 1

g2 : simple, non-complex,dim(g2) = 3.


Now, from the dimensional data of the above g1 and g2, we exploit the completeclassification of simple Lie algebras (see e.g. Helgason [27]). For g2 which is threedimensional, simple and non-complex, the only possible Lie algebras are sl(2,R) andsu(2), up to isomorphism. Contrastly for g1, there are many possibilities since g1 canbe any real form of a complex simple Lie algebra with dimension d2/4 − 1. However,assuming that the rank of s0

Σ; which is defined as the dimension of a Cartan subalgebraof (s0

Σ)C; is d/2, the number of possible Lie algebras for g1 can be significantly reduced.Precisely, if the rank of (s0

Σ)C is d/2, the full classification of simple Lie algebras [27]implies that g1 has to be any real form of sl(d/2,C), i.e. g1 must be isomorphic toeither sl(d/4,H), sl(d/2,R) or su(p, d/2 − p) for 0 ≤ p ≤ d/4. Therefore, s0

Σ can bereduced to any combination of the following Lie algebras (up to isomorphism).

g1 g2

sl(d/4,H) su(2)sl(d/2,R) sl(2,R)

su(p, d/2 − p) , 0 ≤ p ≤ d/4

Checking the dimensions for g1, we have dim(sl(d/4,H)) and dim(sl(d/2,R)) are equalto d2/4 − 1. Next, recall that any X ∈ su(p, q) can be expressed as

X =

[A CC† B

], A ∈ u(p), B ∈ u(q), C ∈ Cp×q,

with Tr(A + B) = 0, such that the dimension of su(p, q) can be calculated as p2 +q2 + 2pq − 1 = (p + q)2 − 1. So su(p, d/2 − p) has also the real dimension d2/4 − 1,for all 0 ≤ p ≤ d/4. To check the rank of g1 ⊕ g2, recall that su(2) and sl(2,R)are real forms of sl(2,C) which has a one-dimensional Cartan subalgebra. Similarly,sl(d/4,H), sl(d/2,R) and su(p, d/2− p) are all real forms of sl(d/2,C) whose Cartansubalgebra has dimension d/2 − 1. So any combination g1 ⊕ g2 of the above g1 andg2 has rank d/2.

We remember that in Algorithm 4.23 (Type C - Step 1), it was already assuredthat s0

Σ has a maximal compact subalgebra of dimension 2(d/4)2 + d/4 + 3. This willreduce the number of possible Lie algebras for s0

Σ as follows,

1. sl(d/4,H) ⊕ su(2) is possible; as already checked.

2. sl(d/4,H)⊕ sl(2,R) is not possible since sl(2,R) has only one-dimensional max-imal compact subalgebra so(2).

3. sl(d/2,R) ⊕ su(2) is not possible since the maximal compact subalgebra ofsl(d/2,R) is so(d/2), which has smaller dimension than sp(d/4), the maximalcompact subalgebra of sl(d/4,H), i.e,

dim(so(d/2)) =(d/2)(d/2− 1)

2= 2(d/4)2−d/4 < 2(d/4)2+d/4 = dim(sp(d/4)).


4. sl(d/2,R) ⊕ sl(2,R) is not possible for the same reason as No 3.

5. su(p, d/2 − p) ⊕ su(2) is possible only for p = (d/4) −√

(d/4 + 1)/2.

6. su(p, d/2 − p) ⊕ sl(2,R) is possible only for p = (d/4) −√

(d/4 + 3)/2.

Note that the maximal compact subalgebra of su(p, q) is of the form

[A 00 B

] ∣∣∣ A ∈ u(p), B ∈ u(q) , Tr(A+B) = 0

,

which has real dimension p2 + q2 − 1. This was used to calculate p in No 5 and 6above such that su(p, d/2 − p) ⊕ su(2) and su(p, d/2 − p) ⊕ sl(2,R) have a maximalcompact subalgebra with the same dimension as that of sl(d/4,H) ⊕ su(2).

Finally, we have only three possible Lie algebras left. For the following argument,we use a result of Tits [59]. The Lie algebra su(p, q) has a faithful real representationwhen

(p+ q)/2 = p (mod 2).

So in our case, we have p+ q = d/2 and thus, d/4 = p (mod 2). For the case of p inNo 5 and 6 above, it turns out that

√(d/4 + 1)/2 = 0 (mod 2) and

√(d/4 + 3)/2 = 0 (mod 2),

respectively. Using the size relation for N -level quantum systems with d = N2 − 1,we have, respectively,

√(N2 + 3)/8 = 0 (mod 2) and

√(N2 + 11)/8 = 0 (mod 2).

This implies in both cases that

N2 + 3 = 0 (mod 8).

It is not difficult to see that there does not exist any integer N ∈ N satisfying theequation above. Hence, su(p, d/2 − p) ⊕ su(2) and su(p, d/2 − p) ⊕ sl(2,R) do nothave a faithful representation on Rd and thus, can not be possible Lie algebras for s0

Σ.Therefore, the only one Lie algebra left is sl(d/4,H) ⊕ su(2) and we are done.

Recall that this exclusion so far was made possible by the final assumption thatthe dimension of a Cartan subalgebra of (s0

Σ)C (or the rank of s0Σ) is d/2. So finally

we conclude with an algorithm [12] to determine whether sΣ is transitive of TypeC by computing the dimension of a Cartan subalgebra of (s0

Σ)C. This is done bydetermining the algebraic multiplicity of the zero eigenvalues of a generic element in(s0

Σ)C.

Algorithm 4.26. (Type C - Step 4). Given the matrix representation of adb′i,

where b′i, i = 1, . . . , D0, is the basis of s0Σ already obtained in Algorithm 4.23.


1. Construct the characteristic polynomial

p(λ, x1, . . . , xD0) := det

(λID0

−D0∑

i=1

xiadbi

).

2. Write the polynomial p(λ, x1, . . . , xD0) into the form

p(λ, x1, . . . , xD0) =

D0∑

k=0

λk fk(x1, . . . , xD0).

3. Find the largest integer r such that all fk(x1, . . . , xD0) are identically zero for

k < r when written as linear combinations of monomials xN1

1 . . . xND0

D0, for some

indices N1, . . . , ND0. Note that fk(x1, . . . , xD0

) is identically zero when all coef-ficients are zero. Then, sΣ is transitive of Type C if and only if r = d/2.

Appendix A

Lie Algebras and Lie Groups

We first recall some basic facts and definitions about Lie algebras and Lie groups.Specifically, and according to the content of this thesis, we focus on matrix Lie groups.Details on Lie theory can be consulted in [27] and [41]. For introductory purposesespecially dealing with matrix Lie groups one can follow for instance [9] and [26]. LetK denote either field R or C of real or complex numbers, respectively.

Definition A.1. A K-vector space g with a bilinear product

[·, ·] : g × g −→ g,

is called a Lie algebra over K if

(i) [X, Y ] = −[Y,X] for all X, Y ∈ g

(ii) [[X, Y ], Z] + [[Y, Z], X] + [[Z,X], Y ] = 0 (Jacobi identity).

Example A.2. Some examples of classical Lie algebras are

sl(n,K) := X ∈ Kn×n | Tr(X) = 0sp(n,K) := X ∈ K2n×2n | X⊤J + JX = 0so(n) := X ∈ Rn×n | X⊤ +X = 0su(n) := X ∈ Cn×n | X† +X = 0,Tr(X) = 0,

where

J :=

[0 In

−In 0

].

A Lie algebra g over R (C) is called real (complex). A Lie subalgebra h is aK-linear subspace of g for which [h, h] ⊂ h holds, where

[h1, h2] := [H1, H2] | H1 ∈ h1, H2 ∈ h2.

132

133

It is called an ideal of g, if [h, g] ⊂ h. A Lie algebra g is called simple, if its only idealsare g and 0, it is called abelian if [g, g] = 0. In the sequel, g is always assumed tobe finite dimensional. Denote the endomorphisms of g, i.e. the K-linear mappingsg → g by End(g) and let GL(g) denote the subset of all invertible endomorphisms.For any X ∈ g, the adjoint transformation is the linear map

adX : g −→ g, Y 7−→ [X, Y ] (A.1)

andad: g −→ End(g), Y 7−→ adY (A.2)

is called the adjoint representation of g.

By means of (A.1) and (A.2), the properties (i) and (ii) of Definition A.1 areequivalent to

adXY = −adYX

andad[X,Y ] = adXadY − adY adX ,

respectively. It follows immediately from property (i) that adXX = 0 for all X ∈ g.

Definition A.3. Let g be a finite dimensional Lie algebra over K. The symmetricbilinear form

κ : g × g −→ K, κ(X, Y ) 7−→ Tr(adXadY ) (A.3)

is called the Killing form of g.

A Lie group is defined as a group together with a manifold structure such thatthe group operations are smooth maps. For an arbitrary Lie group G, the tangentspace T1G at the unit element 1 ∈ G possesses a Lie algebraic structure. This tangentspace is called the Lie algebra of the Lie group G, denoted by g. The tangent mappingof the conjugation mapping in G at 1,

conjx(y) := xyx−1, x, y ∈ G,

is given byAdx := T1(conjx) : g −→ g

and leads to the so-called adjoint representation of G in g, given by

Ad: x 7−→ Adx.

In this thesis, we exclusively deal with matrix Lie groups, i.e. G consists ofinvertible real or complex matrices, such that the elements of the corresponding Liealgebra can also be regarded as matrices. In this case the adjoint representation ofg ∈ G applied to X ∈ g is given by

AdgX = gXg−1, (A.4)

134 Appendix A. Lie Algebras and Lie Groups

i.e., by the usual similarity transformation of matrices, and the adjoint transformationis given by

adYX = Y X −XY.

Example A.4. Some examples of classical matrix Lie groups are

SL(n,K) := g ∈ Kn×n | det g = 1 corresponding Lie algebra: sl(n,K),

Sp(n,K) := g ∈ K2n×2n | g⊤Jg = J corresponding Lie algebra: sp(n,K),

SO(n) := g ∈ Rn×n | g⊤g = 1, det g = 1 corresponding Lie algebra: so(n),

SU(n) := g ∈ Cn×n | g†g = 1, det g = 1 corresponding Lie algebra: su(n),

where J is defined as in Example A.2.

A basic property of the Killing form κ defined by (A.3) is its Ad-invariance, i.e.

κ(AdgX,AdgY ) = κ(X, Y ) for all X, Y ∈ g and g ∈ G. (A.5a)

Differentiating this equation with respect to g immediately yields

κ(adXY, Z) = −κ(Y, adXZ) for all X, Y, Z ∈ g. (A.5b)

Definition A.5. A matrix Lie group G is said to be compact if it is a closed andbounded subset of Cn×n. The corresponding Lie algebra of a compact matrix Liegroup is also called a compact Lie algebra.

Example A.6. Some examples of compact Lie algebras are,

so(n) := S ∈ Rn×n | S⊤ = −Su(n) := X ∈ Cn×n | X† = −X,su(n) := X ∈ Cn×n | X† = −X,Tr(X) = 0,sp(n) := u(2n) ∩ sp(n,C),

with the corresponding compact matrix Lie groups

SO(n) := g ∈ Rn×n | g⊤g = 1U(n) := g ∈ Cn×n | g†g = 1,SU(n) := X ∈ Cn×n | g†g = 1, det g = 1,Sp(n) := U(2n) ∩ Sp(n,C).

A compact Lie algebra g admits a positive definite Ad-invariant bilinear form,cf. [41], Ch. IV., Prop. 4.24. This property is used to show that the Killing form oncompact Lie algebras is negative semi-definite.

Proposition A.7. Let g be a compact Lie algebra. Then the Killing form is negativesemi-definite.

135

Proof. [41], Ch. IV, Cor. 4.26.

In our purpose, it is convenient to use Cartan’s Criterion, cf. [41], Thm. 1.45, todefine semisimple Lie algebras. In general, other equivalent definitions are possible.

Definition A.8. A finite dimensional Lie algebra is called semisimple, if its Killingform is nondegenerate.

Together with Proposition A.7 this yields the following theorem.

Theorem A.9. A semisimple Lie algebra is compact if and only if its Killing formis negative definite. On the other hand, the Killing form of a Lie algebra g is negativedefinite if and only if g is compact and semisimple.

Proof. Cf. [41], Ch. IV, Prop. 4.27.

As another characterization of semisimple Lie algebras we have the followingtheorem.

Theorem A.10. The Lie algebra g is semisimple if and only if

g = g1 ⊕ ...⊕ gm

with simple ideals gi. This decomposition is unique. Moreover, every ideal i ⊂ g isthe sum of various gi.

Proof. [41], Ch. I, Thm. 1.54.

Example A.11. Some examples of semisimple Lie algebra are sl(n,K), so(n),sp(n,K), su(n) and sp(n). Furthermore, let

Ip,q :=

[Ip 00 −Iq

].

Then the following Lie algebras are also semisimple.

so(p, q) := X ∈ R(p+q)×(p+q) | X⊤Ip,q + Ip,qX = 0, p+ q ≥ 3,

su(p, q) := X ∈ C(p+q)×(p+q) | X†Ip,q + Ip,qX = 0,Tr(X) = 0, p+ q ≥ 2.

Let g0 be a real Lie algebra. Then forming the tensor product with C yields acomplex Lie algebra

gC

0 : = g0 ⊗R C = g0 ⊕ ig0.

It is called the complexification of g0.

136 Appendix A. Lie Algebras and Lie Groups

Definition A.12. Let g be a complex Lie algebra. Then any real Lie algebra g0

with the propertyg = g0 ⊕ ig0

is called a real form of g. The Lie algebra g0 is said to be a compact real form, if it isa compact Lie algebra.

Note that dimC g = dimR g0. If g0 is a real form of the complex Lie algebra g,then an R-basis of g0 is also a C-basis of g. Consequently, if X, Y ∈ g0, then thematrix of adX adY is the same for g0 as it is for g and the respective Killing formsare related by

κg0= κg|g0×g0

. (A.6)

Complexification preserves semisimplicity. Moreover, the following result holds.

Proposition A.13. Let g be a complex Lie algebra with real form g0. Then g issemisimple if and only if g0 is semisimple.

Proof. [41], Ch. I, Cor. 1.53.

Definition A.14. A Lie subalgebra h of a finite dimensional Lie algebra g over C

is called a Cartan subalgebra if

(i) h is nilpotent; i.e. the sequence of subalgebra hk+1 = [h, hk], where h0 = h,eventually vanish for some k.

(ii) h is its own normalizer; i.e. h is exactly the set of those elements X ∈ g suchthat [X, h] ⊂ h.

Proposition A.15. Let g be a semisimple finite dimensional Lie algebra over C.Then h is a Cartan subalgebra of g if and only if h is a maximal abelian subalgebra ofg and adH is diagonalizable for all H ∈ h.

Proof. [41], Ch. II, Prop. 2.13.

Appendix B

List of Transitive Lie Algebras

We recall the classification results of Lie groups which act transitively on the Grass-mann manifold Gk,N(C), on the complex projective space CPN−1, and on the punc-tured Euclidean space Rd \ 0. Note that the results are stated in terms of thecorresponding Lie algebras.

Theorem B.1. (Volklein [60]) A Lie subgroup G ⊂ SL(N,C) acts transitivelyon the Grassmann manifold Gk,N(C), for k ≥ 2, if and only if the corresponding Liesubalgebra g ⊂ sl(N,C) is conjugate to sl(N,C) or su(N).

Theorem B.2. (Kramer [44], Volklein [60]) A Lie subgroup G ⊂ SL(N,C) actstransitively on the complex projective spaces CPN−1 if and only if the correspondingLie subalgebra g ⊂ sl(N,C) is conjugate to one of the following types:

(1) sl(N,C)

(2) su(N).

(3) sp(N/2,C), sp(N/2) for N even.

(4) sl(N/2,H) for N even.

Theorem B.3. (Boothby [11], Kramer [44]) A Lie subgroup G ⊂ GL(d,R) actstransitively on Rd \ 0 if and only if the corresponding Lie subalgebra g ⊂ gl(d,R) isconjugate to one of the following types:

(1) so(d) ⊕ R, if d ≥ 2.

(2) su(d/2) ⊕ eiαR or su(d/2) ⊕ C, if d is even and d ≥ 3.

(3) sp(d/4) ⊕ eiαR, sp(d/4) ⊕ C or sp(d/4) ⊕ H, if d = 4k.

(4) g2 ⊕ R, if d = 7.

137

138 Appendix B. List of Transitive Lie Algebras

(5) spin(7) ⊕ R, if d = 8.

(6) spin(9) ⊕ R, if d = 16.

(7) sl(d,R) or gl(d,R), if d ≥ 2.

(8) sl(d/2,C), sl(d/2,C)⊕ eiβR or gl(d/2,C), if d is even and d ≥ 2.

(9) sl(d/4,H), sl(d/4,H) ⊕ eiβR or sl(d/4,H) ⊕ C, if d = 4k.

(10) sl(d/4,H) ⊕ sp(1) or sl(d/4,H) ⊕ H, if d = 4k.

(11) sp(d/2,R) or sp(d/2,R) ⊕ R, if d is even and d ≥ 3.

(12) sp(d/4,C), sp(d/4,C)⊕ eiβR or sp(d/4,C) ⊕ C, if d = 4k.

(13) spin(9, 1,R) or spin(9, 1,R) ⊕ R, if d = 16.

Here α and β have to satisfy α ∈ (−π2, π

2) and β ∈ [−π

2, π

2], respectively.

Appendix C

Some Calculations for the Vectorof Coherence Representation

C.1 Gell-Mann Matrices

The Gell-Mann matrices are given by

B1 :=1√2

0 1 01 0 00 0 0

, B2 :=

1√2

0 −i 0i 0 00 0 0

, B3 :=

1√2

1 0 00 −1 00 0 0

B4 :=1√2

0 0 10 0 01 0 0

, B5 :=

1√2

0 0 −i0 0 0i 0 0

, B6 :=

1√2

0 0 00 0 10 1 0

B7 :=1√2

0 0 00 0 −i0 i 0

, B8 :=

1√6

1 0 00 1 00 0 −2

.

The Gell-Mann matrices are a generalization of the Pauli matrices to N = 3. Theyform an orthonormal basis of her0(3), i.e. Tr(BjBk) = δjk. Similarly, iBj8

j=1 formsan orthonormal basis of su(3). The commutation relations of the Gell-Mann basis,

[Bj , Bk] = i8∑

r=1

fjkrBr,

as well as their anti-commutation relations,

Bj, Bk =2

3I3δjk +

8∑

r=1

gjkrBr,

are summarized in the following tables.

139

140 Appendix C. Some Calculations for the Vector of Coherence Representation

fjkr

√2 1/

√2 −1/

√2

√6/2

123 147 156 458jkr 246 367 678

257345

gjkr 2/√

6 1/√

2 −1/√

2 −1/√

6 −2/√

6

118 146 247 448 888228 157 366 558

jkr 338 256 377 668344 778355

C.2 Calculations for LH, Ljk and pjk

Consider the Lindblad-Kossakowski master equation in the GKS form. We write theevolution of each components of the vector of coherence vl, l = 1, 2, . . . , d as follows

vl =d

dt(Tr(Blρ)) = Tr(Blρ).

For the Hamiltonian part LH , use the fact that the Hamiltonian matrixH is Hermitianand has zero trace, we can write H in the Bj basis as

H =

d∑

m=1

hmBm.

Calculating the l-th element of LHv,

(LHv)l = −iTr(Bl

(∑

m

hmBm

)ρ− Blρ

(∑

m

hmBm

))

= −i∑

m

hmTr([Bl, Bm]ρ) = −i∑

m

hmTr(i∑

r

flmrBrρ)

=∑

m

hm

∑

r

flmrvr,

Hence,

LHlr =

d∑

m=1

hmflmr.

C.2. Calculations for LH , Ljk and pjk 141

For the dissipation part LaffD we calculate the l-th element of Laff

D v,

(LaffD v)l =

∑

jk

ajkTr

(BlBjρBk −

1

2BlBkBjρ−

1

2BlρBkBj

)

=∑

jk

ajkTr

((BkBlBj −

1

2BlBkBj −

1

2BkBjBl

)ρ

)

=∑

jk

ajkTr

((1

2BkBlBj −

1

2BkBjBl +

1

2BkBlBj −

1

2BlBkBj

)ρ

)

=∑

jk

ajkTr

((1

2Bk[Bl, Bj] + [Bk, Bl]Bj

)ρ

).

By using the relations (2.45) and (2.46) we obtain

BjBk =1

NINδjk +

i

2

∑

m

(fjkm − igjkm)Bm

=1

NINδjk +

i

2

∑

m

zjkmBm , zjkm = fjkm + igjkm,

for any index j, k = 1, 2, . . . , d, we calculate

12Bk[Bl, Bj ] =

1

2Bk i

∑

m

fljmBm =i

2

∑

m

fljmBkBm

=i

2

∑

m

fljm

(1

NINδkm +

i

2

∑

r

zkmrBr

)

=iIN2N

∑

m

fljmδkm − 1

4

∑

m

fljm

∑

r

zkmrBr

=iIN2N

fljk −1

4

∑

m

fljm

∑

r

zkmrBr,

and similarly

12[Bk, Bl]Bj =

i

2

∑

m

fklmBmBj

=i

2

∑

m

fklm

(1

NINδmj +

i

2

∑

r

zmjrBr

)

=iIN2N

∑

m

fklmδmj −1

4

∑

m

fklm

∑

r

zmjrBr

=iIN2N

fklj −1

4

∑

m

fklm

∑

r

zmjrBr.

Add all terms to obtain

1

2Bk[Bl, Bj] +

1

2[Bk, Bl]Bj =

iINNfjkl −

1

4

∑

m

(fljm

∑

r

zkmrBr + fklm

∑

r

zmjrBr

).


Now, multiply the above terms with ρ and take the trace. Since vr = Tr(Brρ), wehave

(LaffD v)l =

∑

jk

ajk

(i

Nfjkl −

1

4

∑

m

(fljm

∑

r

zkmrvr + fklm

∑

r

zmjrvr

)).

Writing compactly in terms of matrices Ljk we have

(LaffD v)l =

d∑

j,k=1

ajk(Ljkv + pjkv0),

with

Ljklr = −1

4

d∑

m=1

fljm(fkmr − igkmr) + fklm(fmjr − igmjr)

= −1

4

d∑

m=1

fjml(fkmr − igkmr) − fkml(−fjmr − igmjr)

= −1

4

d∑

m=1

fjml(fkmr − igkmr) + fkml(fjmr + igjmr)

pjk =i√N

[fjk1 . . . fjkd]⊤ , v0 = 1/

√N.

C.3 Calculations for Tr(LD)

By using the decomposition of LD ∈ gl(d,R) in Eq.(2.54), we can also calculate thetrace of LD directly as the sum of the diagonal elements of LD. We start with thefollowing lemma as a tool to calculate the trace of LD.

Lemma C.1. Let BjN2−1j=1 be an orthonormal basis of her0(N) which satisfies the

commutation relation [Bj, Bk] = BjBk − BkBj = i∑

m fjkmBm. Then for any j 6= k

we have∑N2−1

m,l fjmlfkml = 0.

Proof. For any j 6= k, we calculate that

N2−1∑

m

Tr ([Bj , Bm][Bk, Bm]) =∑

m

Tr((

i∑

l

fjmlBl

)(i∑

r

fkmrBr

))

= −∑

m

Tr(∑

l,r

fjmlfkmrBlBr

)= −

∑

m,l,r

fjmlfkmrTr(BlBr)

= −∑

m,l,r

fjmlfkmrδlr = −∑

m,l

fjmlfkml.

C.3. Calculations for Tr(LD) 143

So we have to show that

N2−1∑

m

Tr ([Bj , Bm][Bk, Bm]) = 0

for any j 6= k. By a straightforward calculation,

N2−1∑

m

Tr ([Bj , Bm][Bk, Bm]) =∑

m

Tr ((BjBm − BmBj)(BkBm − BmBk))

=∑

m

Tr(BjBmBkBm − BjBmBmBk

−BmBjBkBm +BmBjBmBk)

=∑

m

2 Tr(BjBmBkBm)︸︷︷︸(A)

−∑

m

Tr(BmBk, BjBm)︸︷︷︸(B)

.

Note that Lemma 2.3 can be used to calculate part (A) and (B), since we can use an

orthonormal basis BmN2−1j=1 of her0(N) plus BN2 = IN/

√N as an orthonormal basis

of CN×N when complex span is considered.

Part (A) : Tr(2Bj

∑N2−1m BmBkBm + 2Bj

IN√NBk

IN√N

)= Tr(2BjIN Tr(Bk)︸︷︷︸

=0

)

Tr(2Bj

∑N2−1m BmBkBm

)+ 2/N Tr(BjBk)︸︷︷︸

=0, j 6=k

= 0

∑N2−1m 2Tr(BjBmBkBm) = 0

Part (B) : Tr(∑N2−1

m BmBk, BjBm + IN√NBk, Bj IN√

N

)= Tr(INTrBk, Bj)

Tr(∑N2−1

m BmBk, BjBm

)+ 1/N Tr(Bk, Bj︸︷︷︸

=0, j 6=k

) = Tr(IN2 Tr(BkBj)︸︷︷︸=0, j 6=k

)

∑N2−1m Tr(BmBk, BjBm) = 0.

So,∑N2−1

m Tr ([Bj, Bm][Bk, Bm]) = −∑N2−1m,l fjmlfkml = 0, for j 6= k, as desired

Proposition C.2. (Trace of LD). The matrix representation LD ∈ gl(d,R),d = N2 − 1 of the Lindblad-Kossakowski master equation in the vector of coherencerepresentation has strictly negative trace.

Proof. Recall the explicit form of LD in Eq.(2.54)

LD =d∑

j,k=1

ajkLjk

=d∑

j

ajjLjj + 2d∑

j<k

Re(ajk)Re(Ljk) − 2d∑

j<k

Im(ajk)Im(Ljk).


Now calculate Tr(LD) explicitely via the formula of Ljk in Eq.(2.51).

(1). Tr(Ljj) =∑

l

Ljjll = −1

4

∑

l,m

fjml(fjml − igjml) + fjml(fjml + igjml)

= −1

2

∑

l,m

f 2jml < 0 , for all j.

(2). Tr(Re(Ljk)) =∑

l

Re(Ljkll ) = −1

4

∑

l,m

fjmlfkml + fkmlfjml

= −1

2

∑

l,m

fjmlfkml = 0 , for j 6= k, (by Lemma C.1).

(3). Tr(Im(Ljk)) =∑

l

Im(Ljkll ) = −1

4

∑

l,m

(fkmlgjml − fjmlgkml)

=∑

m<l

(fkmlgjml − fjmlgkml) + (fklmgjlm − fjlmgklm)

=∑

m<l

(fkmlgjml − fjmlgkml) + ((−fkml)gjml − (−fjml)gkml)

= 0 , for j 6= k.

Therefore we have,

Tr(LD) =∑

j

ajjTr(Ljj) = −1

2

∑

j

ajj

∑

l,m

f 2jml < 0,

since the nonegativity of each diagonal entry of the GKS matrix A, i.e. ajj ≥ 0 forall j, is necessary to guarantee positive semidefiniteness of the GKS matrix A.

Appendix D

Some Related Algorithms forTransitivity

Generating a basis for a real Lie algebra

Algorithm D.1. Given A0, A1, . . . , Am ∈ gl(d,R) and suppose they are linearlyindependent. Otherwise, use Gaussian elimination to select linearly independent ele-ments from them.

1. Set g1 = span〈A0, A1, . . . , Am〉, set k = 2.

2. Set V = span〈gk−1, [gk−1, gk−1]〉, where [gk−1, gk−1] is the span of all matrices[A,B], where A,B ∈ gk−1.

3. Using Gaussian elimination, select a basis of V . Then, set gk as the span of thisbasis.

4. If gk = gk−1 thensΣ := gk = 〈A0, A1, . . . , Am〉LA,

is the Lie algebra generated by A0, A1, . . . , Am. Else, set k = k + 1 and repeatback to Step 2.

Diagonalization of a Killing-form

Algorithm D.2. Given w1, w2, . . . , wD where wi = adbiand bi is a basis of the

Lie algebra sΣ. Let κ(a, b) = Tr(adaadb), for a, b ∈ sΣ. We assume that κ(wi, wj) 6= 0for some or possibly all i 6= j. We suppose that w1 is such that κ(w1, w1) 6= 0 (reorderwi if necessary). Define

Qku := u−∑

i≤k

κ(u, vi)

κ(vi, vi)vi,

145

146 Appendix D. Some Related Algorithms for Transitivity

where κ(vi, vi) 6= 0 for i ≤ k. Set k = 1 and v1 = w1.

1. Compute Qkwk+1. Ifκ(Qkwk+1, Qkwk+1) 6= 0

orκ(Qkwk+1, Qkwj) = 0 , for all j ≥ k + 1,

then set vk+1 = Qkwk+1 and repeat this step for k = k + 1. Else, go to nextstep.

2. Choose the first j > k + 1 such that κ(Qkwk+1, wj) 6= 0. Set

vk+1 = Qk(wk+1 + cwj)

for c any scalar such that κ(vk+1, vk+1) 6= 0. Back to step 1 for k = k + 1.

For a finite number of step, v1, v2, . . . , vD is a basis of sΣ, which satisfies κ(vi, vj) = 0,for all i 6= j.

For the explanation of Algorithm D.2, see [12].

Checking if a real subspace is isomorphic (as a field) to R, C

or H

Algorithm D.3. Consider c1, c2, . . . , cr ∈ gl(d,R) as a basis for a real subspaceV ⊂ gl(d,R).

1. If r 6= 1, 2, or 4, then clearly V is not isomorphic to R, C, or H.

2. If r = 1 and c1 = αId for α ∈ R, then V ∼= R.

3. If r = 2, use Gaussian elimination to convert c1 and c2 to Id and b, respectively,where Tr(b) = 0. Then V ∼= C if and only if b2 = −αId for some α > 0.

4. If r = 4, use Gaussian elimination to convert c1, c2, c3, c4 to Id, b1, b2, b3where Tr(bi) = 0. Suppose there exists a symmetric matrix A = [aij ], 1 ≤ i, j ≤2, whose elements satisfy the condition

bibj + bjbi = 2aijId,

after a suitable numbering of bi. Then V ∼= H if and only if A is real andnegative definite. Otherwise, V can not be isomorphic to H.

Note that if there exists such matrix A in step 4, then one can construct an isomor-phism from H to V , see [12] for a concrete form of this isomorphism. Conversely, givenany isomorphism from H to some subspace of gl(d,R), the pure quaternions must cor-respond to elements in gl(d,R) with trace zero. Moreover, for any pure quaternionsq1 and q2, the matrix A = [aij ], with aij := (qiqj + qjqi)/2 for 1 ≤ i, j ≤ 2, is realsymmetric and negative definite [12].

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Index of Notations

R, C, H field of real numbers, complex numbers and quaternions, respectivelyR+, R+

0 the set of positive and non-negative real numbers, respectivelyRe(a) real part of a ∈ C

Im(a) imaginary part of a ∈ C

IN identity matrix with size Nρ density operatorσ reduced density operator, σ = ρ− IN/NP the set of density operatorsBd the Bloch ball of size dΦ(P) ⊂ Bd the set of vector of coherence representation of density operatorsX† conjugate transpose of XX⊤ transpose of XMp(C

N×N) the set of all p× p block matriceswith N ×N complex matrices as block entries

gl(V ) the set of all linear operators on a vector space Vso(V ) the set of skew-adjoint linear operators on a vector space Vsym(V ) the set of self-adjoint linear operators on a vector space VLH , LD Hamiltonian part and dissipative part of the generator of the

Lindblad-Kossakowski master equation, respectivelyLH , LD Hamiltonian part and dissipative part of the generator of the

Lindblad-Kossakowski master equation in the vector ofcoherence representation, respectively

her(N) the set of Hermitian matricesher0(N) the set of Hermitian matrices with trace zeroher1(N) the set of Hermitian matrices with trace one

MjN2

j=1 an orthonormal basis of MN(C) = CN×N

BjN2−1j=1 an orthonormal basis of her0(N),

also used for an orthonormal basis of sl(N,C)

GjN2

j=1 an orthonormal basis of her(N)Lu the set of all admissibile unital Lindblad-Kossakowski generatorsLn the set of all admissibile non-unital Lindblad-Kossakowski generators

152

Index of Notations 153

fijk commutation structure constants of su(N)gijk anti commutation structure constants of su(N)(Σ) control system on Lie group(Σi) induced system of (Σ)sΣ system Lie algebra of (Σ)cΣ control Lie algebra of (Σ)RΣ(X) reachable set of (Σ) from XR(ρ0) reachable set from ρ0

SΣ system semigroup of (Σ)GΣ system group of (Σ)S(RN), S(CN) unit sphere in RN and CN , respectivelyO(ρ) unitary orbit of ρPk,N(C) complex GrassmannianGk,N(C) Grassmann manifoldCPN−1 complex projective spacead adjoint representation of su(N)

ad a representation of su(N) acting on her0(N)

ksu(N) a matrix representation of adsu(N)

Ksu(N) the corresponding Lie group of the Lie algebra ksu(N)

g0 semisimple part of transitive Lie algebra g

c center of g

z centralizer of g in gl(d,R)

Sp(L) non-zero eigenvalues of adC

L

EC0 generalized eigenspace of adC

L with respect to zero eigenvalue

EC

a eigenspace of adC

L with respect to non-zero eigenvalue a ∈ Sp(L)〈·, ·〉 inner product〈X, Y 〉LA Lie algebra generated by X and Y〈X〉G the smallest group generated by the set X

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