TitleFROM QUANTUM INFORMATION TO QUANTUMTELEPORTATION(The 8th Symposium on Non-EquilibriumStatisitical Physics)

Author(s) OHYA, MASANORI

Citation 物性研究 (2001), 75(5): 888-900

Issue Date 2001-02-20

URL http://hdl.handle.net/2433/96953

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

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[1] M.Ohya, Fundamentals of quantum mutual entropy and capacity, Open Systems and

Information Dynamics, 6, No.1, 69-78, 1999.

[2] M.Ohya, Complexities and their applications to charcterization of chaos,

International Journal of Theoretical Physics, 37, No.1, 495-505, 1998.

[3] K.Inoue, A.Kossakowski and M.Ohya, A description of quantum chaos and its

application to spin systems, SUT Preprint.

[4] K-H.Fichtner and M.Ohya, Quantum teleportation with entangled states given by

beam splittings, to appear in Commun. Math. Phys.

[5] K-H.Fichtner and M.Ohya, Quantum teleportation and beam splitting, quant-phi

0005040.

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FROM QUANTUM INFORMATION TO QUANTUMTELEPORTATION

MASANORI OHYA

1. INTRODUCTION

After Shannon, great development has been made in communica­tion of information. Shannon's information theory is now linked toseveral different fields such as mathematics, physics, economics, lifescience. In the same vein, the information theory on quantum me­chanics, so-called quantum information theory, is an indiespensablefoundation in recent science and technology, in particular, to studyoptical communication, quantum computer and quantum chaos[23,27, 25].

In this report, we discuss the fundamentals of quantum information[1 ,17, 19,27] and apply them to study the chaos of quantum dynamicalsystems[25, 13] and quantum computer[31 , 6, 26, 29].

We first review the fundamentals of quantum information, and weexplain how to apply them to construct a "chaos degree" measur­ing the chaos of dynamics[21, 25]. Further we discuss the quantumteleportation process[2, 14, 9, 10] in a mathematical vein of quantumchannel.

2. FUNDAMENTALS IN QUANTUM INFORMATION

Fundamental mathematical concepts in quantum information the­ory are quantum entropy describing the amount of information andquantum channel describing the dynamics associated with informa­tion communication.

The concept of channel is not only useful in information theory butalso a convenient mathematical tool to treat several physical dynamicsin a unified way[17].

In classical systems, an input (or initial) system is described by theset of all random variables A = M (0) and its state space 6 = P (0)and an output (or final) system is by M (0) and P (0) .

A quantum input system is described by the set A =B (1i) of allbounded linear operators on 1i and 6 is the set T(1i) of all densityoperators on 1i . An output system is A = B (R) and 6= T (1i) .

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More general quantum system is described by a C*-algebra A and itsstate space 6 .

In any case, a channel is a mapping from 6 to 6. Almost allphysical transformations are described by this mapping.

DEFINITION 2.1. :Let A* be a channel from 6 to 6 ..

(1) A* is linear if A*(Aep+ (1- A)'l/J) = AA*ep+ (1- A)A*'l/J holds forall ep, 'l/J E 6 and any A E [0, 1].

(2) A* is completely positive (C. P.) if A* is linear and its dual A :A -+ A satisfies

n

L A;A(A;Aj)Aj > 0i,j=l

for any n E N and any {Ai} C A, {Ai} cA.Most of channels appeared in physical processes are the C.P. channels[19].

For instance, the open system dynamics can be written as follows: If asystem E1 interacts with an external system E2 described by anotherHilbert space K and the initial states of E1 and E2 are p and u, re­spectively, then the combined state Ot of E1 and E2 at time t after theinteraction between two systems is given by Ot = Ut(p Q9 u)Ut ,whereUt = exp(-itH) with the total Hamiltonian H of E1 and E2 • Achannel is obtained by taking the partial trace w.r.t. K such asp -+ A; p =trICOt.

The quantum entropy was introduced and developed to study somephysical problems such as irreversible behavior, symmetry breaking,and it is an expression of the amount of information carried by astate. Here we review three fundamental quantum entropies whichare important to study the information transmission processes.

Consider an quantum system described by a density operator ona Hilbert space 1-l. The entropy of a state p was introduced by vonNeumann [16, 27] as

S(p) =.-trplogp.

The second entropy is the relative emtropy for two states p, u E 6(1-l),which is defined by

S( u) = { trp (log p -logu) (p« u)p, +00 (otherwise) ,

where p« u means that truA = 0 ~ trpA = 0 for any A > 0 [32].

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Let A* : 6(Ji) ~ 6(Ji) be a channel and define the compound stateby BE = l::k PkEk ® A*Ek with a Schatten decomposition l::kPkEk ofp, which expresses the correlation between the initial state p and thefinal state A*p [17,19]. Another useful entropy is the mutual entropy[17] for a stat~ p E 6(Ji) and a channel A* , which is given by

I(p;A*) sup{S(BE,p®A*p);E = {Ek}}

sup { "2;PkS(A*Ek, A*p); E = {Ek} } ,

where the supremum is taken over all Schatten decompositions. Themutual entropy expresses the amount of information trasmitted from.the initial state p to the final state A*p, so that it satisfies the funda­mental inequality of Shannon type [30, 17]:

o< I(p; A*) < min{S(p), S(A*p)}.

In Shannon's communication theory in classical systems, p is aprobability distribution P = (Pk) and A* is a transition probability(t iJ ) , so that the Schatten decomposition of p is unique and the com­pound state of p and its output p (= P = (Pi)) is the joint distributionr = (ri,j) with ri,j =ti,jpj. Then the above entropies become theShannon entropy and mutual entropy, respectively;

Such quantum entropies have been used to define the capacity ofchannel [24, 28]and to study quantum communication processes [19].

3. CHAOS DEGREE

In the context of information dynamics, a chaos degree associatedwith a dynamics in classical systems was introduced in [22]. It hasbeen applied to several dynamical maps such logistic map, Baker'stransformation and Tinkerbel' map with succesful explainations oftheir chaotic characters[23, 13]. This chaos degree has several meritscompared with usual measures such as Lyapunovexponent.

Here we discuss the quantum version of the classical chaos degree,which is defined by quantum entropies in Section 2, and we call thequantum chaos degree the entropic quantum chaos degree. In orderto contain both classical and quantum cases, we define the entropicchaos degree in C*-algebraic terninology. This setting will not be

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used in the sequel application, but for mathematical completeness wefirst discuss the C*-algebraic setting.

Let (A, 6) be an input C* system and (A, 6) be an output C*system; namely, A is a C* algebra with unit I and 6 is the set of allstates on A. We assume A = A for simplicity. For a weak* compactconvex subset S (called the reference space) of 6, take a state c.p fromthe set S and let

'P = 1wdJ.t",

be an extremal orthogonal decomposition of c.p in S, which describesthe degree of mixture of c.p in the reference space S. The measure J-Lcpis not uniquely determined unless S is the Schoque simplex, so thatthe set of all such measures is denoted by Mcp (S) . The entropic chaosdegree with respect to c.p E S and a channel A* is defined by

d ('P; A*) =inf {1 SS (A*'P) dJ.t",; J.t", E M", (S)} (3.1)

where SS (A*c.p) is the mixing entropy of a state c.p in the referencespace S [24]. When S =6, VS (c.p; A*) is simply written asD (c.p; A*) .This DS (c.p; A*) contains the classical chaos degree and the quantumone.

The classical entropic chaos degree is the case that A is abelian andc.p is the probability disribution of an orbit generated by a dynamics(channel) A*; c.p = ~kPk8k, where 8k is the delta measure such as

6k (j) =-= {~ ~~ =F;\ .Then the claasical entropic chaos degree is

Dc (c.p; A*) = LPkS(A*8k).k

We explain the entropic chaos degree of a quantum system de­scribed by a density operator. Let F* be a channel sending a state toa state and p(no) be an intial state. After time no, the state is F*np(no) ,

whose Schatten decomposition is denoted by L:k A~no) E~no). Then wedefine a channel A:n on ®f11, by

A':na = F*a ® ... ® F*ma , a E 6(11,),

from which the entropic chaos degree (3.1) for the channels F* andA:n are written as

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Dq (p(no); F*) - inf {~A~no)S (F*Ekno ») ;{Ekno )} } ,

Dq (p(no); 1\;,.) = inf { ~ ~ A~no)S ( A;"Ekno») ;{Ekno )} } ,

where the infimum is taken over all Schatten decompositions of F*no p.We can judge whether the dynamics F* causes a chaos or not by

the value of D as

D > 0 and not constant ¢:=;> chaotic,

D - constant ¢:=;> quasi-chaotic,

DO¢:=;> stable.

The classical version of this degree was applied to study the chaoticbehaviors of several nonlinear dynamics [13, 20]. The quantum en­tropic chaos degree is applied to the analysis of quantum spin system[12]and quantum Baker's type transformation[15] , and we could measurethe chaos of these systems. The information theoretical meaning ofthis degree was explained in [22, 23].

4. QUANTUM CHAOS DEGREE

In this section, we apply it to study the appearance of chaos inquantum spin systems[12]. The chaos degree defined in the previoussection has following properties.

THEOREM 4.1. For any p = L:k AkEk E (5 (1-l), F* : (5 (1-l) ~(5 (1-l) , and A:n : (5 (&l{"1-l) ~ (5 (®i1-l) , the following statementshold.

(1) Let Ut be a unitary operator satisfying Ut = exp (itH) for anyt E R.

If F* p = AdUt (p) =UtpUt , then Dq (p; F*) = Dq (p; A:n) = O.(2) Let a be a fixed state on 1-l. If F*p = a, then Dq (p; F*) =

D q (p; A':n) = S (a).(3) Let Abe a fixed positive real number. If F*p = e-Ap+ (1 - e-A) a,

then limA_ooDq (p; F*) = limA_ oo Dq (p; A:n) = S (a) .(4) Let {Pn } be the positive operated measure and F* p = L:k PkPPk.

Then Dq (p; F*i) = Dq (p; A:n) = constant for any j, mEN,that is, F* is quasi-chaotic. If [Pk , p] = 0, then D q (p; F*i) =Dq (p; A:n) = 0, that is, F* is stable.

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The proof of this theorem is given in [12]. We will apply the quan­tum entropic chaos degree to spin 1/2 system. See [12] again for thedetails.

Let X = (Xl, X2, X3) be a vector in R,3 satisfying IIXII = JE~=1 x~< 1 and I be the identity 2 x 2 matrix. Any state p in a spin 1/2system is expressed as

p=HI+O'oX),where jJ = (a1' a2, a3) is the Pauli spin matrix vector;

CTI = (~ ~), CT2 = (~ ~i), CT3 = (~ ~1 ) .

Let f be a non-linear map from R 3 to R 3 satisfying II f (X) II < 1

for any X E R 3 with IIXII < 1. A channel F* is defined by

for any state p.We now define Baker's type map and see whether this map produces

the chaos. For any vector X = (Xl, X2, X3) on R 3 , we consider thefollowing map f : R3 ~ R3 :

where,

( 2a (Xl + ~) - ~ !a (X2 + ~) - ~ 0)~ ~'2 ~~'

( (1 ) 1 1 ( 1) 1 1)'2a Xl + - - v12a - - -a X2 + - + -a - - 0~ ~'2 ~ ~ ~'

Whenever~ < IX11 < 1 (resp.~ < IX21 < 1), we replace xlwith

o(resp.x2 = 0).The entropic chaos degree D (p(no); F*)and D (p(no); A:n) can be

computed [12], and the result of D (p(no); A:n) is shown in Fig. 4.1 for

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an initial value X = (0.3,0.3,0.3). We took 740 different a's betweeno and 1 with m = 1000, no = 2000.

0.5 ---------------r.--0.4 _

03 ......._

0.2 _

OJ ----------+'t----------OL..----L_--'-_...._ ......=-a._.......L._-&-_...I....----lI...----I0.0 0.1 0.2 03 0.4 O.s 0.6 0.7 O.S 0.9 a

. v( (,,> All< )Ftgure 4.1: The change of p ~ til w.r.t. a

The result shows that the quantum dynamica constructed by Bak­er's type transformation is stable in 0 < a < 0.5 and chaotic in 0.5< a < 1.0. Though there are several approaches to study chaoticbehaviors of quantum systems, we used a new quantity to measurethe degree of chaos for a quantum system. Our chaos degree has thefollowing merits:(l) once the channel A*, describing the dynamics ofa quantum system, is given, it is easy to compute this degree numer­ically; (2) the argorithm computing the degree is easily set for anyquantum state.

5. QUANTUM TELEPORTATION

Quantum teleportation has been introduced by Benett et al. [3] anddiscussed by a number of authors in the framework of the singlet state[4]. Recently, a rigorous formulation of the teleportation problem ofarbitrary quantum states by means of quantum channel was givenin [13] based on the general channel theoretical formulation of thequantum information theory. Here we report some results by rigrousstudies of the teleportation processes in [2, 9, 10].

The following is a generalization of the channel theoretical approachto the teleportation problem proposed by [13]:

Step 0:: A girl named Alice has an unknown quantum state pon (a N-dimensional) Hilbert space 'Hl and she was asked toteleport it to a boy named Bob.

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Step 1:: For this purpose, we need two other Hilbert spaces 1i2and 1i3, 1i2 is attached to Alice and 1i3 is attached to Bob. Pre­arrange a so-called entangled state u on 1i2® 1i3 having certaincorrelations and prepare an ensemble of the combined system inthe state p ® u on 1i1® 1i2® 1-£3.

Step 2:: One then fixes a family of mutually orthogonal projec­tions (Fnm)~m=1 on the Hilbert space 1i1® 1i2 corresponding toan observable F := L: zn,mFn,m, and for a fixed one pair of in-

n,mdices n, m, Alice performs a first kind incomplete measurement,involving only the 1i1® 1i2 part of the system in the state p ® u,which filters the value Znm, that is, after measurement on thegiven ensemble p ® u of identically prepared systems, only thosewhere F shows the value Znm are allowed to pass. Accordingto the von Neumann rule, after Alice's measurement, the statebecomes

(123)._ (Fnm ® l)p ® u(Fnm ® 1)Pnm .- tr123(Fnm ® l)p ® u(Fnm ® 1)

where tr123 is the full trace on the Hilbert space 1i1 ® 1i2 ® 1i3.Step 3:: Bob is informed which measurement was done by Alice.

This is equivalent to transmit the information that the eigenvalueZnm was detected. This information is transmitted from Alice toBob without disturbance and by means of classical tools.

Step 4:: Making only partial measurements on the third part onthe system in the state p~~3) means that Bob will control a stateAnm(p) on 1i3 given by the partial trace on 1i1® 1i2 of the statep~;,3) (after Alice's measurement)

Anm(p) tr12 p~~3)

(Fnm ® l)p ® u(Fnm ® 1)- tr12 --'--------'----"-----'---tr123(Fnm ® l)p ® u(Fnm ® 1)

Thus the whole teleportation scheme given by the family (Fnm)and the entangled state u can be characterized by the family(Anm ) of channels from the set of states on 1i1 into the set ofstates on 1i3 and the family (Pnm) given by

Pnm(P) := tr123(Fnm ® l)p ® u(Fnm ® 1)

of the probabilities that Alice's measurement according to theobservable F will show the value Znm.

The teleportation scheme works perfectly with respect to a certainclass 6 of states P on 1i1 if the following conditions are fulfilled.

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(El): For each n, m there exists a unitary operator Unm : 11,1 ~ 11,3such that

Anm(p) = Unm P U~m (p E 6)

(E2):

LPnm(P) = 1 (p E 6)nm

(El) means that Bob can reconstruct the original state P by uni­tary keys {Unm} provided to him.

(E2) means that Bob will succeed to find a proper key with cer­tainty.

Such a teleportation process can be classified into two cases [2],where we discussed to find the solutions of the teleportation in eachcase and the conditions of the uniqueness of unitary key. Along thepaper [2], we here report the results constructing more realistic tele­portation models.

In the papers [3,4]' the authors used EPR spin pair to construct ateleportation model. In order to have a more handy model, we hereuse coherent states to construct a model. One of the main points forsuch a construction is how to prepare the entangled state. The EPRentangled state used in [3] can be identified with the splitting of aone particle state, so that the teleportation model of Bennett et al.can be described in terms of Fock spaces and splittings, which makesus possible to work the whole teleportation process in general beamsplitting scheme. Moreover to work with beams having a fixed num­ber of particles seems to be not realistic, especially in the case of largedistance between Alice and Bob, because we have to take into accountthat the beams will lose particles (or energy). For that reason oneshould use a class of beams being insensitive to this loss of particles.That and other arguments lead to superpositions of coherent beams.

In this report, we discuss the construction, given in [9], of a telepor­tation model being perfect in the sense of conditions (El) and (E2),where we take the Boson Fock space r(L2(G)) := 11,1 = 11,2 = 11,3with a certain class p of states on this Fock space. Then we con­sider a teleportation model where the entangled state (7 is given bythe splitting of a superposition of certain coherent states. Unfortu­nately this model doesn't work perfectly, that is, neither (E2) nor (El)hold. However this model is more realistic than the perfect model,

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and we show that this model provides a nice approximation to be per­fect. To estimate the difference between the perfect teleportation andnon-perfect teleportation, we add a further step in the teleportationscheme:

Step 5:: Bob will perform a measurement on his part of the sys­tem according to the projection

F+ := 1 -lexp(O) >< exp(O) I

where lexp(O) >< exp(O) I denotes the vacuum state (the coher­ent state with density 0).

Then our new teleportation channels (we denote it again by Anm )have the form

A ()(Fnm ® F+)p ® a(Fnm ® F+)nm P := tr12 --=------.,;...-=-------.:.-------:.-

tr123(Fnm ® F+)p ® a(Fnm ® F+)

and the corresponding probabilities are

Pnm(P) := tr123(Fnm ® F+) P® a(Fnm ® F+)

For this teleportation scheme, (E1) is fulfilled. Furthermore we get

(1 - e-~)2LPnm(P) = 1 + (N -l)e-d (--> 1 (d --> +00))nm

Here N denotes the dimension of the Hilbert space and d is the expec­tation value of the total number of particles (or energy) of the beam,so that in the case of high density (or energy) "d ~ +00" of the beamthe model works perfectly. Mathematical details of the above resultsare given in [9].

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DEPARTMENT OF INFORMATION SCIENCES, SCIENCE UNIVERSITY OF TOKYO,278-8510, NODA CITY, CHIBA, JAPAN

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