8.4.2 Quantum process tomography8.5 Limitations of the quantum

operations formalism

量子輪講2003 年 10 月 16 日担当：徳本　晋tokumoto@is.s.u-tokyo.ac.jp

Motivations

How do quantum operations relate to experimentally measurable quantities?

What measurements should an experimentalist do if they wish to characterize the dynamics of a quantum system?

(Classical) System identification(Quantum) Quantum process tomography

Quantum state tomography

The procedure of experimentally determining an unknown quantum state.

Ex. Distinguish non-orthogonal quantumstates like and with certainty.

it is possible to estimate ρ if we have a large number of copies of ρ.

0 2/)10(

Case of a single qubit

Suppose we have many copies of a single qubit density matrix ρ. The set forms an orthonormal set of matrices with respect to the Hilbert-Schmidt inner product,

so ρ may be expanded as

Expressions like tr(Aρ) have an interpretation as the average value of observables.

2 , 2 , 2 , 2I X Y Z

†( , ) tr( )A B A B

tr( ) tr( ) tr( ) tr( )

2

I X X Y Y Z Z

Average value of observations

To estimate tr(Zρ) we measure the observable Z a large number of times, m, obtaining outcomes

. Empirical average of these quantities, , is an estimate for the true value of tr(Zρ).

It becomes approximately Gaussian with mean equal to tr(Zρ) and with standard deviation

In similar way we can estimate tr(Xρ), tr(Yρ).

1 2, , , { 1, 1}mz z z /ii

z m

Central limit theorem

( )Z m

Case of more than one qubit

Similar to the single qubit case, an arbitrary density matrix on n qubits can be expanded as

where the sum is over vectors with entries chosen from the set 0,1,2,3.

1 2 1 2

n

tr( )

2n nv v v v v v

v

1( , , )nv v v

iv

How can we use quantum state tomography to do quantum process tomography?

dimension: dChose d2 pure quantum state so that the corresponding density matricesform a basis set. For each state we prepare andoutput from process. We use quantum state tomography to determine the state .Since the quantum operation ε is now determined by a linear extension of ε to all state, we are now done.

21 , ,d

2 21 1 , ,d d

j j j j j

j j

Way of determining a useful representation of ε

Our goal is to determine a set of operation elements {Ei} for ε,

To determine the Ei from measurable parameters, we consider an equivalent description of ε using a fixed set of operators , for some set of complex numbers eim.So , where .

†( ) i ii

E E

iE i im mm

E e E

†( ) m n mnmn

E E mn im ini

e e Entries of a positive Hermitian matrix χ

chi matrix representation

chi matrix representation

χ will contain d4-d2 independent real parameters, because a general linear map of d by d matrices to d by d matrices is described by d4 independent parameters, but there are d2 additional constraints due to the fact that ρ remains Hermitian with trace one.

Any d×d matrix can be written as a linear combination of the basis

: fixed, linearly independent basis for the space of d×d matrices.

any d×d matrix can be written as a unique linear combination of the .Input:

Thus, it is possible to determine by state tomography.

2 (1 )j j d

j

, , 2 , 2n m n m n i m

1 1

2 2

i in m i n n m m

Linear combination of the basis( )j

Calculating and

Each may be expressed as a linear combination of the basis states,

and since is known from the state tomography, can be determined by standard linear algebra algorithms. To proceed, we may write

where are complex numbers which can be determined by standard algorithms from linear algebra.

( )j

( )j jk kk

( )j

jk

† mnm j n jk k

k

E E mnjk

jk mnjk

Combining equations

Combining the last two expressions and (8.152) we have

From the linear independence of the it follows that each k,

This relation is a necessary and sufficient condition for the matrix χ to give the correct quantum operation ε.

mnmn jk k jk k

k mn k

k

mnjk mn jk

mn

Calculating χ

One may think of χ and λ as vectors, and β as a d4×d4 matrix with columns indexed by mn, and rows by jk. To show how χ may be obtained, let κ be the generalized inverse for the matrix β, satisfying relation

Most computer packages for matrix manipulation are capable of finding such generalized inverses. We now prove that χ defined by

satisfies the relation (8.158).

,

mn st xy mnjk jk st xy

st xy

mnmn jk jk

jk

Calculating Ei

Let the unitary matrix diagonalize χ,

From this it can easily be verified that

are operation elements for ε.

†U* .mn mx x xy ny

xy

U d U

i i ji jj

E d U E

Process tomography for a single qubit (1)

We use

input states:output states:

These correspond to , where

We may determine β, and similarly determines λ.

0 1 2 3, , ,E I E X E iY E Z

0 , 1 , 0 1 2 , 0 1 2i

1

4

2 1 4

3 1 4

0 0

1 1

(1 )( ) 2

(1 )( ) 2

i i

i i

j

( )j j

1 2 3 40 0 , 1 0 , 0 1 , 1 1

Process tomography for a single qubit (2)

However, due to the particular choice of basis, and the Pauli matrix representation of , we may express the β matrix the Kronecker product , where

so that χ may be expressed conveniently as

in terms of block matrices.

iE

1

2

I X

X I

1 2

3 4

8.5 Limitations of the quantum operations formalism

Motivation: Are there interesting quantum systems whose dynamics are not described by quantum operations?

In this section, we will construct an artificial example of a system whose

evolution is not described by a quantum operation, and try to understand the circumstances under which this is

likely to occur.

an artificial example of a system whose evolution is not described by a quantum operation

A single qubit ρ is prepared in some unknown quantum state. The preparation of this qubit involves certain procedures to be carried out in the laboratory in which the qubit prepared. The state of the system after preparation is

if ρ is a state on the bottom half of the Bloch sphere, and

if ρ is a state on the top half of the Bloch sphere.This process is not an affine map acting on the Bloch sphere, and therefore it cannot be a quantum operation.

0 0 other degrees of freedom

1 1 other degrees of freedom

the circumstances under which this is likely to occur

Quantum system which interacts with the degrees of freedom used to prepare that system after the preparation is complete will in general suffer a dynamics which is not adequately described within the quantum operations formalism.

Summary of Chapter 8

The operator-sum representation Environmental models for quantum operations Quantum process tomography Operation elements for important single qubit

quantum operations