Download 8.4.2 Quantum process tomography 8.5 Limitations of the quantum

8.4.2 Quantum process tomography
8.5 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 quantum
states like 0 and ( 0  1 ) / 2 with certainty.
it is possible to estimate ρ if we have a large
number of copies of ρ.
Case of a single qubit
Suppose we have many copies of a single qubit
density matrix ρ. The set I 2 , X 2 , Y 2 , Z 2
forms an orthonormal set of matrices with respect
to the Hilbert-Schmidt inner product,
( A, B)  tr( A† B)
so ρ may be expanded as
tr(  ) I  tr( X  ) X  tr(Y  )Y  tr( Z  ) Z

2
Expressions like tr(Aρ) have an interpretation as
the average value of observables.
Average value of observations
To estimate tr(Zρ) we measure the observable Z
a large number of times, m, obtaining outcomes
z1 , z2 , , zm {1, 1} . Empirical average of these
quantities, i zi / m , is an estimate for the true
value of tr(Zρ).
Central limit theorem
It becomes approximately Gaussian with mean
equal to tr(Zρ) and with standard deviation ( Z )
In similar way we can estimate tr(Xρ), tr(Yρ).
m
Case of more than one qubit
Similar to the single qubit case, an arbitrary density
matrix on n qubits can be expanded as
tr( v   v    v  ) v   v    v
 
n
1
v
2
n
2
1
2
n
where the sum is over vectors v  (v1 , , vn ) with
entries vi chosen from the set 0,1,2,3.
How can we use quantum state tomography
to do quantum process tomography?
dimension: d
Chose d2 pure quantum state  1 , ,  d so that the
corresponding density matrices  1  1 , ,  d  d
form a basis set.
For each state  j we prepare  j  j and    j  j 
output from process. We use quantum state
tomography to determine the state    j  j  .
Since the quantum operation ε is now determined by a
linear extension of ε to all state, we are now done.
2
2
2
Way of determining a useful
representation of ε
Our goal is to determine a set of operation elements
{Ei} for ε,  (  )   Ei  Ei†
i
To determine the Ei from measurable parameters,
we consider an equivalent description of ε using a
fixed set of operators Ei , Ei   eim Em for some set of
m
complex numbers eim.

†


e
e

(

)

E

E

 m n mn , where mn i im in .
So
mn
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
 j (1  j  d 2 ) : 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  j .
Input: n , m ,    n  m  2 ,    n  i m 
  n m         i     
2
1 i
1 i
  n n 
 m m
2
2
Linear combination of the basis
Thus, it is possible to determine  (  j ) by state
tomography.

mn


Calculating jk and jk
Each  (  j ) may be expressed as a linear combination
of the basis states,
 (  j )    jk  k
k
and since  (  j ) is known from the state tomography,
 jk can be determined by standard linear algebra
algorithms. To proceed, we may write
Em  j En†    jkmn k
k
where  are complex numbers which can be
determined by standard algorithms from linear
algebra.
mn
jk
Combining equations
Combining the last two expressions and (8.152) we have
mn


 mn jk k    jk k
k
mn
k
From the linear independence of the  k it follows that
each k,
mn

jk
 mn   jk
mn
This relation is a necessary and sufficient condition for
the matrix χ to give the correct quantum operation ε.
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
 jkmn    jkst  stxy  xymn
st , xy
Most computer packages for matrix manipulation are
capable of finding such generalized inverses. We now
prove that χ defined by
 mn    mn
jk  jk
jk
satisfies the relation (8.158).
Calculating Ei
Let the unitary matrix U † diagonalize χ,
 mn  U mx d x xyU ny* .
xy
From this it can easily be verified that
Ei  di U ji E j
j
are operation elements for ε.
Process tomography for a single qubit (1)
We use E0  I , E1  X , E2  iY , E3  Z
input states: 0 , 1 ,    0  1  2 ,    0  i 1
output states: 1    0 0 

 4    1 1 
 2        i      (1  i )( 1   4 ) 2
3        i      (1  i )( 1   4 ) 2
These correspond to  j   (  j ) , where
1  0 0 , 2  1 0 , 3  0 1 , 4  1 1
We may determine β, and similarly  j determines λ.
2
Process tomography for a single qubit (2)
However, due to the particular choice of basis, and
the Pauli matrix representation of Ei , we may express
the β matrix the Kronecker product      , where
1I
 
2 X
X
 I 
so that χ may be expressed conveniently as
 1 2 
  




 3  4 
in terms of block matrices.
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
  0 0  other degrees of freedom
if ρ is a state on the bottom half of the Bloch sphere, and
  1 1  other degrees of freedom
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.
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

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The operator-sum representation
Environmental models for quantum operations
Quantum process tomography
Operation elements for important single qubit
quantum operations