Optimal Bounds on Functions of Quantum States under Quantum Channels

0.18 MB·

by  Chi-Kwong Li

Optimal Bounds on Functions of Quantum States
under Quantum Channels
Chi-Kwong Li ∗1, Diane Christine Pelejo †1,2, and Kuo-Zhong Wang ‡3
1Department of Mathematics, College of William and Mary
2Department of Applied Science,College of William and Mary
6 3Department of Mathematics, National Chiao Tung University
1
0
2

r
p
A Abstract

1 Let(σ1,σ2) D(σ1,σ2)be a scalarfunctionsuchasthe tracedistance,the fidelity, andthe
1 7→
relative entropy, etc. For two given quantum states ρ1 and ρ2, we determine optimal bounds

] for D(ρ1,Φ(ρ2)) for Φ for different classes of functions D(, ), where is the set of unitary
h ∈S · · S
quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and
p
- the set of all quantum channels. Furthermore, we determine states σ = Φ(ρ2) that attain the
t
n optimal values and discuss the uniqueness of such states.
a
u Keywords: Quantumstates,densitymatrices,quantumchannels,unitalchannels,mixedunitary
q
channels, fidelity, relative entropy, majorization, eigenvalues.
[

AMS Classification. 15A18, 15A42, 81P40
2
v
7
1 Introduction
2
7
6
In quantum sciences research, one often compares a pair of quantum states ρ ,ρ by considering
0 1 2
. some scalar functions D(ρ ,ρ ). For instance, in quantum information and quantum control, one
1 1 2
0 would like to measure the ‘distance’ between a state ρ1 and another state ρ2 which go through a
6
quantum channel or a quantum operation Φ. Suppose quantum states are represented as n n
1 ×
: density matrices, i.e., positive semi-definite matrices with trace one. The following measures are
v
often used [1, 2]:
i
X 1
(tr ρ ρ 2)1/2, tr ρ ρ , √2 1 tr √ρ √ρ , (1)
ar | 1− 2| 2 | 1− 2| q − | 1 2|
which are known as the Hilbert-Schmidt (HS) distance, the trace distance and the Bures distance,
respectively. Here ρ is the positive semidefinite square root of ρ†ρ. In particular, the Bures
| |
distance is a function of the fidelity [3]
F(ρ ,ρ ) = tr √ρ √ρ . (2)
1 2 1 2
| |
The purpose of this paper is to study the following.
∗[email protected]
†[email protected]
‡[email protected]
1
Problem 1.1 Let D be a scalar function on a pair of quantum states. Suppose ρ ,ρ are two
1 2
quantum states and is a set of quantum channels. Determine the optimal bounds for D(ρ ,Φ(ρ ))
1 2
S
for Φ , and also the states σ = Φ(ρ ) attaining the optimal bounds.
2
∈ S
These optimal bounds provide insight on the geometry of certain sets of quantum states [4, 5]
and play an important role in quantum state discrimination [6, 7, 8]. Physically, if a quantum state
ρ goes through some quantum channel Φ, one would like to know D(ρ ,Φ(ρ )) for another fixed
2 1 2
quantum state ρ . If Φ is under our control, a solution to this problem can help us select Φ to
1
attain the maximum or minimum value for D(ρ ,Φ(ρ )). On the other hand, if we only know that
1 2
Φ lies in a certain class of quantum channels, then thesolution will tell us therange of values where
D(ρ ,Φ(ρ )) lies.
1 2
Denote by , , the set of n n complex matrices, the set of n n Hermitian matrices,
n n n
M H D × ×
and the set of n n density matrices, respectively. Then quantum channels are trace preserving
×
completely positive map Φ : with the operator sum representation
n n
M → M
r
†
Φ(X) = F XF for all X , (3)
j j ∈ Mn
Xj=1
where F ,...,F M satisfy r F†F = I . The map Φ is a unitary channel if r = 1 and F
1 r ∈ n j=1 j j n 1
is unitary; it is a mixed unitaryPchannel if every F is a multiple of a unitary matrix; it is unital if
j
Φ(I ) = I .
n n
In the next two sections, we will obtain results for two general classes of functions D(, ). The
· ·
first type of functions will cover the Hilbert-Schmidt (HS) distance and the trace distance. The
second type will cover the fidelity, the Bures distance, and also the relative entropy defined by
S(ρ ρ ) = trρ (logρ logρ ). (4)
1 2 1 1 2
|| −
For each class of functions, we will give the complete solution of Problem 1.1 when is the set
S
of unitary quantum channels, the set of mixed unitary channels and the set of unital quantum
channels. These will be done in the next two sections. We also consider the set of all quantum
channels and obtain a complete answer for the first class of functions, and partial results for the
secondclassoffunctions. Someconcludingremarksandfutureresearchdirectionswillbementioned
in Section 4.
In our discussion, we will let E ,E ,...,E denote the standard basis for . By the
11 12 nn n
{ } M
following result [9, Theorem 3.6], the solutions of Problem 1.1 are the same for the set of mixed
unitary channels and the set of unital channels.
Lemma 1.2 Let ρ,σ . The following are equivalent.
n
∈D
1. There exists a mixed unitary quantum channel Φ such that Φ(ρ) = σ.
2. There exists a unital quantum channel Φ such that Φ(ρ) = σ.
3. λ(σ) λ(ρ).
≺
4. There are unitary U ,...,U M such that σ = 1(U†ρU + +U†ρU ).
1 n ∈ n n 1 1 ··· n n
2
2 Schur-Convex Functions
For p 1, define the Schatten p-norm of a Hermitian matrix A by
≥
A = (tr Ap)1/p. (5)
p
k k | |
It is not hard to see that if A has eigenvalues a ,...,a , then
1 n
n
A = ℓ (a ,...,a ) = a p 1/p. (6)
p p 1 n j
k k | |
(cid:8)Xj=1 (cid:9)
The Hilbert Schmidt norm is and, up to a multiple, the trace norm is .
2 1
k·k k·k
In [4, Theorem 4], the authors observed that
max ρ Uρ U† = Λ↓(ρ ) Λ↑(ρ ) , (7)
1 2 1 1 2 1
U is unitary k − k k − k
and
min ρ Uρ U† = Λ↓(ρ ) Λ↓(ρ ) , (8)
1 2 1 1 2 1
U is unitary k − k k − k
where Λ↓(X) (respectively, Λ↑(X)) denotes the diagonal matrix having the eigenvalues of X as
diagonal entries arranged in descending order (respectively, ascending order).
Actually, the same result holds if one replaces by any unitary similarity invariant (u.s.i.)
1
k·k
norm , i.e., a norm satisfying UXU† = X for all X and unitary U . To
n n
k·k k·k k k k k ∈ H ∈ M
describethefullgeneralizationoftheresult,weneedthenotionofmajorizationandSchur-convexity.
One may see the excellent monograph [10] for the background. We give the basic definition in the
following to facilitate our discussion.
Recall that for x,y Rn, we say that x is majorized by y, denoted by x y if the sum of the
∈ ≺
k largest entries of x is not larger than that of y for j = 1,...,n 1, and the sum of the entries of
−
x equals to that of y. A function f : Rn R is Schur-convex if f(x) f(y) whenever x y. It is
→ ≤ ≺
strictly Schur-convex if f(x) < f(y) whenever x y and x = y.
≺ 6
Denote by λ(X) = (λ (X),...,λ (X)) the vector of eigenvalues of X arranged in de-
1 n n
∈ H
scending order. In particular, if X , then λ(X) is a vector in the set
n
∈ D
Ω = (x ,...,x ): x x 0, x + +x = 1 . (9)
n 1 n 1 n 1 n
{ ≥ ··· ≥ ≥ ··· }
We have the following.
Theorem 2.1 Suppose the function D : R is defined by D(σ ,σ )= d(λ(σ σ )) for
n n 1 2 1 2
D ×D → −
a Schur-convex function d: Rn R. Then
→
max D(ρ ,Uρ U†) = D(Λ↓(ρ ),Λ↑(ρ )), (10)
1 2 1 2
U is unitary
and
min D(ρ ,Uρ U†) = D(Λ↓(ρ ),Λ↓(ρ )). (11)
1 2 1 2
U is unitary
The maximum is attained at Uρ U† if there exists a unitary V such that Vρ V† = Λ↓(ρ ) and
2 1 1
VUρ U†V† = Λ↑(ρ ). The maximum is attained at Uρ U† if there exists a unitary V such that
2 2 2
VUρ U†V† = Λ↓(ρ ). The converses of the two preceding statements are also true if d is strictly
2 2
Schur-convex.
3
Theorem 2.1 provides a complete solution to Problem 1.1 for the the set of unitary channels
S
if D(σ ,σ )= d(λ(σ σ )) for a Schur-convex function d(). In particular, it provides information
1 2 1 2
− ·
about the state σ = Φ(ρ ) that attains the maximum and minimum values. For example, take
2
ρ = diag(.55,.45,0) and ρ = diag(.35,.33,.32), and let be any u.s.i. norm. Since all u.s.i.
1 2
k·k
norms are Schur-convex, then ρ ρ and ρ diag(.32,.33,.35) will yield the minimum and
1 2 1
k − k k − k
maximum values in the set ρ Uρ U† : U unitary . Furthermore, if we choose a norm
1 2
{k − k } ||·||
that corresponds to a strictly Schur-convex function such as the Schatten p-norm for p (1, ),
∈ ∞
then the lower bound and upper bound can only occur at the matrices ρ and diag(.32,.33,.35),
2
respectively. On the other hand, for the Schatten 1-norm, i.e., the trace norm, the minimum may
occur at other matrices such as Uρ U† = diag(.33,.35,.32) and the maximum may occur at other
2
matrices such as Uρ U† = diag(.33,.32,.35). Another situation where the optimal is attained by
2
multiple states may arise when ρ has repeated eigenvalues. For example, if ρ = 1I , then for any
1 n n
Φ , Φ(ρ ) attains the maximum/minimum.
2
∈ S
Next, we turn to Problem 1.1 for the set of mixed unitary channels and unital channels. By
S
Lemma 1.2, and the results in [11], we have the following solution of Problem 1.1 if D(σ ,σ ) =
1 2
d(λ(σ σ )) for a Schur-convex function d() and is the set of mixed unitary channels or the
1 2
− · S
set of unital channels. Furthermore, as shown in Lemma 1.2, we can always construct the mixed
unitary channel of the form
1
σ (U ρU†+ +U ρU†) (12)
7→ n 1 1 ··· n n
for some unitary U ,...,U M .
1 n n
∈
Theorem 2.2 Suppose the function D : R is defined by D(σ ,σ ) = d(λ(σ σ ))
n n 1 2 1 2
D ×D → −
for a Schur-convex function d : Rn R. Let be the set of mixed unitary channels or the set of
→ S
unital channels acting on . Then
n
M
maxD(ρ ,Φ(ρ )) = D(Λ↓(ρ ),Λ↑(ρ )) (13)
1 2 1 2
Φ∈S
and
n
minD(ρ1,Φ(ρ2)) = DΛ↓(ρ1), djEjj, (14)
Φ∈S
Xj=1
 
where (d ,...,d ) is determined by the following algorithm:
1 n
Step 0. Set (∆ ,...,∆ ) = λ(ρ ) λ(ρ ).
1 n 1 2
−
Step 1. If ∆ ∆ , then set (d ,...,d ) =λ(ρ ) (∆ ,...,∆ ) and stop.
1 n 1 n 1 1 n
≥ ··· ≥ −
Else, go to Step 2.
Step 2. Let 1 j < k ℓ n be such that
≤ ≤ ≤
∆ ∆ > ∆ = = ∆ < ∆ = = ∆ = ∆ .
1 j−1 j k−1 k ℓ ℓ+1
≥ ··· ≥ ··· ··· 6
Replace each ∆ ,...,∆ by (∆ + +∆ )/(ℓ j +1), and go to Step 1.
j ℓ j ℓ
··· −
The maximum is attained at Φ if there exists a unitary V satisfying Vρ V† = Λ↓(ρ ) and
1 1
∈ S
VΦ(ρ )V† = Λ↑(ρ ). The minimum is attained at Φ if there exists a unitary V satisfying
2 2
∈ S
Vρ V† = Λ↓(ρ ) and VΦ(ρ )V† = ( n d E ). The converses of the above two statements also
1 1 2 j=1 j jj
hold if d is strictly Schur-convex. P
Here is an example illustrating the construction of the vectors (d ,...,d ) in the theorem.
1 n
4
Example 2.3 Let ρ = 1 diag(4,3,3,0) and ρ = 1 diag(5,2,2,1).
1 10 2 10
Apply Step 0. Set (∆ ,...,∆ ) = 1 (4,3,3,0) 1 (5,2,2,1) = 1 ( 1,1,1, 1).
1 4 10 − 10 10 − −
Apply Step 2. Change (∆ ,...,∆ ) to 1 (1/3,1/3,1/3, 1).
1 4 10 −
Apply Step 1. Set (d ,...,d ) = 1 (4,3,3,0) 1 (1/3,1/3,1/3, 1) = 1 (11,8,8,3).
1 4 10 − 10 − 30
Finally, we consider the set of all quantum channels. It is known that for any two quantum
S
states, there is a quantum channel sending the first one to the second one. We have the following.
Theorem 2.4 Suppose the function D : R is defined by D(σ ,σ ) = d(λ(σ σ ))
n n 1 2 1 2
D ×D → −
for a Schur-convex function d :Rn R. Let be the set of all quantum channels acting on .
n
→ S M
Then
maxD(ρ ,Φ(ρ )) = D(Λ↓(ρ ),E ) and minD(ρ ,Φ(ρ )) = D(ρ ,ρ ). (15)
1 2 1 nn 1 2 1 1
Φ∈S Φ∈S
The minimum is attained at Φ if Φ(ρ ) = ρ . The maximum is attained at Φ if there
2 1
∈ S ∈ S
exists a unitary V satisfying Vρ V† = Λ↓(ρ ) and VΦ(ρ )V† = E . If, in addition, d is strictly
1 1 2 nn
Schur-convex, then the converses of the two preceding statements are also true.
Proof. The conclusion on the minimum is clear. For the maximum, note that for any σ ,
n
∈ D
k k k k k k
λ (ρ σ) λ (ρ )+ λ ( σ) λ (ρ )+ λ ( E )= λ (Λ↓(ρ ) E )
j 1 j 1 j j 1 j nn j 1 nn
− ≤ − ≤ − −
Xj=1 Xj=1 Xj=1 Xj=1 Xj=1 Xj=1
for j = 1,...,n 1, and n λ (ρ σ) = 0. Because d() is Schur-convex, the result follows. 2
− j=1 j 1− ·
P
3 Fidelity, relative entropy, and other functions
In this section, we consider Problem 1.1 for other functions including the fidelity
1/2 1/2
F(ρ ,ρ )= tr √ρ ρ √ρ = √ρ √ρ = tr ρ ρ , (16)
1 2 (cid:18)q 2 1 2(cid:19) k 1 2k1 | 1 2 |
and the relative entropy
S(ρ σ) = tr(ρ(logρ logσ)) = tr(ρlogρ) tr(ρlogσ). (17)
|| − −
In [12, Theorem 2.1], it was shown that if is the set of unitary channels, then
S
n
maxF(ρ ,Φ(ρ )) = F(Λ↓(ρ ),Λ↓(ρ )) = λ (ρ ) λ (ρ ), (18)
1 2 1 2 j 1 j 2
Φ∈S Xj=1q q
and
n
minF(ρ ,Φ(ρ )) = F(Λ↓(ρ ),Λ↑(ρ )) = λ (ρ ) λ (ρ ). (19)
1 2 1 2 j 1 n−j+1 2
Φ∈S Xj=1q q
If is the set of unital channels, it was also shown in [5, Corollary 2.4] that the above minimum is
S
also valid, but determining the maximum is an open problem.
In the following, we consider different functions f and g on quantum states and study upper
bounds and lower bounds for a function D : R of the form
n n
D ×D →
D(ρ ,ρ ) = trf(ρ )g(Φ(ρ )) and D(ρ ,ρ )= tr f(ρ )g(Φ(ρ )) (20)
1 2 1 2 1 2 1 2
| |
5
with Φ for different sets of quantum channels. The results will cover a number of important
∈ S S
functions in quantum information research, and the techniques based on the theory of majorization
can be further extended to other functions.
To present our results, we need some more definitions and results in majorization (see [10]) to
present our general theorem.
A scalar function f : [0,1] R can be extended to f : such that f(σ) =
n n
→ D → H
U†diag(f(µ ),...,f(µ ))U if σ = U†diag(µ ,...,µ )U, where U is unitary and µ µ 0.
1 n 1 n 1 n
≥ ··· ≥ ≥
For two vectors x,y Rn, x is weakly majorized by y, denoted by x y if the sum of the k
w
∈ ≺
largest entries of x is not larger than that of y for k = 1,...,n. Furthermore, for x,y Rn with
∈
nonnegative entries, x is log majorized by y, denoted by x y if the product of the entries of x
log
≺
is the same as that of y, and the product of the k largest entries of x is not larger than that of y
for k = 1,...,n 1. It is known that x y then x y.
log w
− ≺ ≺
We have the following.
Theorem 3.1 Let f,g : [0,1] R, ρ ,ρ .
1 2 n
→ ∈ D
(a) If f(ρ ) and g(ρ ) have eigenvalues a a and b b , then
1 2 1 n 1 n
≥ ··· ≥ ≥ ··· ≥
n n
min tr(f(ρ )g(U†ρ U)) = a b , max tr(f(ρ )g(U†ρ U)) = a b . (21)
1 2 j n−j+1 1 2 j j
U unitary Xj=1 U unitary Xj=1
The minimum is attained at a unitary U if and only if there exists a unitary V such that
V†f(ρ )V = diag(a ,...,a ) and V†g(U†ρ U)V = g(V†U†ρ UV) = diag(b ,...,b ); The
1 1 n 2 2 n 1
maximumisattainedataunitaryU ifandonlyifthereexistsaunitaryV suchthatV†f(ρ )V =
1
diag(a ,...,a ) and V†g(U†ρ U)V = g(V†U†ρ UV)= diag(b ,...,b ).
1 n 2 2 1 n
(b) If f(ρ ) and g(ρ ) have singular values α α and β β , then
1 2 1 n 1 n
≥ ··· ≥ ≥ ··· ≥
n n
min tr f(ρ )g(U†ρ U) = α β , max tr f(ρ )g(U†ρ U) = α β . (22)
1 2 j n−j+1 1 2 j j
U unitary | | Xj=1 U unitary | | Xj=1
The minimum is attained at a unitary U if and only if there exists a unitary V such that
V†f(ρ )V = diag(α ,...,α ) and V†g(U†ρ U)V = g(V†U†ρ UV) = diag(β ,...,β );
1 1 n 2 2 n 1
| | | | | |
the maximum is attained at a unitary U if and only if there exists a unitary V such that
V†f(ρ )V = diag(α ,...,α ) and V†g(U†ρ U)V = g(V†U†ρ UV) = diag(β ,...,β ).
1 1 n 2 2 1 n
| | | | | |
Proof. Let f(ρ ),g(ρ ) have eigenvalues a a and b b , respectively. For any
1 2 1 n 1 n
≥ ··· ≥ ≥ ··· ≥
unitary V M satisfying V†f(ρ )V = diag(a ,...,a ), we have
n 1 1 n
∈
tr(f(ρ )g(U†ρ U)) =tr(diag(a ,...,a )V†g(U†ρ U)V) = tr(diag(a ,...,a )g(V†U†ρ UV)).
1 2 1 n 2 1 n 2
By [10, II.9 Theorem H.1.g-h], we have
n n n
a b a d a b . (23)
j n−j+1 j j j j
≤ ≤
Xj=1 Xj=1 Xj=1
Evidently, the bounds are attained if the unitary matrices U have the said properties. Assertion
(a) follows.
Next, suppose f(ρ ),g(ρ ) have singular values α α 0 and β β 0,
1 2 1 n 1 n
≥ ··· ≥ ≥ ≥ ··· ≥ ≥
respectively. Suppose f(ρ )g(U†ρ U) has singular values s ,...,s . By [10, II.9 Theorem H.1],
1 2 1 n
(α β ,...,α β ) (s ,...,s ) (α β ,...,α β ), (24)
1 n n 1 log 1 n log 1 1 n n
≺ ≺
6
and tr f(ρ )g(U†ρ U) = n s satisfies
| 1 2 | j=1 j
P
n n n
α β s α β . (25)
j n−j+1 j j j
≤ ≤
Xj=1 Xj=1 Xj=1
Suppose V M is unitary such that V†f(ρ )V = diag(ξ α ,...,ξ α ) with ξ ,...,ξ 1,1 .
n 1 1 1 n n 1 n
∈ ∈ {− }
One easily construct the unitary U M so that g(U†ρ U) attaining the lower and upper bounds.
n 2
∈
Evidently, only those unitary matrices having the said properties will yield the optimal bounds.
Assertion (b) follows. 2
If is the set of all unitary channels, then the lower bounds and upper bounds in Theorem
S
3.1 are attainable by trf(σ )g(Ψ(σ )) for some Ψ . There are no restrictions to the real valued
1 2
∈ S
functionsf andg inTheorem3.1. So,itcanbeappliedtoawidevariety ofsituations. Forexample,
if f(x) = g(x) = √x, we obtain the result for the fidelity function F(σ ,σ ) = tr f(σ )g(σ ) and
1 2 1 2
| |
conclude that for any unitary U M ,
n
∈
n n
[λ (ρ )λ (ρ )]1/2 F(ρ ,U†ρ U) [λ (ρ )λ (ρ )]1/2. (26)
j 1 n−j+1 2 1 2 j 1 j 2
≤ ≤
Xj=1 Xj=1
If f(x)= x and g(x) = log(x), then for any unitary U M ,
n
∈
n n
λ (ρ )logλ (ρ ) tr(ρ log(U†ρ U)) λ (ρ )logλ (ρ ). (27)
j 1 n−j+1 2 1 2 j 1 j 2
≤ ≤
Xj=1 Xj=1
Here we use the convention that 0log0 = 0 and alog0 = if a (0,1]. Applying this result to
−∞ ∈
S(σ σ )= trσ (logσ logσ ), we have
1 2 1 1 2
|| −
n n
λ (ρ )log(λ (ρ )/λ (ρ )) S(ρ U†ρ U) λ (ρ )(log(λ (ρ )/λ (ρ )) (28)
j 1 j 1 j 2 1 2 j 1 j 1 n−j+1 2
≤ || ≤
Xj=1 Xj=1
for any unitary U M .
n
∈
Next, we consider the set of mixed unitary channels and the set of unital channels. Given
S
ρ ,ρ , from Lemma 1.2, the following statements are true.
1 2 n
∈D
(i) For any Φ , we have λ(Φ(ρ )) λ(ρ ).
2 2
∈ S ≺
(ii) If f(ρ ) has eigenvalues a a 0, then for any (x ,...,x ) λ(ρ ), there is Φ
1 1 n 1 n 2
≥ ··· ≥ ≥ ≺ ∈ S
such that
n n
tr(f(ρ )g(Φ(ρ ))) = a g(x ), and tr f(ρ )g(Φ(ρ )) = a g(x ). (29)
1 2 j j 1 2 j j
| | | |
Xj=1 Xj=1
Hence, we have the following.
Theorem 3.2 Let f,g : [0,1] R, ρ ,ρ , and Φ be a unital channel. Suppose f(ρ ) have
1 2 n 1
→ ∈ D
eigenvalues a a , singular values α α , and ρ has eigenvalues b b .
1 n 1 n 2 1 n
≥ ··· ≥ ≥ ··· ≥ ≥··· ≥
(a) The best lower upper and upper bounds of n tr(f(ρ )g(Φ(ρ ))) equal
j=1 1 2
P
n
inf a λ (g(σ)) : σ , λ(σ) (b ,...,b ) , (30)
j n−j+1 n 1 n
∈ D ≺
(cid:8)Xj=1 (cid:9)
7
and
n
sup a λ (g(σ)) :σ , λ(σ) (b ,...,b ) , respectively. (31)
j j n 1 n
∈ D ≺
(cid:8)Xj=1 (cid:9)
Suppose the function g(x) is increasing concave. Then the infimum value n a g(b )
j=1 j n−j+1
is attainable, and a unital channel Φ will attain the infimum value if andPonly if there is a
unitary V satisfying V†f(ρ )V = diag(a ,...,a ) and V†g(Φ(ρ ))V = diag(g(b ),...,g(b )).
1 1 n 2 n 1
In particular, the infimum can be attained at a unitary channel.
(b) The best lower upper and upper bounds of n tr f(ρ )g(Φ(ρ )), respectively, are
j=1 | 1 2 |
P
n
inf α λ (g(σ)) : σ , λ(σ) (b ,...,b ) , (32)
j n−j+1 n 1 n
| | ∈ D ≺
(cid:8)Xj=1 (cid:9)
and
n
sup α λ (g(σ)) :σ , λ(σ) (b ,...,b ) . (33)
j j n 1 n
| | ∈ D ≺
(cid:8)Xj=1 (cid:9)
If the functions f(x) and g(x) have non-negative values on [0,1], then the lower and upper
bounds are the same as those in (a). If in addition that g is increasing concave, then the
infimumvalue n a g(b )isattainable, andthe infimumwilloccur atΦ(ρ )that satisfy
j=1 j n−j+1 2
the same condiPtions described in (a).
Proof. (a) We may assume that V†f(ρ )V = diag(a ,...,a ). For any Φ , we have
1 1 n
∈ S
n
a d = tr(f(ρ )g(Φ(ρ ))), (34)
j j 1 2
Xj=1
where (d ,...,d ) are the diagonal entries of V†g(Φ(ρ ))V. Hence, (d ,...,d ) is majorized by
1 n 2 1 n
λ(g(Φ(ρ ))), where λ(Φ(ρ )) are majorized by (b ,...,b ). Similar to the proof of Theorem 3.1,
2 2 1 n
n n n
a λ (g(Φ(ρ ))) a d a λ (g(Φ(ρ ))) (35)
j n−j+1 2 j j j j 2
≤ ≤
Xj=1 Xj=1 Xj=1
Hence the forms of the best lower upper and upper bounds of n tr(f(ρ )g(Φ(ρ ))) holds. If g
j=1 1 2
is increasing concave, we can apply (vi) of Table 2 in [10, 3.BP.2] to the negative of the function
ψ : (x ,...,x ) Ω n a g(x ) to show that ψ is Schur-concave. Thus the minimum
1 n ∈ n 7→ j=1 j n−j+1
occurs at (x ,...,x )= (Pb ,...,b ).
1 n 1 n
(b) Note that the singular values of g(Φ(ρ )) are γ γ , which is a rearrangement of
2 1 n
≥ ··· ≥
g(x ), g(x ), wherex ,...,x are theeigenvalues of Φ(ρ ) satisfies (x ,...,x ) (b ,...,b ).
1 n 1 n 2 1 n 1 n
| | ···| | ≺
Now, the eigenvalues of f(ρ )g(Φ(ρ )) are the singular values of f(ρ )g(Φ(ρ )), which is log
1 2 1 2
| |
majorized by (α γ ,...,α γ ) and log majorizes (α γ ,...,α γ ). Thus,
1 1 n n 1 n n 1
n n
α γ tr f(ρ )g(Φ(ρ )) α γ . (36)
j n−j+1 1 2 j j
≤ | | ≤
Xj=1 Xj=1
If f(x) has nonnegative values, then the eigenvalues of f(ρ ) are the its singular values, and the
1
same holds for g(Φ(ρ )). Thus, the results in (a) applies. 2
2
We can specialize the result to the function f(x)= x and g(x) = log(x) to conclude that
n
λ (ρ )logλ (ρ )) trρ logΦ(ρ )
j 1 n−j+1 2 1 2
≤
Xj=1
8
for any unital channel Φ, and hence
n
S(ρ Φ(ρ )) = trρ (logρ logΦ(ρ )) λ (ρ )log(λ (ρ )/λ (ρ )). (37)
1 2 1 1 2 j 1 j 1 n−j+1 2
|| − ≤
Xj=1
For the Fidelity function
F(ρ ,Φ(ρ )) = tr ρ1/2Φ(ρ )1/2 (38)
1 2 | 1 2 |
we can deduce the following result in [5]
n
minF(ρ,Φ(σ)) = F(Λ↓(ρ),Λ↑(σ)) = λ (ρ) λ (σ). (39)
i n−i+1
Φ∈S
Xi=1p p
Itwas noted in [5]that the maximum value is noteasy to determine. As shownin Theorem 3.2, the
upper boundof F(ρ ,Φ(ρ )) = tr ρ1/2Φ(ρ )1/2 is the same as the upper boundof tr(ρ1/2Φ(ρ )1/2),
1 2 | 1 2 | 1 2
and one needs to determine
n
sup λ (ρ )1/2x1/2 : x x 0, (x ,...,x ) λ(ρ ) . (40)
{ j 1 j 1 ≥ ··· ≥ n ≥ 1 n ≺ 2 }
Xj=1
By the continuity of the function f(x) = g(x) = √x and the compactness of the set =
R
(x ,...,x ) : x x 0, (x ,...,x ) λ(ρ ) , we see that supremum is attainable.
1 n 1 n 1 n 2
{ ≥ ··· ≥ ≥ ≺ }
On the other hand, the determination of the maximum depends heavily on λ(ρ ) and λ(ρ ). For
1 2
instance, if λ(ρ ) = (1/n,...,1/n), then is a singleton and F(ρ ,Φ(ρ )) = tr ρ1/2 /√n. If
2 1 2
R | |
ρ = diag(1,0,...,0), then contains all quantum states, and F(ρ ,Φ(ρ )) = 1 if Φ(ρ )= ρ . On
2 1 2 2 1
R
the other hand, if ρ = I /n, then I /n for any ρ so that F(ρ ,Φ(ρ )) = 1 for some unital
1 n n 2 1 2
∈ R
channel Φ.
In the following, we describe how to determine the unital channel Φ that gives rise to max
F(ρ ,Φ(ρ )) for given ρ ,ρ . The result actually covers a larger class of functions.
1 2 1 2 n
∈ D
Theorem 3.3 Let D : R be defined as follows.
n n
D ×D →
(a) D(σ ,σ ) = tr(f(σ )g(σ )) or D(σ ,σ ) = tr f(σ )g(σ ), where f(x) = xp and g(x) = xq
1 2 1 2 1 2 1 2
| |
with p,q > 0 such that p+q = 1, or
(b) D(σ ,σ )= tr(f(σ )g(σ )) with f(x)= x and g(x) = logx.
1 2 1 2
Suppose is the set of mixed unitary channels or the set of unital channels acting on . If
n
S M
ρ ,ρ have eigenvalues a a 0 and b b 0, respectively, then
1 2 n 1 n 1 n
∈D ≥ ··· ≥ ≥ ≥ ··· ≥ ≥
n
maxD(ρ ,Φ(ρ )) = f(a )g(d ), (41)
1 2 j j
Φ∈S
Xj=1
where (d ,...,d ) is determined by the algorithm below.
1 n
IfΦ suchthatthereexistsaunitaryV satisfyingV†ρ V = Λ↓(ρ )andV†Φ(ρ )V = ( n d E ),
∈ S 1 1 2 j=1 j jj
then the upper bound is attained. P
Algorithm for determining d d :
1 n
≥ ··· ≥
Step 0. If a > 0 and a = = a = 0, let a = (a ,...,a ) and b = (b ,...,b ), and set
r r+1 n 1 r 1 r
···
(d ,...,d ) = (b , ,b ). (Here, if a > 0, then r = n and (d ,...,d ) is vacuous.)
r+1 n r+1 n n r+1 n
···
Step 1. Let k 1...,r be the largest integer such that
∈{ }
1 1
(a ,...,a ) (b ,...,b ). (42)
1 k 1 k
a + +a ≺ b + +b
1 k 1 k
··· ···
9
Step 2. Set (d ,...,d ) = b1+···+bk(a ,...,a ). Stop if k = r. Otherwise, change r to
1 k a1+···+ak 1 k
r k, a =(a ,...,a ),b = (b ,...,b ); repeat Steps 1 and 2.
k+1 r k+1 r
−
Note that in Step 0 of the algorithm above, we can alternatively choose (d ,...,d ) =
r+1 n
(b ,...,b )S for any doubly stochastic matrix S. Also, in Step 1, a + + a = 0. This
r+1 n 1 k
··· 6
implies that b + +b = 0 because otherwise, the maximality of the choice for k in the previous
1 k
··· 6
iteration will be contradicted.
By Theorem 3.3, we see that S(ρ Φ(ρ )) tr(λ (ρ )log(λ (ρ )/d )), where we use the usual
1 2 j 1 j j j
|| ≥
convention that 0log0= 0 and alog0 = if a > 0. The proof of Theorem 3.3 is quite involved,
−∞
and will be presented in Section 4. we illustrate the results in Theorem 3.3 and Theorem 3.2 in the
following example.
Example 3.4 Let ρ = 1 diag(4,3,3,0) and ρ = 1 diag(5,2,2,1).
1 10 2 10
Apply Step 0. Set d = 0.1, a = (.4,.3,.3) and b = (.5,.2,.2).
4
Apply Step 1. Because (0.4,0.3)/0.7 (0.5,0.2)/0.7, (0.4,0.3,0.3) (0.5,0.2,0.2)/0.9,
≺ ≺
we set (d ,d ,d ) = (0.36,0.27,0.27), and stop.
1 2 3
Hence, (d ,d ,d ,d )= (0.36,0.27,0.27,0.1). For the set of unital channels,
1 2 3 4
S
minF(ρ ,Φ(ρ )) = (√4,√3,√3,0)(1,√2,√2,√5)t/10 =(2+2√6)/10,
1 2
Φ∈S
maxF(ρ ,Φ(ρ )) = (√4,√3,√3,0)(√3.6,√2.7,√2.7,1)t/10 = 3/√10
1 2
Φ∈S
and
minS(ρ Φ(ρ )) = (4,3,3)(log(10/9),log(10/9),log(10/9))t/10,
1 2
Φ∈S ||
maxS(ρ Φ(ρ )) = (4,3,3)(log4,log(3/2),log(3/2))t/10.
1 2
Φ∈S ||
Next we consider the set of all quantum channels. It is known that for any σ ,σ , there
1 2 n
S ∈ D
is a quantum channel Φ such that Φ(σ ) = σ . Similar to Theorem 3.2, we have the following.
1 2
Theorem 3.5 Suppose Ω is defined as in (9), f,g : [0,1] R, and ρ ,ρ are such that
n 1 2 n
→ ∈ D
f(ρ ) have eigenvalues a a and singular values α α . Let Φ be a quantum
1 1 n 1 n
≥ ··· ≥ ≥ ··· ≥
channel.
(a) The best lower and upper bounds of n tr(f(ρ )g(Φ(ρ ))) equal
j=1 1 2
P
n
inf a λ (g(σ)) : λ(σ) = (x ,...,x ) Ω (43)
j n−j+1 1 n n
∈
(cid:8)Xj=1 (cid:9)
and
n
sup a λ (g(σ)) : λ(σ) = (x ,...,x ) Ω , respectively. (44)
j j 1 n n
∈
(cid:8)Xj=1 (cid:9)
Suppose g(x) is increasing concave, then the infimum value equal to g(0) n−1a +a g(1)
j=1 j n
is attainable, and Φ attains the infimum if and only if there is a unPitary V such that
∈ S
V†f(ρ )V = diag(a ,...,a ) and V†g(Φ(ρ ))V =diag(g(0),...,g(0),g(1)).
1 1 n 2
10