A Shrinking Factor for Unitarily Invariant Norms under a Completely Positive Map

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by  Alexey E. Rastegin

A Shrinking Factor for Unitarily Invariant Norms under a Completely Positive Map
Alexey E. Rastegin
Department of Theoretical Physics, Irkutsk State University, Gagarin Bv. 20, Irkutsk 664003, Russia
A relation between values of a unitarily invariant norm of Hermitian operator before and after
action of completely positive map is studied. If the norm is jointly defined on both the input and
outputHilbertspaces,onedefinesashrinkingfactorundertherestrictionofgivenmaptoHermitian
operators. As it is shown, for any unitarily invariant norm this shrinking factor is not larger than
themaximum of two values for thespectral norm and thetrace norm.
Keywords: Ky Fan’s maximum principle, symmetric gauge function, Choi-Kraus representation, Ky Fan’s
norm
0
1
I. INTRODUCTION
0
2
n In many disciplines, linear maps on a space of operatorsprovide key tools for treatmentof the subject. For several
a reasons the class of completely positive maps (CP-maps) is especially valuable [1]. The recent advances in quantum
J information theory have led to a renewed interest in this area [10]. In effect, it seems that all possible changes of

9 quantumstateiscoveredbycontractivecompletelypositivemaps[8],thoughthemonotonicityofrelativeentropycan
2 beprovedinmoregeneralframework[16]. Anyway,alltheimportantexamplesareactuallycompletelypositive. Thus,

studiesofusedquantitativemeasuresunderactionofCP-mapsformaveryactualissue. Ofcourse,otherpropertiesof
]
h CP-mapswithrespectto certainnormsaresubjects ofactiveresearch[4,7]. As a rule,distance measuresaremetrics
p induced by norms with handy properties. Unitarily invariant norms are very useful in this regard [9]. At the same
- time, some variety of measures is typically needed with respect to themes of interest. So, a question on contractivity
h
of given map with respect to applied norm is significant in many different fields of physics (see [12] and references
t
a therein). Hence we may be interested in general results on a problem of contractivity without explicit specification
m
of the measure. Below the result of such a kind will be given for the class of unitarily invariant norms. Namely, the
[ norm of image of Hermitian operator is not greater than the norm of operator itself multiplied by some shrinking

factor. For given CP-map and any norm from the considered class, this factor does not exceed the maximum of two
1
exact values of shrinking factor for the spectral norm and the trace norm. The discussion is carried out entirely in
v
3 finite dimensional setting.
3
3
5 II. DEFINITION AND NOTATION
.
1
0 Let H be d-dimensional Hilbert space. We denote by L(H) the space of all linear operatorson H, and by L (H)
s.a.
0 thespaceofself-adjoint(Hermitian)operatorsonH. ForanyX∈L(H)theoperatorX†Xispositivesemidefinite,and
1
its unique positive square root is denoted by |X|. The eigenvalues of |X| counted with multiplicities are the singular
:
v valuesofoperatorX, insignsσ (X)[6]. Eachunitarilyinvariantnormis generatedbysomesymmetricgaugefunction
i
i of the singular values, i.e. |||X||| =g σ (X),...,σ (X) (see, e.g., theorem 7.4.24in [6]). The determining properties
X g 1 d
for a symmetric gauge function are l(cid:0)isted in [6]. The(cid:1)two families, the Schatten norms and the Ky Fan norms, are
r
a most widely used. For any real p≥1, the Schatten p-norm is defined as [6]
1/p
d
||X|| := σ (X)p .
p i
(cid:18) i=1 (cid:19)
X
This family recovers the trace norm ||X|| for p = 1, the Frobenius norm ||X|| for p = 2, and the spectral norm
tr F
||X|| for p → ∞ [6]. Let us use these signs, although ||X|| ≡ ||X|| and ||X|| ≡ ||X|| as the Ky Fan norms
∞ ∞ (1) tr (d)
though. For integer k ≥1, the Ky Fan k-norm is defined by [6]
||X|| := k σ↓(X)≡g σ (X),...,σ (X) , (1)
(k) i (k) 1 d
i=1
X (cid:0) (cid:1)
where the arrows down show that the singular values are put in the decreasing order. In terms of the norms (1),
the partitioned trace distances have been introduced [15]. These measures enjoy similar properties to the trace norm
distance. Inthefollowing,wewillassumethat||X|| ≡||X|| fork ≥d. Weshallnowdefinethemainobjecttreated
(k) tr
in this paper.
2
Definition 2.1. Let Φ be the restriction of CP-map Φ:L(H )→L(H ) to Hermitian operators. Its shrinking
s.a. A B
factor with respect to given unitarily invariant norm |||(cid:5)||| is defined as
g
η (Φ ):=sup |||Φ(X)||| : X∈L (H ), |||X||| =1 .
g s.a. g s.a. A g
n o
If H = H then on both the spaces a norm |||(cid:5)||| is defined by the same symmetric gauge function. When
A B g
dim(H ) 6= dim(H ), we append zero singular values so that the vectors σ(X) and σ Φ(X) have the same dimen-
A B
sionality equal to max{dA,dB}. In this regard, our consideration is related to those sy(cid:0)mmet(cid:1)ric gauge functions that
are not changed by adding zeros. Only under this condition the same unitarily invariantnormis correctly defined on
the spaces of different dimensionality. The needed property is provided by all the functions assigned to the Ky Fan
normsandtheSchattennorms. Anylinearcombinationofsuchfunctionswithpositivecoefficientsisalsoasymmetric
gauge function that enjoys this property. Indeed, the symmetric gauge functions, providing the above property, form
a convex set.
InDefinition2.1thesupremumistakenoverHermitianinputsX. First,self-adjointoperatorsareveryimportantin
manyapplicationsincludingquantuminformationtopics. Say,thedifferencebetweentwodensitymatricesistraceless
Hermitian,andtherestrictiontosuchoperatorsdeservesattention[12]. Second,aconsiderationofHermitianXallows
to simplify analysis. Third, some relations with positive or self-adjoint operators have later been extended to more
general ones [2, 18]. So, our definition is suitable for such a generalization.
In the seminal paper [5] Ky Fan obtained important results with respect to extremal properties of eigenvalues.
One of his formulations is now known as Ky Fan’s maximum principle. The present author have applied this power
principle for stating the basic properties of the partial fidelities [14], which were originally introduced by Uhlmann
[17], and the partitioned trace distances [15]. Changing the proof of theorem 1 in [5], we can merely prove
k λ↓(X)=max Tr(PX): 0≤P≤I, Tr(P)=k , (2)
i
i=1
X (cid:8) (cid:9)
where the maximum is taken over those positive operators P with trace k that satisfy P ≤ I. Alternately, the
maximizationmaybeoverallprojectorsofrankk,asintheoriginalstatement[5]. IfoperatorXispositivesemidefinite
then the maximum can be taken under the condition Tr(P) ≤ k or, for projectors, rank(P) ≤ k. Using the Jordan
decomposition, we have the following result.
Lemma 2.2. For any X∈L (H) and k ≥1, there exist two mutually orthogonal projectors P and P such that
s.a. Q R
rank(P +P )≤k and
Q R
||X|| =Tr (P −P )X .
(k) Q R
(cid:2) (cid:3)
Proof. First, we suppose that k ≤ d. We write X = Q−R with positive semidefinite Q and R whose supports are
orthogonal. These operators are positive and negative parts of X respectively. Putting the spectral decomposition
|X|=Q+R= qu u†+ rv v† ,
q q r r
q r
X X
we see that {q}∪{r}={σ (X)}. For given k, we define two subspaces, namely
i
K :=span u : q ∈{σ↓,σ↓,...,σ↓} , K :=span v : r ∈{σ↓,σ↓,...,σ↓} .
Q q 1 2 k R r 1 2 k
If P is projector onto K (cid:8)and P is projector onto(cid:9)K , then we at(cid:8)once get (P −P )X =(cid:9)P|X| for projector
Q Q R R Q R
P = P +P of rank k. By construction, the trace of P|X| sums just k largest singular values of X. The case k > d
Q R
is reduced to the trace norm for that the needed projectors are already built and rank(P +P )=d<k. (cid:3)
Q R
III. MAIN RESULTS
In this section, we will study a change of unitarily invariant norms under action of a CP-map. Since they are
positive-valued, upper bounds are usually indispensable. Let Φ : L(H ) → L(H ) be a completely positive linear
A B
map. We shall use the Choi-Kraus representation [3, 8]
Φ(X)= E XE† , E : H →H .
n n n A B
n
X
From the physical viewpoint, this result is examined in [10]. In the context of Stinespring’s dilation theorem, it is
discussed in [1]. The Choi-Krausrepresentationis not unique, but a freedom is unitary in character(see theorem 8.2
in [10]). Two sets {E } and {G } determine the same CP-map if and only if
n m
G = v E ,
m mn n
n
X
3
where numbers v are entries of some unitary matrix of proper dimensionality. Then for given CP-map the two
mn
positive semidefinite operators
M:= E E† , W:= E†E ,
n n n n
n n
X X
are not dependent on a choice of the set {E }. The second operator has been used for another definition of the trace
n
norm distance via extremal properties of contractive CP-maps [13].
Theorem 3.1. Let Φ:L(H )→L(H ) be a CP-map. For every X∈L (H ) there holds
A B s.a. A
||Φ(X)|| ≤η||X|| , k =1,2,...,max{d ,d } ,
(k) (k) B A
where the factor η :=max{||M|| ,||W|| }.
∞ ∞
Proof. First,weassumethatd ≤d . LetX=Q−RbetheJordandecompositionofX,thenΦ(X)=Φ(Q)−Φ(R).
B A
It follows from Φ(X)† =Φ(X), Lemma 2.2 and properties of the trace that
||Φ(X)|| =Tr Π −Π Φ(Q)−Φ(R) ≤Tr Π +Π Φ(Q)+Φ(R) =ηTr (S+T)|X| (3)
(k) B Q R B Q R A
(cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3) (cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3) (cid:2) (cid:3)
for two mutually orthogonalprojectorswith rank(Π +Π )≤k. In (3) we use Q+R=|X| and positive semidefinite
Q R
operators
S=η−1 E† Π E , T=η−1 E† Π E .
n Q n n R n
n n
X X
Denoting µ ≡ ||M|| and ν ≡ ||W|| , we obviously write µ−1M ≤ I and ν−1W ≤ I . Combining the former with
∞ ∞ B A
properties of the trace, we have
Tr S+T =η−1Tr Π +Π M ≤Tr Π +Π µ−1M ≤k . (4)
A B Q R B Q R
(cid:0) (cid:1) (cid:2)(cid:0) (cid:1) (cid:3) (cid:2)(cid:0) (cid:1) (cid:3)
Using Π +Π ≤I and ν−1W≤I , we also obtain
Q R B A
hu,(S+T)ui=η−1 hu,E† Π +Π E ui≤hu,η−1Wui≤hu,ν−1Wui≤hu,ui (5)
n Q R n
n
X (cid:0) (cid:1)
for each u ∈ H . This implies S+T ≤ I and the truth of using Ky Fan’s principle for the right-hand side of (3).
A A
So, the relations (3) and (4) provide the claim. When d > d , the calculations (5) remain valid for k > d , hence
B A A
the right-hand side of (3) is not greater than η||X|| . (cid:3)
tr
As it is known, the role of particular symmetric gauge functions g ((cid:5)) is that norm inequalities can sometimes
(k)
be extended to all unitarily invariant norms. Let u,v ∈ Cd be given vectors with d = max{d ,d }. In accordance
A B
with theorem 7.4.45 in [6], the inequality g(u) ≤ g(v) holds for all symmetric gauge functions g((cid:5)) on Cd if and only
if g (u)≤g (v) for k=1,2,...,d. By Theorem 3.1, for any symmetric gauge function we then obtain
(k) (k)
g σ (Φ(X)) ≤η g σ (X) ,
i i
(cid:0) (cid:1) (cid:0) (cid:1)
or merely |||Φ(X)||| ≤ η|||X||| , whenever X ∈ L (H ). In terms of shrinking factors, the norm inequality can be
g g s.a. A
reformulated as follows.
Theorem 3.2. For each unitarily invariant norm |||(cid:5)||| , defined on both the spaces H and H , a corresponding
g A B
shrinking factor satisfies
η (Φ )≤max{||M|| ,||W|| } .
g s.a. ∞ ∞
In the next sectionwe will show that||M|| is the exactvalue ofshrinking factorfor the spectralnormand||W||
∞ ∞
is the one for the trace norm. So, a degree of non-contractivity of Φ is quite revealed by these two values.
s.a.
IV. THE SPECTRAL NORM AND TRACE NORM
Let X be Hermitian operator such that ||X|| =1. Using the Jordan decomposition X=Q−R, we get
∞
||Φ(X)|| = Tr Π Φ(Q)−Φ(R) ≤Tr Π Φ(Q)+Φ(R) (6)
∞ B B
(cid:12) (cid:12)
(cid:12) (cid:2) (cid:0) (cid:1)(cid:3)(cid:12) (cid:2) (cid:0) (cid:1)(cid:3)
(cid:12) (cid:12)
4
for correspondingprojectorΠ ofrank one. Due to |X|≤I andKy Fan’s maximumprinciple (2), the right-handside
A
of (6) can be treated as
Tr E† ΠE |X| ≤Tr E† ΠE =Tr MΠ ≤||M|| .
A n n A n n B ∞
(cid:16)Xn (cid:17) (cid:16)Xn (cid:17) (cid:0) (cid:1)
So, we have ||Φ(X)|| ≤ ||M|| for any X ∈ L (H ) with ||X|| = 1. Noting Φ(I ) = M, the inequality between
∞ ∞ s.a. A ∞ A
norms is saturated. Hence we obtain the exact value of shrinking factor
η (Φ )=||M|| . (7)
∞ s.a. ∞
Note that this is a particular case of the Russo-Dye theorem (see, e.g., corollary 2.9 in [11]). The above calculation
is given here due to its simplicity and illustration of the method.
In line with (3), for the trace norm there holds
||Φ(X)|| ≤Tr Π +Π Φ(Q)+Φ(R) =Tr W|X| , (8)
tr B Q R A
(cid:2)(cid:0) (cid:1)(cid:0) (cid:1)(cid:3) (cid:0) (cid:1)
since Π +Π = I by rank(Π +Π ) = d . If ||X|| = 1 then the right-hand side of (8) does not exceed ||W|| .
Q R B Q R B tr ∞
This value can actually be reached. Let Y be projector onto 1-dimensional eigenspace corresponding to the largest
eigenvalue of operator W. Then Φ(Y) is positive semidefinite and
||Φ(Y)|| =Tr Φ(Y) =Tr WY =||W|| .
tr B A ∞
(cid:0) (cid:1) (cid:0) (cid:1)
In other words, the exact value of shrinking factor is given by
η (Φ )=||W|| . (9)
tr s.a. ∞
Thus, for the spectral and trace norms the exact value of shrinking factor is simply calculated. For other norms a
task is more difficult but the bound of Theorem 3.2 is useful for many aims. So, this bound can be rewritten as
η (Φ )≤max η (Φ ),η (Φ ) . (10)
g s.a. ∞ s.a. tr s.a.
(cid:8) (cid:9)
To sum up, we havea valuable conclusion. If the restrictionΦ is contractivewith respectto both the spectraland
s.a.
tracenormthenitiscontractivewithrespecttoallunitarilyinvariantnorms. Moreover,adegreeofnon-contractivity
can be measured by using these two norms.
Finally,we apply ourresultsto the operationofpartialtrace. This operationisespecially importantinthe context
ofquantuminformationprocessing. Henceweareinterestedinrelationsbetweennormsbeforeandafterpartialtrace.
The writersof [9]resolveda questionfor those unitarily invariantnorms that aremultiplicative overtensorproducts.
TheexplicitChoi-Krausrepresentationofpartialtraceisgivenin[10]. However,theoperatorsMandWcanbefound
directly. Let us take H =H ⊗H with partial tracing over H , that is
A B C C
Ψ(X):=Tr (X) (11)
C
for any X∈L H ⊗H . First, this operation preserves trace, because
B C
(cid:0) (cid:1)
Tr Ψ(X) =Tr Tr (X) =Tr (X) .
B B C A
(cid:0) (cid:1) (cid:8) (cid:9)
Combining this with Tr Ψ(X) =Tr WX finally gives W=I . Second, the right-hand side of definition for M is
B A A
rewritten as (cid:0) (cid:1) (cid:0) (cid:1)
E I E† =Ψ I ⊗I =I Tr (I ) .
n A n B C B C C
n
X (cid:0) (cid:1)
So we obtain M=d I , where d =dim(H ). Because ||M|| =d and ||W|| =1, the statement of Theorem3.2
C B C C ∞ C ∞
gives
|||Ψ(X)||| ≤d |||X||| (12)
g C g
for X∈L (H ) and any unitarily invariant norm. For the spectral norm this relation coincides with the one given
s.a. A
in [9]. For the Frobenius norm the method of [9] provides a more precise bound. On the other hand, the validity of
(12) is not restricted to those norms that are multiplicative over tensor product.
5
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