Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture

0.25 MB·English

by  Nengkun Yu

Bounds on the distance between a unital quantum channel and the convex hull of
unitary channels, with applications to the asymptotic quantum Birkhoff conjecture
Nengkun Yu
∗
State Key Laboratory of Intelligent Technology and Systems,
Tsinghua National Laboratory for Information Science and Technology,
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
and Centre for Quantum Computation and Intelligent Systems (QCIS),
Faculty of Engineering and Information Technology,
University of Technology, Sydney, NSW 2007, Australia
Runyao Duan
†
Centre for Quantum Computation and Intelligent Systems (QCIS),
2 Faculty of Engineering and Information Technology,
1
University of Technology, Sydney, NSW 2007, Australia
0
and State Key Laboratory of Intelligent Technology and Systems,
2
Tsinghua National Laboratory for Information Science and Technology,

n Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
a
J
Quanhua Xu
‡
5 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
and Laboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 25030 Besanc¸on cedex, France
]
(Dated: December 28, 2011)
h
p
Motivated by the recent resolution of Asymptotic Quantum Birkhoff Conjecture (AQBC), we
-
t attempt to estimate the distance between a given unital quantum channel and the convex hull of
n unitary channels. We provide two lower bounds on this distance by employing techniques from
a quantuminformation andoperatoralgebras, respectively. Wethenshowhowtoapplytheseresults
u
to construct some explicit counterexamples to AQBC. We also point out an interesting connection
q
between the Grothendieck’sinequality and AQBC.
[

1 PACSnumbers: 03.67.-a,3.65.Ud
v
2
7 I. INTRODUCTION be made noiseless for quantum informationtransmission
1 with the help of a friendly environment even in one-shot
1 case. Furthermore, it turns out that these channels are
1. Suppose we are given a quantum system with a d- the only quantum channels having this desirable prop-
0 dimensional Hilbert space , and the state (or density erty [3]. Surprisingly, if arbitrarily large number of uses
d
2 H of the channels are allowed, unital quantum channels,
operator) of the system is given by a trace one posi-
1 those channels Φ with identity operator a fixed point,
tive operator ρ from the linear operator space L( ).
v: Quantum channels, or trace-preserving completelyHpods- say Φ(I) = I, can also achieve maximum capacity and
Xi itive maps, are all possible deterministic quantum op- act exactly like noiseless channel [4].
erations one can perform over the system [1, 2]. Let Clearly, any mixture of unitary channels remains uni-
r
a Φ be such a quantum channel over L( d) with Kraus tal. An interesting question is to ask whether one can
H reversethisprocedure,i.e.,decomposinganyunitalquan-
operator sum representation Φ = kEk · Ek†, and let tum channel Φ T( ) into a mixture of unitary chan-
K(Φ)=span{Ek}beitsKrausoperPatorspace. Thecon- nels from U( ∈). THhdis was called “quantum Birkhoff
vexhullofunitarychannels(noiselesschannels)onL( d) Hd
is given by Conv(U( )). So any Ψ Conv(U( ))Hcan conjecture”(QBC),originatedfromBirkhoff’scelebrated
d d
H ∈ H characterizationoftheextremepointsofdoublystochas-
be written as a mixture (convex combination) of uni-
tic matrices. Unfortunately, this conjecture is only true
tary channels. (The number of unitary channels in the
for d 2, and counterexamples exist whenever d 3
mixture can be made finite due to the Carath´eodory’s ≤ ≥
[5–7]. This suggests the following quantity to measure
theorem on convex hull). The mixture of unitary chan-
the distance between Φ and the convex hull of unitary
nels plays a special role in environment-assisted quan-
channels.
tumcommunicationmodel. Actually,these channelscan
D(Φ,Conv(U( )))=inf D(Φ,Ψ):Ψ Conv(U( )) ,
H { ∈ H }
where D(Φ,Ψ) will be given by the diamond norm of
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] Φ Ψ. Since Conv(U( )) is acompactconvexset,“inf”
− H
‡Electronicaddress: [email protected] in the above equation can be replaced by “min”.
2
Motivated by some results in about the environment- Motivated by these progresses and in order to better
assisted quantum capacity and in an attempt to remedy understand the structure of unital channels, in this pa-
the conjecture in certain way, Smolin, Verstraete, and perweareinterestedinestimatingthetracedistancebe-
Winter proposed the following tween a unital quantum channel and the convex hull of
unitary channels, say D(Φ,Conv(U( ))). We find that
Conjecure 1. (Asymptotic Quantum Birkhoff Conjec- this distance is interesting even fromHthe perspective of
ture [4]) Let Φ T( ) be a unital channel, then Φ⊗n quantum channel discrimination: Suppose we are given
∈ H
can be approximated by a mixture of unitary channels an unknown quantum channel, which is secretly chosen
from U( ⊗n) with arbitrary precision. That is betweenΦandsomeΨ Conv(U( )))withequalproba-
H ∈ H
bility 1/2. Thendue to the operationalmeaning oftrace
limn D(Φ⊗n,Conv(U( ⊗n)))=0. distance, we can conclude that the success probability
→∞ H
ofdiscriminationisatleast1/2+1/4D(Φ,Conv(U( ))),
Thisrevisedconjectureseemshighlyreasonableasone H
whichisstrictlylargerthan1/2wheneverΦisnotamix-
couldnaturallyexpectthatmanycopiesofaunitalchan-
tureofunitarychannels. Anotherpurposeofthispaperis
nel will be better approximated by a mixture of unitary
to provide some relatively elementary and self-contained
channels on a higher-dimensional space. If this is true,
disproofsto AQBC.This ispartiallydue tothe factthat
it will provide a very satisfactory interpretation to the
the elegant disproof of AQBC in Ref. [11] makes use of
following result: The environment-assisted quantum ca-
some basic properties of factorizable maps which cannot
pacityofanyunitalchanneloverL( )isgivenbylog d
Hd 2 be easily appreciated by readers who do not have deep
qubits,themaximumcapacityonecanachieveunderthis
backgroundin operator algebras.
model. Amuchmoredeepconsequenceisthatthestruc-
InSectionIIwecollectsomepreliminariesaboutsuper-
ture of unital channels will be greatly simplified. Due to
operatorsandSchurchannels. TheninSectionIIIweex-
itssignificance,theasymptoticquantumBirkhoffconjec-
plainindetailthe operationalmeaningoftracedistance.
turewaslistedasoneofmajoropenproblemsinquantum
InSectionIVwe firstprovideacomputable lowerbound
information theory [8].
forD(Φ,Conv(U( ))) whenthe Krausoperatorspaceof
Some supporting evidences wereobtained inRef. [10], H
Φ does not contain any unitary operator. This enables
whereMendlandWolfpresentedaunitalchannelΦsuch
us to derive many counterexamples for AQBC, includ-
that Φ 2 is a mixture of unitary channels although Φ
⊗
ing some factorizable maps presented in Ref. [11]. It
itself is not. Furthermore, they showed that it is pos-
is worth pointing out that this proof only employs some
sible that the tensor of Φ and a constant unital channel
basic techniques from quantum information theory. We
(acompletelydepolarizingchannelthatmapseverystate
believe that it may interest readers with quantum infor-
intothecompletelymixedstateI/d)maybecomeamix-
mationbackground. In SectionV we gofurther to study
ture of unitary channels. One may naturally conjecture
the class of Schur channels. In this special case, we are
these properties might be true for any unital quantum
able to provide a lower bound and an upper bound for
channels.
D(Φ,Conv(U( ))). Roughly speaking, we show that up
Recently Haagerup and Musat disproved this asymp- H
to a factor of 1/2, any Schur channel can be approxi-
totic version by exhibiting a class of so-called non-
mated by a mixture of diagonal unitary channels, and
factorizable maps as counterexamples [11]. Actually the
the later has a simpler structure. As a direct applica-
results obtained in Ref. [11] shows that any such non-
tion, we obtain a new proof of the fact that any Schur
factorizable map Φ is a very strong counterexample to
channelthatdoesnotsatisfytheQBCwillautomatically
AQBC in the following sense:
violate the AQBC. Our proof for this part has employed
some powerful tools from operator algebras. In Section
D(Φ Ψ, (L( )) D(Φ, (L( ))),
d m d
⊗ FM H ⊗H ≥ FM H VIwepresenttwoexplicitexamplesofSchurchannelsto
demonstrate the utility of our results: the first example
where Ψ is any unital channel over L( ), and
m
H has only two Kraus operators and is a non-factorizable
(L( )) denotes the set of factorizable maps over
d
FM H map, and the second one is a factorizable map. As an-
L( ). Inotherwords,anynon-factorizablemaptensor-
d
H otherinterestingapplication,inSectionVIIwepointout
ing with a unital channel could not reduce the distance
aconnectionbetweenAQBCandGrothendieck’sinequal-
tothesetoffactorizablemaps,whichisasuper-setofthe
ity in the metric theory of tensor products.
convex hull of unitary channels. See also Shor’s talk in
Ref. [12]foranalternativeapproachtoAQBCandanex- Remarks on related results: After we obtained the
cellentdiscussionoftheresultsinRef. [11]. Theinterest- results in Section IV, and were working on the proof of
ing thing here is that all these counterexamples are non- theTheorem3inSectionV,thesecondauthorR.D.hap-
factorizablemaps,anditremainedunknownwhetherany pened to learn from Prof. M. B. Ruskai that Haagerup
facterizable map would fulfill AQBC. This problem was and Musat had made further progress on the connec-
signified in the arXiv version of Ref. [11] by establish- tion between Schur channels and AQBC. Namely, they
ingthefollowingsurprisingconnection: Ifallfactorizable obtained Theorem 5 and thus showed that any Schur
maps satisfy AQBC, then the Connes embedding prob- channel that violates QBC (including some factorizable
lem has a positive answer. maps) should also be a counterexample to AQBC [13].
3
They also provided a modified version of the connection Ifwespecifyanorthonormalbasis k of ,wecan
between factorizable maps satisfying AQBC and Connes rewrite Φ in Eq (1) into the following{|foZrmi}: Z
embedding problem. The proof of Theorem 3 has em-
ployed some similar techniques in [13]. dim( )
Z
Φ(X)= AkXBk†, (3)
kX=1
II. PRELIMINARIES
where A = k A and B = k B are linear operators
k k
in L( , ). ShimZi|larly, when ΦhisZa| quantum channel, we
We will use symbols , , etc to represent finite di- H K
H K can choose A =B = k V so that
mensional Hilbert spaces over complex numbers. A d- k k h Z|
dimensional Hilbert space , which is essentially the
same as d, will be explicitHly represented as when- dim(Z)
ever the Cdimension matters. L( , ) denotesHthde set of Φ(X)= AkXA†k, A†kAk =I , (4)
linearoperators(ormappings)frHomK to ,andL( )is kX=1 Xk H
H K H
shorthand for L( , ). For any X L( ), X L( )
† which is the famous Kraus operator sum representation
H H ∈ H ∈ H
denotes the adjoint operator (or complex conjugate) of
of a quantum channel [1].
X. X isHermitian(orself-adjoint)ifX =X. U L( )
† Now we tend to introduce norms ofsuper-operatorsin
∈ H
is said to be unitary if U U =I . We denote the set of
† T( , )basedonthenormsoflinearoperators. Werefer
unitary operators on by U( H). X L( ) is (semi- H K
to Refs. [14, 15] for some detailed discussion on norms
H H ∈ H
definite) positive, write X 0, if the quadratic form
ofsuper-operatorsandhowtocomputethemusingsemi-
≥
ψ X ψ 0 for any ψ . In particular, X is
definite programming techniques. We will briefly review
h | | i ≥ | i ∈ H
said to be a density operator (or a quantum state) if
some basic results for later use. For any X L( ) and
d
X is positive and with trace one. T( , ) is the set ∈ H
p 1, the p-th norm of X is given by
H K
of linear mappings from L( ) to L( ). Again, T( ) ≥
H K H
is shorthand for T(H,H). Elements in T(H,K) are nor- X p =(TrX p)p1,
mallycalledsuper-operators. NotethatL( )isaHilbert || || | |
H
space with the standard Hilbert-Schmidt inner product
where X = √X X. The trace and the operator norms
< A,B >= Tr(A†B). Then the adjoint operator of of X ar|e s|pecial c†ases of p=1 and p , respectively,
Φ T( , ) is defined as the unique super-operator →∞
∈ H K
Φ T( , ) such that
† ∈ K H X 1 =TrX , X = max X ψ .
|| || | | || ||∞ ψψ =1|| | i||
<Y,Φ(X)>=<Φ†(Y),X >, X L( ),Y L( ). h | i
∀ ∈ H ∈ K
A super-operator Φ T( , ) is said to be positive if The trace norm and the operator norm of a super-
∈ H K
it preserves the positivity, say, Φ(X) 0 whenever X operator Φ T( , ) are given respectively as follows:
≥ ≥ ∈ H K
0. Φ is said to be a quantum channel if it satisfies: i)
(trace-preserving)Tr(Φ(X))=Tr(X)foranyX L( ), Φ 1 = sup Φ(X) 1, Φ = sup Φ(X) .
and ii)(completely positive) for any n 1, the ∈induHced || || ||X||1≤1|| || || ||∞ ||X||∞≤1|| ||∞
≥
super-operator Φ = Φ I T( , )
is positive, wherenI ⊗isL(tHhne)id∈entiHty⊗suHpenr-Kop⊗erHatnor In the above equation we can replace“sup” with “max”
on L( ). We callL(ΦHna) quantum unital channel if it whenonlyfinite dimensionalHilbertspacesareinvolved.
Hn The completely bounded trace norm (or diamond norm)
further satisfies: iii) (unital condition) Φ(I )=I . Any
and operator norm (simply completely bounded norm)
unitary operator U U( ) induces a unitHary quKantum
∈ H are given respectively as follows:
channel T( ) in the followingway: (X)=UXU .
†
U ∈ H U
TheclassofunitarychannelsonL( )willbe denotedas
Φ =sup Φ I , Φ =sup Φ I .
U( ). H || ||⋄ n 1|| ⊗ L(Hn)||1 || ||cb n 1|| ⊗ L(Hn)||∞
H ≥ ≥
Any super-operator Φ T( , ) can be represented
∈ H K
by a pair of linear operators A,B L( , ) such Proposition 1. For any Φ T( , ), the diamond
∈ H K⊗Z ∈ H K
that norm and the completely bounded norm satisfy the fol-
lowing properties:
Φ(X)=Tr AXB†, X L( ), (1)
Z ∈ H
i) The dimension of theauxiliary system toachieve
where is an auxiliary Hilbert space with dim( )
•
Z Z ≤ the norms can be restricted to that of , Φ =
dim( )dim( ),andTr representsthepartialtraceover
. FHor the sKpecial caseZof quantum channels, the above Φ IL( ) 1 and Φ cb = Φ IL( ) H.|| ||⋄
Z || ⊗ H || || || || ⊗ H ||∞
form can be greatly simplified. Actually, in Eq. (1) we
ii) The following duality relation holds for Φ and
can choose A = B = V L( , ) for some isom- •
etry V and obtain the fol∈lowinHg Kwe⊗ll-Zknown Stinespring Φ†, ||Φ||1 =||Φ†||∞ and ||Φ||⋄ =||Φ†||cb.
unitaryembeddingrepresentationofaquantumchannel:
iii) If Φ is completely positive, then Φ = Φ
1
Φ(X)=TrZVXV†, V†V =IH. (2) • and ||Φ||cb =||Φ||∞ =||Φ(IH)||∞. || ||⋄ || ||
4
The norms defined above enable us to introduce dis- iv) Φ is trace-preserving if s = 1 for k =
S kk
•
tance between quantum states and quantum channels. 1, ,d;
···
The trace distance between two quantum density opera-
v) Φ is unital iff s =1 for k =1, ,d.
tors ρ and σ in L( ) is given by S kk
• ···
H
Proof: iv) and v) follow directly by evaluating
D(ρ,σ)= ρ σ .
|| − ||1 Tr(ΦS(k j )) = skjδkj. We shall see that ii) and iii)
| ih |
are simple corollaries of i). So we first prove i). In fact,
In the following discussion we also need the fidelity be-
letAandBbeanytwod dmatricessuchthatS =AB .
tween ρ and σ, †
×
We may assume A = [a , ,a ] and B = [b , ,b ],
1 d 1 d
··· ···
F(ρ,σ)=Tr ρ1/2σρ1/2. where ak and bk are all d-dimensional column vectors.
p Then S = kakb†k. Set Ak = Diag(ak), Bk =Diag(bk).
The so-called Uhlmann theorem makes the meaning of That is, APk and Bk are diagonal matrices with diago-
fidelity more transparent: nals ak and bk, respectively. By some routine calcula-
tions we directly verify that ΦS(X)= dk=1AkXBk† for
F(ρ,σ)= ψma,φx |hψ|φi|, anyX ∈L(Hd). Inparticular,whenSPispositivewe can
| i| i writeS =AA†forsomeA L( d). Hencewecanchoose
∈ H
where ψ , φ range over all purifications of ρ Ak =Bk in this specialcase. That provesboth the posi-
and σ,|reisp|eciti∈veHly,⊗saKy Tr ψ ψ = ρ and Tr φ φ = tivity andcompletely positivity ofΦS. Conversely,if ΦS
σ. Most notably, the abovKe|eqiuhat|ion remainsKtr|ueihev|en is positive. Then by choosing e = d k , we have
when one of ψ or φ is fixed. This fact plays a crucial ΦS(e e)=S is positive. | i Pk=1| i (cid:3)
role in our la|teir dis|cuission. Trace distance and fidelity T|heihfo|llowing proposition gives another fundamental
areequivalentincharacterizingthedistancebetweentwo property of Schur multiplier. Relevant discussions can
states in the following sense: be found in Page 110 of Ref. [9].
Proposition 3. For any Schur multiplier Φ, the dia-
2(1 F(ρ,σ)) D(ρ,σ) 2 1 F2(ρ,σ).
− ≤ ≤ − mond norm, the trace norm, completely bounded norm
p
Following the same idea, we can define the trace dis- and operator norm all coincide, that is, Φ = Φ 1 =
tance between two quantum channels Φ and Ψ via the Φ cb = Φ . || ||⋄ || ||
|| || || ||∞
following way:
So a Schur multiplier Φ is a quantum channel iff S is
S
positive and with all diagonal entries one. In particular,
D(Φ,Ψ)= Φ Ψ .
|| − ||⋄ whenever ΦS is a quantum channel, it is also unital. We
Letusnowintroduceaspecialclassofsuper-operators. shall denote
For any S L( ), we can define a super-operator Φ
d S
∈ H S( )= Φ :S L( ),S 0,s =1,1 k d ,
via the following way: Hd { S ∈ Hd ≥ kk ≤ ≤ }
and call the elements from S( ) (or simply S ) Schur
ΦS(X)=S X, X L( d), Hd d
◦ ∀ ∈ H channels. Note that the difference of two Schur multi-
where S X = [s x ] is the entry-wise product or pliers is still a Schur multiplier. Applying Proposition3,
kj kj
Hadamard◦product. (Hereweassumethatwehavespec- we obtain an immediate consequence that auxiliary sys-
ified an orthonormal basis k : k = 1, ,d for . tems are not required to distinguish between two Schur
d
Thus any linear operator fr{o|miL( ) is·e·x·pre}ssed aHs a channels.
d
H
matrix under the standard matrix basis k j . For
{| ih |}
instance, S = s k j ). Such Φ is called Schur
k,j kj| ih | S III. OPERATIONAL INTERPRETATION OF
multiplier induPced by S. Schur multipliers have been
TRACE DISTANCE
extensively studied in the literatures of operator alge-
bras. We refer to Chapters 3 and 8 of Ref. [9] for some
highly accessible introductions, and Ref. [11] for recent We have introduced trace distance between quantum
advances. For later use, some basic properties of Schur states and quantum channels, and will study the trace
multipliers are listed as follows: distance betweenaunital quantumchannelandthe con-
vex hull of unitary channels in greater detail. Before
Proposition 2. Let S L( d). Then ΦS satisfies the we proceed, we need justify the importance of this mea-
∈ H
following: sure from the perspective of quantum information. In
one word, the trace distance characterizes some sort of
• i) ΦS = dk=1Ak·Bk†, where all Ak,Bk are diago- stochasticdistinguishabilityofquantumstatesandquan-
nal matrPices; tum channels. Actually, the trace distance naturally oc-
curs when we study the following state discrimination
ii) Φ is positivity-preserving iff S is positive;
S
• problem. Suppose we are given an unknown quantum
iii) Φ is completely positive iff S is positive; system whose state is secretly prepared in one of ρ and
S 0
•
5
ρ , with equal priori probability 1/2. The task here is appropriate form of Sion’s minimax theorem [17] to ex-
1
to determine the identity of the system with a success change the order of “max” and “min”, and obtain Eq.
probability as high as possible. To do so we need ap- (5) immediately.
ply a two-outcome quantum measurement E ,E to Nowwetrytogeneralizetheaboveresulttothecaseof
0 1
{ }
the system, and to maximize the success probability of quantumchannels. Thesimplestcaseistodistinguishbe-
discrimination, i.e., tweentwo quantumchannels Φ ,Φ T( , ). The ba-
0 1
∈ H K
sicstrategyhereistochooseaninputstateρ L( ),
′
1 ∈ H⊗H
and then to distinguish between the respective output
P (ρ ,ρ )= max (Trρ E +Trρ E ),
succ 0 1 0 0 1 1
{E0,E1}2 satuaxtielsiarIyL(sHt′a)te⊗sΦpai(cρe).,WwehehraeveH′ is a finite-dimensional
whereE 0andE +E =I. Bysomesimplealgebraic
i 0 1
≥
manipulations, one can verify that the optimal success D(Φ ,Φ ;ρ)=D((I Φ )(ρ),(I Φ )(ρ)).
0 1 L( ′) 0 L( ′) 1
probability of discrimination is given by [16] H ⊗ H ⊗
To achieve the maximum success probability, we need
1 1 take “sup” over all possible input states, and have
P (ρ ,ρ )= + D(ρ ,ρ ).
succ 0 1 0 1
2 4
D(Φ ,Φ )=supD(Φ ,Φ ;ρ).
0 1 0 1
Thus a larger trace distance between ρ0 and ρ1 implies ρ
a higher success probability of discrimination. This in-
One can readily verify that the RHS of the above equa-
terpretation can be extended to compact convex sets of
tion gives us the diamond norm Φ Φ , and ρ can
densityoperators. LetA0andA1 betwocompactconvex be restricted to density operators||on0− 1||⋄(thus “sup”
setsofdensityoperators. ThetracedistancebetweenA0 H⊗H
can be replaced as “max”). To generalize the trace dis-
and A is given by
1
tance to compact convex sets of quantum channels, we
firstneedthetracedistancewithinputstateρasfollows:
D(A ,A )=min D(ρ ,ρ ):ρ A ,i=0,1 .
0 1 0 1 i i
{ ∈ }
D(C ,C ;ρ)= min D(Φ ,Φ ;ρ).
Then the optimaldiscrimination probabilitybetween A 0 1 0 1
and A is given as 0 Φi∈Ci
1 e
Then the final resulting operational trace distance be-
1 1 tween C and C is given by
P (A ,A )= + D(A ,A ). (5) 0 1
succ 0 1 0 1
2 4
D(C ,C )=supD(C ,C ;ρ)=sup min D(Φ ,Φ ;ρ),
0 1 0 1 0 1
The above formula indicates that we can operationally ρ ρ Φi∈Ci
distinguish between two compact convex sets of density e e (6)
operators by performing a universal quantum measure- where ρ ranges over all possible bipartite density opera-
ment, and the success probability of discrimination is tors on , and it is not clear whether we can re-
′
H ⊗H
completely characterized by the trace distance between place “sup” with “max” as the dimension of may be
′
H
A and A . The most surprising thing here is that the arbitrarily large. The optimal success probability of dis-
0 1
quantum measurement we perform does not depend on crimination between C and C is given by
0 1
the exactform of the unknown state except the assump-
1 1
tion that it is from one of A and A .
0 1 P (C ,C )= + D(C ,C ).
succ 0 1 0 1
It seems that Eq. (5) was first obtained by Gutoski 2 4
andWatrousinRef. [18]byusing the convexsetsepara- e
Interestingly, the (ordinary) trace distance between C
tiontheorem. Jainprovidedadifferentwaybasedonthe 0
and C is given by
minimax theorem [19]. For completeness, we will out- 1
line the later approach as follows. Let E ,E be the
0 1 D(C ,C )= min D(Φ ,Φ )= min maxD(Φ ,Φ ;ρ).
{ } 0 1 0 1 0 1
quantum measurement we need perform, and ρ0 and ρ1 Φi∈Ci Φi∈Ci ρ
be two states from A and A , respectively. Then the
0 1
optimal success probability is given by The major difference between D(C0,C1) and D(C0,C1)
is that the orders of “max” (“sup”) and “min” has been
e
1 reversed. It is not obvious that whether the orders of
P (A ,A )=max min (Trρ E +Trρ E ).
succ 0 1 {Ei}ρi∈Ai 2 0 0 1 1 “min” and “max” (“sup”) are exchangeable or not as
it is unclear whether the objective function D(Φ ,Φ ;ρ)
0 1
The crucial point here is that we first take “min” over satisfies the requirements of minimax theorem. Conse-
all possible pair of states ρ0 and ρ1 according to a fixed quently,itseemsnotclearwhetherD(C0,C1)isthesame
smibelaesumreemaseunrtem{Een0,tsEt1o},manadxitmhieznettahkees“umcacexs”sopvreorbaallbpiloitsy- as D(C0,C1). Nevertheless, we stillehave
ofdiscrimination. Noticing thatthe objectivefunctionis D(C ,C ;ρ) D(C ,C ) D(C ,C ).
0 1 0 1 0 1
linear in (E ,E ) and(ρ ,ρ ) when one of them is fixed, ≤ ≤
0 1 0 1
and all involving sets are compact convex, we can apply In particuelar, we have thee following simple property.
6
Property 1. Let C ,C T( , ) be two compact con- NowwecansummarizetherelationbetweenD(C ,C )
0 1 0 1
⊆ H K
vex sets of quantum channels, and let ρ be a bipartite and D(C ,C ) as follows:
0 1
e
pure entangled state over with full Schmidt rank.
H⊗H Theorem 1. Let C and C be two compact convex sets
Then the following are equivalent: 0 1
of quantum channels in T( , ). Then
H K
i). C C = ;
0 1
∩ ∅
D(C ,C )=D(C ,C ).
0 1 0 1
ii). D(C ,C )>0;
0 1
Proof: Let usefirst denote
iii). D(C ,C )>0; and
0 1
C=C C = Φ Φ :Φ C ,Φ C .
0 1 0 1 0 0 1 1
iv). De(C ,C ;ρ)>0. − { − ∈ ∈ }
0 1
Then C is a compact convex set, and completely deter-
Proeof: We only need to establish the equivalence
mines D(C ,C ) and D(C ,C ). We also write
between i) and iv). By definition, iv) means that we 0 1 0 1
can distinguish between C0 and C1 using ρ as an in- Re= (X,ρ):0 X I ρ,ρ 0,Trρ=1 .
put. This immediately implies that C and C should { ≤ ≤ ⊗ ≥ }
0 1
be disjoint. In other words, i) should hold. The direc- Clearly, R is also a compact convex set.
tion that i) iv) is a little bit tricky, and the key here By Lemma 1, we can rewrite
⇒
is to apply a generalized form of Choi isomorphism [2]
D(C ,C )=min max 2Trρ X.
between super-operators and bipartite linear operators. 0 1 Φ
Φ C(X,ρ) R
By contradiction, assume that C and C are disjoint ∈ ∈
0 1
but D(C ,C ;ρ)=0. It follows from the definition that Noticing that both C and R are compact convex sets,
0 1
there exist Φ0 C0 and Φ1 C1 such that and the objective function 2Tr(ρΦX) is linear both in Φ
e ∈ ∈ and (X,ρ), by Sion’s minimax theorem we can exchange
D((IL( ) Φ0)(ρ),(IL( ) Φ1)(ρ))=0. the order of “max” and “min” as follows:
H ⊗ H ⊗
Equivalently, we have D(C ,C )= max min2Trρ X.
0 1 Φ
(X,ρ) RΦ C
(I Φ )(ρ)=(I Φ )(ρ). (7) ∈ ∈
L( ) 0 L( ) 1
H ⊗ H ⊗
Now we proceed to prove D(C ,C )=D(C ,C ). We
Noticing that ρ is a bipartite pure state with full 0 1 0 1
onlyneedtoshowD(C ,C ) D(C ,C )astheopposite
Schmidt rank, we have the following generalized Choi- 0 1 ≥e 0 1
direction is obvious according the definitions. By the
isomorphism: e
above equation and Eq. (6), it suffices to show that for
J:Φ7→(IL(H)⊗Φ)(ρ). asuncyh(Xth,aρt)∈Rthereisadensityoperatorσ ∈L(H⊗H′)
(The standard Choi-isomorphism is to choose ρ as the
maximally entangled state Ω = 1/√d d k k ). (Φ I)(σ) 1 TrρΦX, Φ C.
| i k=1| i| i || ⊗ || ≥ ∀ ∈
Applyingthisisomorphism,wededucefromPEq. (7)that
Indeed, we can choose σ = u u to be the following
Φ(cid:3)0 = Φ1. This contradicts the assumption C0∩C1 = ∅. bipartite pure state | ih |
So whenever two compact convex sets of quantum
u =(I A)α and A†A=ρ,
channels are disjoint, we can operationally distinguish | i ⊗ | i
between them with a success probability strictly larger where α isagaintheunnormalizedmaximallyentangled
than 1, and any bipartite pure state with full Schmidt state o|veir .
2 H⊗H′
rank can be used as input. Notethatwehavethefollowingwell-knownfactabout
The really interesting thing here is that the equality the trace norm:
of D(C ,C ) = D(C ,C ) does hold. The key to this
0 1 0 1
Y = max 2TrPY,
ifsolltoehweinagppsleimcait-idoenfinoifteSpiorno’gsramminmiminagxchtahreaocrteemrizaantidonthoef || ||1 0≤P≤I
the diamond norm recently discovered by Watrous [20]. where Y is any traceless (TrY =0) Hermitian operator.
Applying the above fact to (Φ I)(σ), we have
Lemma 1. (Watrous [15]) For any super-operator Φ = ⊗
Φ Φ such that Φ and Φ are quantum channels in (Φ I)(σ) =max2TrP(I A)ρ (I A )=max2Trρ Q,
0 1 0 1 1 Φ † Φ
T( −, ), we have the following || ⊗ || P ⊗ ⊗ Q
H K
Φ =max2Trρ X, X I ρ,Trρ=1,ρ 0,X 0, where 0 P I ′ and Q = (I A†)P(I A).
|| ||⋄ Φ ≤ ⊗ ≥ ≥ Noticingt≤hat0≤XH⊗HI ρ=I A†A,w⊗ecaneasil⊗yfind
≤ ≤ ⊗ ⊗
where ρΦ = (Φ I ′)(α α) is the Choi operator of 0 P′ I ′ such that X =Q′ =(I A†)P′(I A)
Φ, α = d ⊗k H k| =ih √|dΩ is the unnormalized [21≤]. Th≤usHw⊗eHhave ⊗ ⊗
| i k=1| i⊗| i | i
maximallyPentangled state over ′, and ′ is an
isomorphic copy of . H ⊗ H H mQax2TrρΦQ≥2TrρΦQ′ =2TrρΦX,
H
7
which completes the proof. (cid:3) Proof: Let Ψ = N p with p a finite prob-
Remarks: After wefinishedthe aboveproof,wewere ability distribution aPndk=1kkUkU( d).{Wk}e need to show
U ∈ H
informedbyGutoskithatinarecentworkhegeneralized that
the results in Ref. [18] to the discrimination oftwo com-
D(Φ,Ψ)=D(Φ, p ) C .
pactconvexsetsofquantumstrategies,andobtainedthe kUk ≥ Φ
results for the case of quantum channels as an immedi- Xk
ate corollary [23]. It is interesting to note that his main Note that
proof technique is a separationtheorem of compact con-
D(Φ, p )=D(Φ I , p I )
vex sets from convex analysis, quite similar to that in Xk kUk ⊗ L(Zd) Xk kUk⊗ L(Zd)
Ref. [18]. Instead, here we employ a different method
bgryamusminigngSicohna’rsamctienriimzaatxiotnhoefodreiammaonnddnseomrmi-d,ienfinaitseimpirlaor- ≥D(Φ⊗IL(Zd)(Ω),Xk pkUk⊗IL(Zd)(Ω)),
spiritofRef. [19]. Hopefully,ourproofmayprovidesome where Ω =1/√d d k k is a maximally entangled
newinsightintothis problem. Gutoski’spaper,however, | i k=1| i| i
stateon d d. NPowapplyingtheinequalityD(ρ,σ)
contains many other interesting results about the trace H ⊗Z ≥
2(1 F(ρ,σ)), we have
norms. −
It is also worth noting that with minor changes the D(Φ,Ψ) 2(1 F(Φ I (Ω), p I (Ω)))
same technique in the above proof can be used to derive ≥ − ⊗ L(Zd) Xk kUk⊗ L(Zd)
Lemma 1, as first shown by Watrous in Ref. [15].
=2(1 max ψ φ ),
All the above discussions are applicable to the case − ψ |h | i|
of C = Φ and C = Conv(U( )). An interesting
ftaainocdntaCi0lsloytnhdvai({stUtwi(n}igtuh))iosuehtvbeane1utxwwiehlieeannryathsuyensifttoearmHlmsqe,urwainestcunamontnccohotannotanpieenlreaΦd- owILfh(ΦZerd⊗e)(|IΩψL)i(Z=gdi)vP(eΩnk)b,√y|φqki|iψskai|fikKxeidrapnugreifiscoavteironallopfuPrikficpaktUiokn⊗s
H
in the latter. To see this, let ρ L( ) be any density φ = √p (U I )Ω k ,
operator. Since Φ is a unital qua∈ntumHchannel, it is also | i Xk k k⊗ Zd | i⊗| Ki
adoublystochasticmap. Thuswehavethemajorization
k is a fixed orthonormal basis for an auxiliary sys-
relationΦ(ρ)≺ρ [22]. By anotherTheoremofUhlmann {te|mKi}, q is a probability distribution, and ψ are
[24], we know there exist a probability distribution {pk} unit Kvect{orks}in . An important observat|ionkihere
and a set of unitary operators {Uk} such that is that ψ is iHndth⊗eZsdupport of Φ I (Ω) which is
Φ(ρ)= pkUkρUk†. spanned| bkyi a set of vectors of the⊗formL(Z(dE)j ⊗IZd)|Ωi,
Xk where we assume that Φ= jEj ·Ej†. Hence
So D(Φ,Conv(U( ));ρ) = 0 for any density oper- P
aDt(oΦre,Cρonfrvo(mU(L()H))H).> 0Oenventhwehoenthetrhehiannpdu,t cwaen hoanvlye |ψki=Xj λj(Ej ⊗IZd)|Ωi
H
be chosen from L( ). This indicates that D and D are for some complex numbers λj, from which we readily
quite different wheHn we do not use auxiliary systems. deduce that
e
ψ =(L I )Ω ,
| ki k⊗ Zd | i
IV. A LOWER BOUND FOR THE DISTANCE where
BETWEEN A QUANTUM CHANNEL AND THE
L = λ E K(Φ).
CONVEX HULL OF UNITARY CHANNELS k j j
∈
Xj
quIatnitsugmencehraanllnyedliΦfficiusltatmoidxetcuirdeeowfhuenthitearryagcihvaennnuenlsitoarl STihnucsew|ψekhiaavree unit vectors, Tr|Lk|2 = Tr(L†kLk) = d.
not. One simple sufficient condition is that the Kraus
oerpaetroart,ori.es.p,aKce(ΦK)(Φ)Ud(oe)s =not.co(nNtaoitneatnhyatutnhitearKyraoups- D(Φ,Ψ)≥2(1−Lmka,qxk|Xk √pkqkTr(L†kUk)/d|})
∩ H ∅
operatorspaceK(Φ)=span{Ek}foraquantumchannel ≥2(1−max{Tr(L†kUk)/d:Lk ∈K(Φ)})
Φ= kEk·Ek†). If this is the case, we can actually ob- 2(1 max 1Tr(L U):TrL2 =d,U U( ) )
tain Pan analytical lower bound for the distance between ≥ − {d † | | ∈ Hd }
Φ and Conv(U( )). 1
H =2(1 max TrL :TrL2 =d,L K(Φ) )
Lemma 2. For any quantum channel Φ T( ) such − d { | | | | ∈ }
d
∈ H
that K(Φ) U( )= , we have 1
∩ Hd ∅ =min Tr(L I )2 :TrL2 =d,L K(Φ)
Tr(L I )2 {d | |− Hd | | ∈ }
D(Φ,Conv(U( ))) min | |− d =C >0. 1
Hd ≥L K(Φ) d Φ inf Tr(L I )2 :L K(Φ) .
∈ ≥ {d | |− Hd ∈ }
8
In the last step we have to use “inf” instead of “min” V. BOUNDS ON THE DISTANCE BETWEEN A
as the domain of L has been broadened from a compact SCHUR CHANNEL AND THE CONVEX HULL
set L K(Φ):TrL2 =d to an unbounded set K(Φ). OF UNITARY CHANNELS
{ ∈ | | }
To finish the proof, we need to show that “inf” in the
lastline canbe replacedby “min”. First, noticethatthe The condition that the Kraus operator space K(Φ) of
RHS of the above equation is less than 2, and Φ does notcontainany unitary operatoris a verystrong
constraint. In most cases we may have that Φ is not
Tr(L I )2 (TrL d)2 a mixture of unitary channels but K(Φ) contains some
| |− Hd | |− .
d ≥ d2 unitary operator. Here we deal with this more general
casebutonlyforSchurchannels. Inthiscaseweareable
If TrL (√2+1)d then the right hand side (RHS) of to show that up to a factor of 1/2, any Schur channel
| | ≥
the above equation is greater than 2. Thus can be approximated by a mixture of diagonal unitary
channels.
Tr(L I )2 Tr(L I )2 Let us denote
inf | |− Hd = min | |− Hd .
L∈K(Φ) d Tr|L|≤(1+√2)d d Λ( d)=S( d) Conv(U( d)).
H H ∩ H
As a final remark, we need show that C > 0 under
Φ Intuitively,Λ( )(orsimplyΛ )isthesetofSchurchan-
d d
the assumption K(Φ) U( ) = . Otherwise, C = 0 H
d Φ nels that are also mixtures of diagonal unitary chan-
∩ H ∅
impliesthatthereissomeL K(Φ)suchthat L =I . nels. So any Ψ Λ can be written into the form
In other words, L is unitarey,∈which is a contra|deic|tionH. (cid:3)d Ψ = kpkUk·Uk†,∈whedre Uk are d×d diagonal unitary
matriPces.
e
Theorem 2. Let Φ T( ) be a quantum channel, and
d
let Ψ∈S(Hm) be an∈y SchHur channel. Then Theorem 3. For given Schur channel Φ∈Sd, we have
1
D(Ψ Φ,Conv(U( ))) C . D(Φ,Λ ) D(Φ,Conv(U )) D(Φ,Λ ). (8)
m d Φ d d d
⊗ H ⊗H ≥ 2 ≤ ≤
Proof: The key observation here is that under the Proof: The second inequality follows directly from
assumption Ψ is with diagonal Kraus operators. Thus Λ Conv(U ). We will employ some standard argu-
d d
⊂
any L K(Ψ Φ) can be decomposed as ments in operator algebras to prove the first inequality.
∈ ⊗ Let Ψ= p Conv(U ) such that
L=⊕mk=1Lk, Lk ∈K(Φ). PkDk(ΦUk,Ψ∈)=D(Φ,dConv(U ))=δ.
d
SupposenowthatL= m L achievesthe minimumin
⊕k=1 k We only need to prove that
C . We have
Ψ Φ
⊗ e e
D(Φ,Λ ) 2δ.
d
1 ≤
C = Tr(L I )2
Ψ⊗Φ md | |− Hm⊗Hd For any two diagonal unitary matrices U and V, let us
= 1 Tr( em (L I ))2 introduce a map JU,V :T(Hd)→T(Hd) as follows:
md ⊕k=1 | k|− Hd
JU,V(Φ)=U Φ(U V)V .
= 1 Tr( m (Le I )2) † · †
md ⊕k=1 | k|− Hd It is obvious that JU,V is an isometry over T( ) in the
d
1 m e following sense: H
= Tr(L I )2
mdkX=1 | k|− Hd D(JU,V(Φ ),JU,V(Φ ))=D(Φ ,Φ ) (9)
e 1 2 1 2
C ,
Φ
≥
for any Φ ,Φ T( ).
1 2 d
∈ H
where we have employed the fact that Now we can further introduce a map J : T( d)
H →
T( ) such that
d
H
Tr(L I )2 dC , 1 k m.
| k|− Hd ≥ Φ ∀ ≤ ≤ J(Φ)= JU,V(Φ)dUdV,
NowthedesireedresultfollowsfromLemma2directly. (cid:3) ZUbdZUbd
As a direct corollary, we have the following
wherebothdU anddV areHaarmeasuresoverthe diag-
onalunitarygroupU . ThemapJsatisfiesthe following
Corollary 1. For any Schur channel Φ S( ), if d
∈ Hd properties:
K(Φ)∩U(Hd)=∅, then i). J is a contractionbin the sense
D(Φ⊗n,Conv(U(Hd⊗n)))≥CΦ >0, ∀n≥1. D(J(Φ1),J(Φ2))≤D(Φ1,Φ2),∀Φ1,Φ2 ∈T(Hd),
9
which is a simple consequence of the convexity of the Clearly Ψ Λ . We will show that Ψ satisfies Eq.
′′ d ′′
∈
diamond norm and Eq. (9). (11). First, we find that Ψ Ψ is a CP map. By a
′′ ′
−
ii). J(Φ) = Φ for any Schur multiplier Φ T( ). direct calculation, we have
d
∈ H
This is true simply due to the following observation
p
k
Ψ Ψ = (W V ) (W V ) .
JU,V(Φ)=Φ, ′′− ′ 4 k− k · k− k †
Xk
where U,V U are diagonal unitary matrices. Thus
d
∈
iii). J(Φ) is a Schur multiplier for any Φ T( ). In
d
b ∈ H
particular,J(Φ)is CPwheneverΦ is CP.To seethat, by (Ψ′′ Ψ′)† =Ψ′′† Ψ′†
− −
a direct calculation, we find that for Φ= kEk·Fk†, is also a CP map.
P
Now employing essentially the same techniques first
J(Φ)= Ek′ ·Fk′†, Ek′ =diag(Ek),Fk′ =diag(Fk). introduced by Haagerup and Musat in [13], we have
X
Clearly, J(Φ) is a Schur multiplier. When Φ is CP, we
can choose ||Ψ′′−Ψ′||♦ =||Ψ′′†−Ψ′†||cb
= Ψ Ψ
′′† ′†
E =F =diag(E )=diag(F ), || − ||∞
k′ k′ k k = (Ψ′′† Ψ′†)(I )
|| − Hd ||∞
thus J(Φ) is CP. = I Ψ (I )
Now we can compute that || Hd − ′† Hd ||∞
= Φ†(I ) Ψ′†(I )
|| Hd − Hd ||∞
Ψ′ =J(Ψ)= pkJ(Uk)= pkAk·A†k, ≤||Φ†−Ψ′†||∞
Xk Xk = Φ Ψ′ 1
|| − ||
Φ Ψ .
where Ak =diag(Uk). We have ≤|| − ′||⋄
That means
D(Φ,Λd) D(Φ,Ψ′)+D(Ψ′,Λd). (10)
≤
The first term in the RHS of Eq. (10) satisfies D(Ψ′,Ψ′′) D(Φ,Ψ′) δ.
≤ ≤
(cid:3)
D(Φ,Ψ)=D(J(Φ),J(Ψ)) D(Φ,Ψ)=δ,
′
≤ ItseemsquitelikelythatinEq. (8)thefirstinequality
where we have employed the contraction property of J, shouldbestrictandthesecondoneshouldbeanequality.
item i) above. However,this is still an unsettled issue.
ItremainstoshowthatthesecondtermintheRHSof
Eq. (10) fulfills Theorem 4. For given Schur channel Φ Sd and arbi-
∈
trary Ψ S , we have
m
∈
D(Ψ,Λ ) δ.
′ d
≤ D(Ψ Φ,Λ ) D(Φ,Λ ), Ψ S . (12)
m d d m
⊗ ⊗ ≥ ∀ ∈
Our strategy is to choose Ψ T( ) such that
′′ d
∈ H
Here m d is a shorthand for .
m d
⊗ H ⊗H
D(Ψ,Ψ ) δ, Ψ Λ (11)
′ ′′ ′′ d
≤ ∈ Proof: To show Eq. (12), we first choose Ψ =
′
Then kpkUk·Uk† ∈Λm⊗d such that
P
D(Ψ′,Λd)≤D(Ψ′,Ψ′′)≤δ. D(Ψ⊗Φ,Λm⊗d)=D(Ψ⊗Φ,Ψ′). (13)
The rest of the proof devotes to finding such Ψ′′. Notice Notice that any diagonalunitary matrix Uk canbe writ-
that eachKrausoperatorAk ofΨ′ is adiagonalcontrac- ten into the following form:
tion. Applying a well-known fact in linear algebra, we
can choose two diagonal unitary matrices V , W such m
k k
that Uk = |jihj|⊗Uj(k),
Xj=1
1
A = (V +W ).
k k k
2 where U(k) U are all diagonalunitary matrices. That
j ∈ d
Now define implies
b
p
Ψ′′ = 2k(Vk·Vk†+Wk·Wk†). Ψ′′ =h1|Ψ|1i= pkU1(k) ∈Λd,
Xk Xk
10
where (k) is the unitary channelcorrespondingto U(k). Example 1. Our first example is chosen from Ref. [7]
U1 1
Intuitively, the compressed version Ψ′′ of Ψ′ remains a (Section 4.3). Φ=E1·E1†+E2·E2†, where
mixture of diagonal unitary channels. Now we have
1 1 1 i
E =Diag(1,0, , ), E =Diag(0,1, , ).
D(Ψ⊗Φ,Ψ′)≥ sup ||(Ψ⊗Φ−Ψ′)(|1ih1|⊗X)||1 1 √2 √2 2 √2 −√2
X 1 1
|| || ≤
= sup 1 1 (Φ Ψ′′)(X) 1 ClearlynoneofE1 andE2 is unitary. We nowshowthat
||| ih |⊗ − ||
X 1 1 there is no unitary in K(Φ). By contradiction, assume
|| || ≤
= Φ Ψ that for some complex numbers λ and µ we have that
′′ 1
|| − || λE +µE is unitary. Then
= Φ Ψ 1 2
′′
|| − ||♦
≥D(Φ,Λd), (λE1+µE2)†(λE1+µE2)=I4,
where we have used the fact that Ψ(1 1) = 1 1 and from which we obtain
| ih | | ih |
Ψ = 1Ψ 1 Λ .
′′ ′ d
h | | i∈ 1 1
Combining the above equationwith Eq. (13), we have λ2 =1, µ2 =1, (λ2+ µ2)=1, (λ2 µ2)=1.
proven Eq. (12). (cid:3) | | | | 2 | | | | 2 | | −| |
A somewhat interesting fact is that the above two re-
Clearly, there is no λ and µ satisfying all the above
sults together can be used to derive some results first
equations. Thus we have K(Φ) U( ) = .
4
obtainedby HaagerupandMusatin Ref. [13], whichare ∩ H ∅
It follows from Corollary 1 that Φ is a counterex-
applicable to the AQBC.
ample to AQBC. One can readily verify that the
Theorem 5. (Haagerup and Musat [13]) For given set {E1†E1,E1†E2,E2†E1,E2†E2} is linearly independent.
Schur channel Φ S and arbitrary Ψ S , we have Thus it follows from Corollary 2.3 of Ref. [11] that Φ is
d m
∈ ∈ a non-factorizable map. (cid:3)
1
D(Ψ Φ,Conv(U )) D(Φ,Conv(U )). (14)
⊗ m⊗d ≥ 2 d Example2. OursecondexampleistakenfromRef. [11]
Proof: Actually Eq. (14) is a quite straightforward (Example 3.3). Φ= 3k=1Ek·Ek†, where
P
application of the above two theorems. First notice that
1 2
Ψ⊗Φ∈Sm⊗d. Applying Theorem 3 to Ψ⊗Φ, we have E1 =Diag(1,√5I5),E2 =Diag(0,r5Z5),E3 =E2†,
1
D(Ψ⊗Φ,Conv(Um⊗d))≥ 2D(Ψ⊗Φ,Λm⊗d). where Z5 =Diag(1,2π5i,45πi,65πi,85πi) satisfying Z55 =I5.
In the following we directly write I and Z for I and Z ,
5 5
On the other hand, it is obvious that
respectively.
D(Φ,Λ ) D(Φ,Conv(U )). As shown in [11], one can choose a set of Hermitian
d d
≥ Kraus operators F ,F ,F such that
1 2 3
Thus the left thing is to show
1 1
F =E , F = (E +E ), F = (E E ).
D(Ψ Φ,Λ ) D(Φ,Λ ), 1 1 2 2 2 3 3 2i 2− 3
m d d
⊗ ⊗ ≥
and this is exactly the content of Theorem 4. (cid:3) ItiseasytoseethatΦ= 3k=1Fk·Fk†. ByCorollary2.5
of Ref. [11], Φ is a factoriPzable map.
Corollary 2. (Haagerup and Musat [13]) Let Φ S
be a Schur channel that does not satisfy the quan∈tumd Now we show that K(Φ)∩U(H6)=∅. Again by con-
tradiction, assume there are complex numbers λ ,λ ,λ
Birkhoff property, that is, Φ Conv(U ). Then Φ does 1 2 3
not satisfy the asymptotic q6∈uantum Bdirkhoff property, such that λ1E1 + λ2E2 + λ3E3 is a unitary. In other
and words,Diag(λ1, 15λ1I+ 25λ2Z+ 25λ3Z−1) is a uni-
q q q
tary. This is equivalent to
1
D(Φ n,Conv(U )) D(Φ,Conv(U )).
⊗ d⊗n ≥ 2 d λ 2 =1, λ I +√2λ Z+√2λ Z 1 =√5I.
1 1 2 3 −
| | | |
For simplicity, we may assume λ = 1, a = √2λ , and
VI. SOME EXPLICIT COUNTEREXAMPLES 1 2
b = √2λ . Then we can rewrite the above equation as
TO THE ASYMPTOTIC QUANTUM BIRKHOFF 3
CONJECTURE follows:
(a2+b2 4)I+(a+b )Z+(a +b)Z 1+ab Z2+ab Z 2 =0.
∗ ∗ − ∗ ∗ −
Our results in Section IV enable us to construct coun- | | | | −
terexamples to AQBC easily. Our basic strategy is to Employingthefactthat I,Z,Z 1,Z2,Z 2 arelinearly
− −
construct Schur channel Φ satisfying K(Φ) U( ) = . independent, we have { }
∩ H ∅
ThenthestatementthatΦisacounterexampletoAQBC
follows directly from Corollary 1. a2+ b2 =4,a+b =0,ab =0.
∗ ∗
| | | |