Fidelity between a bipartite state and another one undergoing local unitary dynamics

0.18 MB·English

by  Lin Zhang

Fidelity between one bipartite quantum state and another
undergoing local unitary dynamics
5 Lin Zhang1 , Lin Chen2,3†, Kaifeng Bu4‡
1 ∗
0 1InstituteofMathematics,HangzhouDianziUniversity,Hangzhou310018,PRChina
2
g 2SchoolofMathematicsandSystemsScience,BeihangUniversity,Beijing100191,PRChina
u
3InternationalResearchInstituteforMultidisciplinaryScience,BeihangUniversity,Beijing100191,PRChina
A
4DepartmentofMathematics,ZhejiangUniversity,Hangzhou310027,PRChina
9
2

]
h
Abstract
p
-
t Thefidelityandlocalunitarytransformationaretwowidelyusefulnotionsinquantum physics. We
n
a studytwoconstrainedoptimizationproblemsintermsofthemaximalandminimalfidelitybetweentwo
u bipartitequantum statesundergoinglocalunitary dynamics. Theproblemsarerelatedtothe geometric
q
[ measure of entanglement and the distillabilityproblem. We show that the problems can be reducedto

semi-definiteprogrammingoptimizationproblems. Wegiveclose-formformulaeofthefidelitywhenthe
4
v twostatesarebothpurestates,orapureproductstateandtheWernerstate. Weexplainfromthepoint
2
of view of local unitary actions that why the entanglement in Werner states is hard to accessible. For
4
7 generalmixedstates, we giveupper and lowerbounds of the fidelityusing tools such as affine fidelity,
7 channels and relativeentropyfrominformationtheory. Wealsoinvestigatethe poweroflocalunitaries,
0
andtheequivalenceofthetwooptimizationproblems.
.
1
0
5 Keywords: Fidelity;bipartitestate;localunitarytransformation;Wernerstate
1
:
v
i 1 Introduction
X
r
a Finding suitable quantities for characterizingthe correlationsin a bipartite or multipartite quantum state
has been an important problem in quantum information theory. Three well-known quantities are entan-
glement, fidelity and mutual information [1]. Investigating the quantities under unitary dynamics has
variousphysicalapplications. Thelocalevolutionoffreeentangledstatesintoboundentangledorsepara-
blestatesinfinitetimepresentsthephenomenonofsuddendeathofdistillability. Inthephenomenon,the
fidelity was used to evaluate how close the evolved state is close to the initial state [2]. Next, finding out
the localunitary orbits of quantum statescharacterizestheir propertiesfor various quantum-information
tasks, and it is also mathematically operational [3]-[8]. By searching for the maximally and minimally
∗E-mail:[email protected];[email protected]
†E-mail:[email protected](correspondingauthor)
‡E-mail:[email protected]
1
correlatedstatesonaunitaryorbit,theauthorsin[9]quantifiedtheamountofcorrelationsintermsofthe
quantum mutual information. The correlations in a multipartite state within the construction of unitary
orbitshavebeenalsoexamined[10]. These applicationsoriginate fromthe factthatthe unitarydynamics
influences the interaction of quantum systems. It is thus a widely concerned question to characterize
how heavy the influence can be under certain metric such as the fidelity. The latter has been used to
evaluate the entangled photon pairs obtained by experimental heralded generation [11], and the unitary
gates of experimentally implementing quantum error correction [12]. In contrast to the global unitary
dynamicswhich involvesnonlocalcorrelation,thelocalunitaryactioncanbelocallyperformedanddoes
not change the properties of quantum states. Because of the easyaccessibility in mathematics, the global
unitary dynamics has been studied a lot [13]. In contrast, much less is known about the local unitary
dynamics.
Inthispaper,westudythemaximalandminimalfidelitybetweentwobipartitequantumstates,oneof
whichundergoesarbitrarylocalunitarydynamics. Tobemorespecific,letρ and σ betwobipartitestates
actingonthe Hilbertspace ofdimensions dim = d, i =1,2. LetU( ) bethe unitarygroup
1 2 i i 1
H ⊗H H H
on . We proposetwoconstrainedoptimizationproblemsascomputingthefunctionals
1
H
G (ρ,σ):= max F(ρ,(U U )σ(U U )†) (1.1)
max 1 2 1 2
Ui∈U(Hi):i=1,2 ⊗ ⊗
and
G (ρ,σ):= min F(ρ,(U U )σ(U U )†) (1.2)
min 1 2 1 2
Ui∈U(Hi):i=1,2 ⊗ ⊗
where F(ρ,σ) := Tr √ρσ√ρ is the fidelity between any two semidefinite positive matrices ρ and σ.
Because of the symmetric property of fidelity, the two functionals are unchanged under the exchange of
(cid:0)p (cid:1)
arguments ρ and σ. Note that if the local unitary action is replaced by global unitary action, then the
problemshavebeenanalyticallysolvedin[13].
Intuitively, the functionals respectivelystand for the maximaland minimal distance that local unitary
cancreatebetweenquantum states. The solution to the optimization problems exists becausethe unitary
group isacompactLiegroup. Wewillshowthattheycanindeedbereducedtothewell-knownsemidef-
inite programming(SDP)problems. Sowemayefficientlycompute thefunctionalsformanystates. Then
we derivethe close-formformulaesto the functionals when ρ and σ arebothpure states,or apureprod-
uct state and the Werner state. We show that in contrast with the separable Werner state, the entangled
Werner state of d > 3 is closer to the set of pure separable states under local unitary dynamics. In this
context, the distillability of two-qubit and two-qutrit Werner states may be distinguished by comparing
theirG . Forgeneralmixedstates,wederivetheupperandlowerboundsofthefunctionalsintermsof
max
the monotonicity of fidelity, quantum channel, the affine fidelity, the integral over the unitary group via
Haarmeasure. We alsoinvestigate how localunitariesinfluence the commutativity of quantum states, as
wellastheequivalenceof thetwooptimizationproblems.
Our results straightforwardly make progress towards the following quantum-information problems.
First,G (ρ,σ)reducestothe geometricmeasureof entanglement(GME)when ρor σ isapureproduct
max
state [14, 15]. Mathematically the GME of a quantum state ρ is defined as max ψ ρ ψ where ψ is a
ψ
h | | i | i
product state. It is known that the GME of a bipartite state measures the closest distance between this
2
state and separable states. It coincides with the intuitive interpretation of the functionals. The GME is
a multipartite entanglement measure and has been extensively studied recently [16, 17]. The GME also
applies to the construction of initial states for Grover algorithm [19, 18], the discrimination of quantum
statesunderlocaloperationsandclassicalcommunications(LOCC)[20],andone-wayquantumcomputa-
tion [16]. For a review of GME we refer the readersto [17]. Recall that the fully entangled fraction (FEF)
for any bipartite state ρ in a d d system is defined as the maximal overlap with maximally entangled
⊗
purestates,
max Ω (U V)ρ(U V)† Ω ,
U,V unitaries ⊗ ⊗
D (cid:12) (cid:12) E
where|Ωi = √1d ∑dj=−01|jjiisthemaximallyenta(cid:12)(cid:12)ngledstate. Then,G(cid:12)(cid:12)max(ρ,σ)isthesquarerootoftheFEF
when one state of ρ and σ is a maximally entangled state for d = d [21]. In this case, the other state
1 2
of ρ and σ can be any mixed state. The FEF works as the fidelity of optimal teleportation, and thus has
experimental significance [22]. The close-form for the FEF in a two-qubit system is derived analytically
byusing themethod ofLagrangemultiplier [23]. Second,G (ρ,σ)isrelatedtothe famousdistillability
min
probleminentanglementtheory. Thelatterisrelatedtotheadditivitypropertyofdistillableentanglement
andtheactivationofboundentanglement[24]. Itisknownthatabipartitestateρisdistillableifandonlyif
thereexistsapositiveintegernandaSchmidt-ranktwopurestate ψ ,suchthat ψ (ρ n)Γ ψ <0[25,26].
⊗
| i h | | i
Ouroptimizationproblemsimplythatρisdistillableifandonlyifmin G2 ( φ ,(ρ n)Γ+x1)< x,
λ∈(0,1) min | λi ⊗
where φ = √λ 00 +√1 λ 11 and x isa positive number such thatthe second argument ispositive
λ
| i | i − | i
semi-definite. Westressthatthedifficultyofthedistillabilityproblemmostlyarisesfromthelocalunitary
orbits involved in the optimization problems above. The distillability problem has turned out to be hard,
and a review of recent progress can be found in [27]. All these problems are thus well motivated by the
findingsinthispaper.
The rest of the paper is organized as follows. In Sec. 2, we show that the computation of the two
functionals G and G can be reduced to the SDP problem. Then we derive the close-form formulae
max min
of functionals when ρ and σ are both pure states, or a pure product state and the Werner state. We also
pointoutapotentialconnectionbetweenthedistillabilityproblemandouroptimizationproblemforG .
max
Next,severalconnections, upperandlowerboundsonthefunctionalsarecomputedinSec.3. Wediscuss
inSec.4,andconcludeinSec.5.
2 SDP and analytical formula of functionals
We see that F(ρ,(U V)σ(U V)†) is a continuous function over local unitary groups U( ) U( ).
1 2
⊗ ⊗ H ⊗ H
Since U( ) and U( ) arecompactLie groups, it follows that thereexists U,V U( )(i = 1,2) such
1 2 i i i
H H ∈ H
that
G (ρ,σ)=F(ρ,(U U )σ(U U )†) and G (ρ,σ)=F(ρ,(V V )σ(V V )†).
max 1 2 1 2 min 1 2 1 2
⊗ ⊗ ⊗ ⊗
Denote σ :=(U U )σ(U U )† and σ = (V V )σ(V V )†. Thus
1 2 1 2 1 2 1 2
⊗ ⊗ ⊗ ⊗
b Gmax(ρ,σ)=e F(ρ,σ) and Gmin(ρ,σ)=F(ρ,σ).
The SDP has been extensively used to treat thebdistillability problem [28e], the separability problem [29],
thequantificationofentanglement[30]andsoon[31]. TheSDPforfidelitybetweentwostatesisobtained
3
by Watrous [1]. We show that our problems of computing G and G can be reduced to the SDP
max min
optimizationproblem[32,33],asaprimalproblembelow. Letτ =σ or σ. Then
1
maximize: Tr(X)+Tr Xb† ,e (2.1)
2
(cid:16) (cid:16) (cid:17)(cid:17)
ρ X
subjectto: >0, (2.2)
" X† τ #
where X isaoperatoroforderd d . Undertheaboveconstrainttheoptimalvalueof 1 Tr(X)+Tr X†
1 2 2
isfidelityF(σ,τ). Itsdualproblemis
(cid:0) (cid:0) (cid:1)(cid:1)
1
minimize: ( ρ,Y + τ,Z ), (2.3)
2 h i h i
Y 1
subjectto: − >0, (2.4)
" 1 Z #
−
where Y,Z are Hermitian operators. So we can numerically solve the optimization problems for many
stateswithhighefficiency. Ontheotherhand,wecananalyticallysolvetheproblemsforpurestates.
Theorem 2.1. Let ρ = Φ Φ and σ = Ψ Ψ , where the spectra of reduced density operators ρ and
12 12 12 12 1
| ih | | ih |
σ are a > > a > 0 and b > > b > 0 , respectively, and d = d = d . Then G (ρ,σ) =
1 1 N 1 N 1 2 max
{ ··· } { ··· }
∑d a b andG (ρ,σ)=0.
j=1 j j min
q
Proof. There are two d d matrices A,B such that Φ = vec(A) and Ψ = vec(B) [1]. Then
2 1 12 12
× | i | i
(U U ) Ψ = vec(U BUT) implies F(ρ,(U U )σ(U U )†) = Tr A†U BUT . Since ρ = AA†
1⊗ 2 | 12i 1 2 1⊗ 2 1⊗ 2 1 2 1
and σ1 = BB†, the first assertion follows from the fact maxU,V|Tr(XUY(cid:12)(cid:12)V)(cid:0)| = ∑kN=1s(cid:1)k((cid:12)(cid:12)X)sk(Y) [34]. The
second assertion is equivalent to the fact that the product states in the Schmidt decomposition of Φ
12
| i
canbebylocalunitariesconvertedintostatesorthogonaltothose of Ψ .
12
| i
When Φ isfromamaximallyentangledbasis,theproofforG (ρ,σ)=0issimilartothetechnique
12 min
| i
fortheknownquantumsuper-densecoding. Ontheotherhand,aparticularcaseofthistheoremhasbeen
usedtoconstructafamilyofentanglementwitnesses[35]. Nextifρisapureproductstate,thenG2 (ρ,σ)
max
amountstotheS(1)-norm σ ,whichislowerboundedbythe(d +d 1)-theigenvalueinincreasing
k kS(1) 1 2−
orderof σ [36],wherethe S(k)-normofbipartiteoperator X on isdefinedas
1 2
H ⊗H
X := sup u X v :SR( u ),SR( v )6k .
k kS(k) {|h | | i| | i | i }
|ui,|vi∈H1⊗H2
Here SR( w ) stands for the Schmidt-rank of pure bipartite state w , i.e. the number of
1 2
| i | i ∈ H ⊗H
nonvanishing coefficientsinSchmidtdecomposition of w .
| i
The problem can be analytically solved for the Werner state σ(t) = 1 (1 1 t∑d ij ji ),
d(d t) d⊗ d− i,j=1| ih |
−
t [ 1,1]. Indeed[37]implies
∈ −
1+ min(t,0)
G (ρ,σ)= σ = | |. (2.5)
max k kS(1) s d(d t)
q −
One can easily show that the minimum of this functional over t is achievable when t = 0. That is, the
Werner state becomes the maximally mixed state, which is at the center of the set of separable states.
4
Next, the maximum of the functional maybe reached attwo points, t = 1 and 1 as plotted in Figure 1.
−
The twopoints respectivelycorrespondtoaseparableWerner stateandthe most entangledWerner state.
Figure 1 implies that in contrast with the separable Werner state, the entangled Werner state of d > 3
is closer to the set of pure separable states under local unitary dynamics. That is, the entanglement of
Werner states might be a more unaccessible quantum correlation than the separability. It is known that
two-qubit entangled states are distillable, and it is conjectured that two-qutrit entangled Werner states
may be non-distillable [38]. Our inequalities imply that the distillability of Werner states of d = 2 and
d > 2 may be essentially distinguished by their distance to the pure separable states under local unitary
dynamics.
0.75
d=2
0.7
d=3
0.65 d=4
0.6
0.55
max 0.5
G
0.45
0.4
0.35
0.3
0.25
−1 −0.5 0 0.5 1
t
Figure 1: The figure shows the maximum distance between the
Werner state and the set of pure product states in terms of the
functionalGmax. Theleftmostvalueissmallerthantherightmost
value for d = 2, and bigger than the rightmostvalue for d = 4.
Thetwovaluesareequalford=3.
In Appendix 5, we also have computed G for the pure product state and the isotropic state. In
max
spiteoftheseresults,findingtheanalyticalsolutiontotheoptimizationproblemsisunlikelybecauselocal
unitary actions are much more involved than global unitary action U. Indeed, the extremal values of
F(ρ,UσU†) aredeterminedbythose globalunitarysuchthatthe commutator [ρ,UσU†] =0[13]. For our
purpose we needto replacethe globalunitary action bylocal unitary U U . We donot know whether
1 2
⊗
thereexistU andU suchthatF(ρ,(U U )σ(U U )†) canattainitsextremalvalues. Innextsection
1 2 1 2 1 2
⊗ ⊗
westudytheupperandlowerboundsofG andG ,aswellastheir connections.
max min
3 Upper and lower bounds of functionals
TheoptimizationproblemsasktofindoutthecriticalpointsargG (ρ,σ)andargG (ρ,σ)inthelocal
max min
unitary group, which respectively achieve G and G . They respectively refer to the local unitary
max min
operatorV V suchthatG (ρ,σ)=F(ρ,(V V )σ(V V )†)andG (ρ,σ)=F(ρ,(V V )σ(V
1 2 max 1 2 1 2 min 1 2 1
⊗ ⊗ ⊗ ⊗ ⊗
V )†). Using these facts, we construct several relations between G and G . In Proposition 3.1 and
2 max min
5
3.2, we show that the sum and difference of G and G (or their squares) are both upper and lower
max min
boundedintermsofsomefunctionsoftheirargumentssuchastherankandfidelity. Asaresult,westudy
thefidelityinequalityinProposition 3.3,andtheaffinefidelityofglobalunitaryactioninProposition 3.4.
WeconstructtheboundsofG andG byusingthemonotonicityofthefidelity,theaffinefidelity,the
max max
integraloverthe unitarygroup viaHaarmeasure,andtherelativeentropyinSubsect3.1and3.2.
Proposition3.1. Letρ,σ,andσ = 1 (1 σ)bethreequantumstates. Wehave
′ d1d2−1 d1d2 −
rank(ρ)>Gmax(ρ,σ)2+(d1d2 1)Gmin(ρ,σ′)2 >1. (3.1)
−
The first equality holds if there exist two unitary matrices W ,W such that conditions (i),(ii) and (iii) hold. The
1 2
second equality holds if and only if there exist two unitary matrices V ,V such that conditions (iv),(v), and (vi)
1 2
hold:
(i) W W =argG (ρ,σ)=argG (ρ,σ );
1 2 max min ′
⊗
(ii) √ρ(W1 W2)σ(W1 W2)†√ρand√ρ(W1 W2)σ′(W1 W2)†√ρbothhaveidenticalnonzeroeigenvalues;
⊗ ⊗ ⊗ ⊗
(iii) rank √ρ(W W )√σ =rank √ρ(W W )√σ =rank(ρ);
1⊗ 2 1⊗ 2 ′
(cid:0) (cid:1) (cid:16) (cid:17)
(iv) V V =argG (ρ,σ)=argG (ρ,σ);
1 2 max min ′
⊗
(v) √ρ(V V )σ(V V )†√ρhasrankone;
1 2 1 2
⊗ ⊗
(vi) √ρ(V1 V2)σ′(V1 V2)†√ρhasrankone.
⊗ ⊗
TheproofisgiveninAppendix5. One caneasilyverifythatthe secondequalityin(3.1)holdswhenρ
ispure,or ρ = 1( 00 00 + 01 01 )and (V V )σ(V V )† = 00 00 . Inbothcases, computing G
2 | ih | | ih | 1⊗ 2 1⊗ 2 | ih | max
andG areequivalent. Thisisthefirstconnection wehavebetweenthetwofunctionals, soitisenough
min
toconsideronlyoneofthem. WeshalldiscussthegeneralcaseinSec.4. Furthermore,conditions5and6
implythatρhasrankatmosttwo. Ifithasranktwo,thenconditions5and6implythatσispure. Thus,at
leastoneofρandσispurewhenthesecondequalityin(3.1)holds. Nextweconstructanotherrestriction
between G and G or their squares. This is realized based on the inequalities for the framework of
max min
wave-particleduality[39]andtheensemblesof Holevoquantity[41].
Proposition 3.2. Let ρ,σ, and σ = 1 (1 σ) be three quantum states. Assume that U U =
′ d1d2−1 d1d2 − 1 ⊗ 2
argG (ρ,σ)andV V =argG (ρ,σ ). Then
max 1 2 min ′
⊗
Gmax(ρ,σ)+Gmin(ρ,σ′)6 2+2F(σ,σ′), (3.2)
q
whereσ = (U1 U2)σ(U1 U2)† andσ′ = (V1 V2)σ′(V1 V2)†. Mbobreover,
⊗ ⊗ ⊗ ⊗
b G2max(ρ,bσ)−G2min(ρ,σ′) 6 1−F2(σ,σ′) (3.3)
(cid:12) (cid:12) q
(cid:12) (cid:12)
and (cid:12) (cid:12) b b
Gmax(ρ,σ)−Gmin(ρ,σ′) 6 1−F2(σ,σ′). (3.4)
q
(cid:12) (cid:12)
(cid:12) (cid:12) b b
6
Proof. Theassertionstraightforwardlyfollowfromthreeinequalitiesin[39,40]and[41]:
max(F(M,ρ)+F(N,ρ))= Tr(M)+Tr(N)+2F(M,N), (3.5)
ρ
q
F2(σ,ρ) F2(τ,ρ) 6 1 F2(σ,τ), (3.6)
− −
(cid:12) (cid:12) q
(cid:12) (cid:12)
and (cid:12) (cid:12)
F(σ,ρ) F(τ,ρ) 6 1 F2(σ,τ). (3.7)
| − | −
q
where M,N arepositivesemidefinite operators,and ρ,σ,τ arethreequantumstates.
The results show that the characterization of fidelity is important for obtaining a tight bound for the
functionals. We study the characterization using an inequality for the approximation of Markov chain
property[42].
Proposition3.3. Letρandσ betwoquantumstatesonCd,andΦbeaquantumchanneloverCd. Then
F(ρ,σ)6∑F(M ρM†,M σM†)6F(Φ(ρ),Φ(σ)), (3.8)
j j j j
j
whereΦ( ) =∑ M M† isanyKrausrepresentationofΦ.
∗ j j∗ j
Proof. FromLemmaB.7in[42],weknowthatforanidentityresolution ∑ E =1,
j j
F(ρ,σ)6∑F(E ρE,σ).
j j
j
Since Φ( ) = ∑ M M† is a quantum channel, ∑ M†M = 1. Assuming E = M†M in the above
∗ j j ∗ j k j j j j j
inequality, wehave
F(ρ,σ)6∑F(M†M ρM†M,σ). (3.9)
j j j j
j
Again,byemployingthe followingsimplefact,LemmaB.6in[42],F(W†ρW,σ)=F(ρ,WσW†),weobtain
theinequalityin(3.8). Theotherinequalityisfromthe concavityoffidelity.
Proposition3.4. ForanygiventwoquantumstatesρandσonCd,thereexistsaunitaryoperatorU onCd,which
0
dependsonρandσ,suchthat
F(ρ,σ)=A(ρ,U σU†), (3.10)
0 0
whereA(ρ,σ)iscalledaffinefidelity[43,44],definedbyA(ρ,σ):=Tr √ρ√σ .
Proof. Consideramapdefinedoverthe unitarygroup U(Cd) inthe(cid:0)followi(cid:1)ng:
g(U)=A(ρ,UσU†).
Apparently, g is a continuous map over U(Cd). Furthermore, g(1 ) 6 F(ρ,σ) s a basic matrix inequality.
d
It implies that the affine fidelity is upper bounded by the fidelity. Since the unitary group U(Cd) is a
7
compactandconnectedLiegroup,itfollowsthattheimageofU(Cd)underthemap gisaclosedinterval.
Thus itsufficesto show thatthere existsa unitary operator V suchthat F(ρ,σ)6 g(V). We proceedwith
thefollowing resultobtainedin[13]: thereexistsaunitaryoperatorV U(Cd)suchthat
∈
F(ρ,σ)=Tr exp log√ρ+Vlog√σV† .
(cid:16) (cid:16) (cid:17)(cid:17)
By Golden-Thompson inequality Tr eA+B 6 Tr eAeB , where A and B are Hermitian, we get that
F(ρ,σ) 6 g(V). Now the fidelity F(ρ,σ) im(g), the image of g. Therefore there exists a unitary op-
(cid:0) (cid:1) (cid:0) (cid:1)
∈
eratorU U(Cd)satisfyingthepropertythatwewant. Wearedone.
0
∈
Inthefollowingtwosubsections,wewillrespectivelyderivetheupperandlowerboundsofG and
max
G .
min
3.1 Bounds of G
max
Themonotonicity ofthefidelityimpliesthe upperbound
G (ρ,σ)6min maxF(ρ ,U σ U†),maxF(ρ ,U σ U†),maxF(ρ,U σU† ) . (3.11)
max 1 1 1 1 2 2 2 2 12 12
(cid:18) U1 U2 U12 (cid:19)
Thisboundisanalyticallyderivable,aswehavecomputedexplicitlymax F(ρ,UσU†)andmin F(ρ,UσU†)
U U
[13]. This result directly applies to the computation of max F(ρ,U σU† ). Next we obtain a lower
U12 12 12
bound of G . From a well-known fact in matrix analysis: Tr(A) 6 Tr( A ) for any matrix A, where
max
| | | |
A =√A†A,letting A =√ρ√σ givesriseto
| |
Tr √ρ√σ 6Tr √ρ√σ .
Clearly Tr √ρ√σ is a nonnegative r(cid:12)(cid:12)eal(cid:0)number(cid:1)(cid:12)(cid:12)and F((cid:0)ρ(cid:12)(cid:12),σ) = (cid:12)(cid:12)T(cid:1)r √ρ√σ , thus F(ρ,σ) > A(ρ,σ) for
anytwosta(cid:0)tesρ,σ(cid:1),thenwehave (cid:0)(cid:12) (cid:12)(cid:1)
(cid:12) (cid:12)
G (ρ,σ) > max A(ρ,(U U )σ(U U )†)
max 1 2 1 2
Ui∈U(Hi):i=1,2 ⊗ ⊗
> A(ρ,(U U )σ(U U )†)dµ(U )dµ(U )
1 2 1 2 1 2
ZU(d1)ZU(d2) ⊗ ⊗
Tr √ρ Tr √σ
= . (3.12)
d d
(cid:0) (cid:1)1 2(cid:0) (cid:1)
We have denoted the uniform normalized Haar measure by µ(U) over the unitary group. On the other
hand,theinequality[45]
S(ρ σ)> 2logF(ρ,σ) (3.13)
|| −
whereS(ρ σ):=Tr(ρ(logρ logσ))isthequantumrelativeentropy,implies
|| −
min S(ρ (U U )σ(U U )†)> 2logG (ρ,σ). (3.14)
1 2 1 2 max
Ui∈U(Hi):i=1,2 || ⊗ ⊗ −
Sowehaveobtainedalowerboundof (1.1)
G (ρ,σ)
max
Tr √ρ Tr √σ 1
>max ,exp min S(ρ (U U )σ(U U )†) . (3.15)
1 2 1 2
(cid:26) (cid:0) d(cid:1)1d2(cid:0) (cid:1) (cid:18)− 2Ui∈U(Hi):i=1,2 || ⊗ ⊗ (cid:19)(cid:27)
8
3.2 Bounds of G
min
Clearly
G (ρ,σ) 6 min F(ρ,(U U )σ(U U )†)dµ(U )
min 1 2 1 2 1
U2 ZU(d1) ⊗ ⊗
6 minF(ρ,1 /d U σ U†)
U2 d1 1⊗ 2 2 2
1
6 Tr(√ρ), (3.16)
√d d
1 2
where the last inequality follows from the integral over U(d ). By exchanging ρ and σ in the inequality,
2
weobtainanupperboundof(1.2)
1
G (ρ,σ)6 min Tr(√ρ),Tr √σ . (3.17)
min √d d
1 2
(cid:8) (cid:0) (cid:1)(cid:9)
Next,westudythelowerbound ofG . LetS(ρ):= Tr(ρlogρ) bethevonNeumannentropywith
min
−
the natural logarithm log and 0log0 0. It is obtained in [46] that for full-ranked states ρ,σ D Cd ,
≡ ∈
wehave (cid:16) (cid:17)
1
F(ρ,σ)>Tr(√ρ) exp ∑λ↓(ρ)logλ↑(σ) , (3.18)
× 2 j j j !
where λ (ρ) denotes the set of eigenvalues of ρ in the decreasing order and λ (σ) denotes the set of
↓ ↑
eigenvalues of σ in the increasing order. It is known that the fidelity is unchanged under the exchange
of arguments. Assuming ρ = ρ, σ = (U U )σ(U U )† and exchanging them in (3.18), we obtain a
1 2 1 2
⊗ ⊗
constantlowerboundof(1.2)
1
Gmin(ρ,σ) > max(cid:26)Tr(√ρ)×exp 2∑j λ↓j(ρ)logλ↑j(σ)!,
1
Tr √σ exp ∑λ↓(σ)logλ↑(ρ) ,
× 2 j j j !
(cid:0) (cid:1)
1
exp max S(ρ (U U )σ(U U )†) , (3.19)
1 2 1 2
(cid:18)−2Ui∈U(Hi):i=1,2 || ⊗ ⊗ (cid:19) (cid:27)
wherethelastargumentfollowsfrom(3.13).
4 Discussion
In this section we investigate the power of local unitaries for the commutativity of quantum states, the
quantificationofthecommutativity,andtheequivalenceofthetwooptimizationproblems. Theygeneral-
izethepreviousdiscussion.
Commutative quantum states can be prepared in the same pure state basis. They not only share
operational mathematicalproperties, and also cansave resources in experiments. One might expectthat,
underlocalunitarydynamicswecouldmaketwonon-commutativequantumstatesbecomecommutative.
That is, given two mixed states ρ and σ with [ρ,σ] = 0, are always there local unitaries U and U such
1 2
6
9
that [ρ,(U U )σ(U U )†] = 0? Unfortunately The answer is negative. Indeed, let ρ = ∑ p a a
1⊗ 2 1⊗ 2 i i| iih i|
and σ = ∑ q b b be their respective spectral decomposition, and p,q > 0 for all i,j. If the answer is
j j| jih j| i j
yes, then there exist two sets a and b , such that up to overall phases they are from the same o.
i j
{| i} {| i}
n. basisof . Belowweconstructacounterexampletothisstatement. Hencetheanswerisno. We
1 2
H ⊗H
considertwotwo-qubitstates
1 1 1
ρ = Ψ+ Ψ+ + Ψ− Ψ− + Φ+ Φ+ , (4.1)
2| ih | 3| ih | 6| ih |
2 1
σ = 00 00 + 11 11 , (4.2)
3| ih | 3| ih |
where Ψ = 1 ( 00 11 ) and Φ = 1 ( 01 10 ) are the standard EPR pairs. Since the pos-
| ±i √2 | i±| i | ±i √2 | i±| i
itive eigenvalues all have multiplicity one, the eigenstates of them of ρ and σ are respectively equal to
Ψ , Φ+ and 00 , 11 , up to overall phases on these states. If the answer is yes, then there are
±
{| i {| i} {| i | i}
localunitariesU ,U suchthat Ψ , Φ+ and (U U ) 00 ,(U U ) 11 arefromthesameo. n.
1 2 ± 1 2 1 2
{| i {| i} { ⊗ | i ⊗ | i}
basis of C2 C2. Since the former consists of entangled states and the latter consists of separable states,
⊗
they have to be pairwise orthogonal. It is impossible because the former and latter respectively span a
3-dimensionaland2-dimensionalsubspaceinC2 C2.
⊗
Another interesting problem is whether the two optimization problems are equivalent for general ρ
and σ. The equivalence would imply the sufficiency of solving only one of them. We propose to study
tworelatedfunctionals
max Tr ρ(U U )σ(U U )†
1 2 1 2
Ui∈U(Hi):i=1,2 (cid:16) ⊗ ⊗ (cid:17)
and
min Tr ρ(U U )σ(U U )† ,
1 2 1 2
Ui∈U(Hi):i=1,2 (cid:16) ⊗ ⊗ (cid:17)
astheyalsomeasurethesimilaritybetweenmixedstatesρandσ. Thecomputationofthetwofunctionals
isequivalent, becauseif we cancompute the former for any ρ and σ, thenwe canalsocompute the latter
byreplacingσ by1 σ;andviceversa. Soitsufficestocompute
−
max Tr ρ(U U )σ(U U )† .
1 2 1 2
Ui∈U(Hi):i=1,2 (cid:16) ⊗ ⊗ (cid:17)
Next,itisknown that Tr(UA) 6Tr(√A†A) foranyunitaryU andanymatrix A. Wehave
| |
max Tr ρ(U U )σ(U U )† 6G (ρ2,σ2). (4.3)
1 2 1 2 max
Ui∈U(Hi):i=1,2 (cid:16) ⊗ ⊗ (cid:17)
Sothetwofunctionalsarenotonlyphysically,butalsomathematicallyrelatedtoG andG .
max min
5 Conclusions
In this paper we have studied two optimization problems that are related to many quantum-information
problems. The problems are generally solvable by the SDP, and we manged to work out the analytical
formulae for some states. For mixed states we have constructed many upper and lower bounds of the
two functionals. We have shown that the entanglement of Werner states might be a more unaccessible
quantum correlation than the separability in terms of the local unitary dynamics. We have investigated
10