The separability versus entanglement problem

3.5 MB·

by  Sreetama Das

LecturesonQuantumInformation
Chapter1: Theseparabilityversusentanglementproblem
SreetamaDas1,2,TitasChanda1,2,MaciejLewenstein3,4,AnnaSanpera4,5,AditiSen(De)1,2,andUjjwalSen1,2
1Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
2HomiBhabhaNationalInstitute,TrainingSchoolComplex,Anushaktinagar,Mumbai-400094,India
3ICFO-InstitutdeCie`nciesFoto`niques,TheBarcelonaInstituteofScienceandTechnology,08034Castelldefels(Barcelona),Spain
4 ICREA, Passeig de Lluis Companys 23, E-08010 Barcelona, Spain
5 Departament de F´ısica, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain
Contents
7
1
I. Introduction 2
0
2
II. Bipartitepurestates: Schmidtdecomposition 2
n
a
III. Bipartitemixedstates: Separableandentangledstates 3
J

9
IV. Operationalentanglementcriteria 3

A. PartialTransposition 4
]
h B. Majorization 5
p C. Cross-normormatrixrealignment 6
-
D. Covariancematrix 7
t
n 1. Definitionandproperties 7
a
2. Covariancematrixcriterionforseparability 8
u
q
V. Non-operationalentanglementcriteria 10
[
3. TechnicalPreface 10

1 A. EntanglementWitnesses 10
v
1. EntanglementWitnessfromtheHahn-Banachtheorem 10
7
2. Examples 12
8
1 B. Positivemaps 13
2 1. Introductionanddefinitions 13
0 2. Positivemapsandentangledstates 14
. 3. Choi-JamiołkowskiIsomorphism 15
1
0 C. Rangecriterion 16
7
1 VI. Bellinequalities 17
: A. DetectionofentanglementbyBellinequality 18
v
i
X VII. Quantificationofentanglement 18
r A. Entanglementofformation 19
a
1. Concurrence 20
2. Entanglementcost 20
B. Distillableentanglement 21
C. Relativeentropyofentanglement 21
D. Negativityandlogarithmicnegativity 22
VIII. Classificationofbipartitestateswithrespecttoquantumdensecoding 22
A. TheHolevobound 23
B. Capacityofquantumdensecoding 23
IX. Multipartitestates 25
A. k-separable,fully-separable,andgenuinemultipartiteentangledstates 25
B. Threequbitcase: GHZclassvs. Wclass 26
C. Monogamyofquantumentanglement 27
2
X. Problems 28
Acknowledgements 29
References 29
I. INTRODUCTION
th
Quantumtheory,formalizedinthefirstfewdecadesofthe20 century,containselementsthatareradicallydifferentfromthe
classicaldescriptionofNature. Animportantaspectinthesefundamentaldifferencesistheexistenceofquantumcorrelations
in the quantum formalism. In the classical description of Nature, if a system is formed by different subsystems, complete
knowledgeofthewholesystemimpliesthatthesumoftheinformationofthesubsystemsmakesupthecompleteinformation
forthewholesystem. Thisisnolongertrueinthequantumformalism. Inthequantumworld, thereexiststatesofcomposite
systems for which we might have the complete knowledge, while our knowledge about the subsystems might be completely
random. Intechnicalterms,onecanhavepurequantumstatesofatwo-partysystem,whoselocalstatesarecompletelymixed.
Onemayreachsomeparadoxicalconclusionsifoneappliesaclassicaldescriptiontostateswhichhavecharacteristicquantum
signatures.
During the last two decades, it has been realized that these fundamentally nonclassical states, also denoted as “entangled
states”, can provide us with something else than paradoxes. They may be used to perform tasks that cannot be achieved with
classical states. As benchmarks of this turning point in our view of such nonclassical states, one might mention the spectac-
ular discoveries of (entanglement-based) quantum cryptography (1991) [1], quantum dense coding (1992) [2], and quantum
teleportation(1993)[3].
Inthischapter, wewillconsiderbothbipartiteandmultipartitecompositesystems. Wewilldefineformallywhatentangled
states are, present some important criteria to discriminate entangled states from separable ones, and show how they can be
classifiedaccordingtotheircapabilitytoperformsomepreciselydefinedtasks. Ourknowledgeinthesubjectofentanglement
isstillfarfromcomplete,althoughsignificantprogresshasbeenmadeintherecentyearsandveryactiveresearchiscurrently
underway. Wewillconsidermultipartitequantumstates(statesofmorethantwoparties)inSectionIX,untilthen,weconsider
onlybipartitequantumstates.
II. BIPARTITEPURESTATES:SCHMIDTDECOMPOSITION
Consider a bipartite system in a shared pure state. The two parties in possession of the system are traditionally denoted as
Alice (A) and Bob (B), who can be located in distant regions. Let Alice’s physical system be described by the Hilbert space
H andthatofBobbyH . ThenthejointphysicalsystemofAliceandBobisdescribedbythetensorproductHilbertspace
A B
H ⊗H .
A B
Def. 1Productandentangledpurestates:
Apurestate,i.e. aprojector|ψ (cid:105)(cid:104)ψ |onavector|ψ (cid:105)∈H ⊗H ,isaproductstateifthestatesoflocalsubsystemsare
AB AB AB A B
alsopurestates,thatis,if|ψ (cid:105)=|ψ (cid:105)⊗|ψ (cid:105). However,therearestatesthatcannotbewritteninthisform. Thesestatesare
AB A B
calledentangledstates.
√
An example of an entangled state is the well-known singlet state |Ψ−(cid:105) = (|01(cid:105) − |10(cid:105))/ 2, where |0(cid:105) and |1(cid:105) are two
orthonormal states. Operationally, product states correspond to those states, that can be locally prepared by Alice and Bob at
two separate locations. Entangled states can, however, be prepared only after the particles of Alice and Bob have interacted
either directly or by means of an ancillary system. The second option is necessary due to the existence of the phenomenon
of entanglement swapping [4]. A very useful representation, only valid for pure bipartite states, is the, so-called, Schmidt
representation:
Theorem1Schmidtdecomposition:
Every|ψ (cid:105)∈H ⊗H canberepresentedinanappropriatelychosenbasisas
AB A B
M
(cid:88)
|ψ (cid:105)= a |e (cid:105)⊗|f (cid:105), (1)
AB i i i
i=1
where|e (cid:105)(|f (cid:105))formapartofanorthonormalbasisinH (H ),a >0,and(cid:80)M a2 =1,whereM ≤dimH ,dimH .
i i A B i i=1 i A B
Thepositivenumbersa areknownastheSchmidtcoefficientsandthevectors|e (cid:105)⊗|f (cid:105)astheSchmidtvectorsof|ψ (cid:105).
i i i AB
Notethatproductpurestatescorrespondtothosestates,whoseSchmidtdecompositionhasoneandonlyoneSchmidtcoefficient.
If the decomposition has more than one Schmidt coefficients, the state is entangled. Notice that the squares of the Schmidt
3
coefficientsofapurebipartitestate|ψ (cid:105)aretheeigenvaluesofboththereduceddensitymatricesρ (= tr (|ψ (cid:105)(cid:104)ψ |))
AB A B AB AB
andρ (=tr (|ψ (cid:105)(cid:104)ψ |))of|ψ (cid:105). ThelastfactgivesusaneasymethodtofindtheSchmidtcoefficientsandtheSchmidt
B A AB AB AB
vectors.
III. BIPARTITEMIXEDSTATES:SEPARABLEANDENTANGLEDSTATES
Asdiscussedinthelastsection,thequestionwhetheragivenpurebipartitestateisseparableorentangledisstraightforward.
One has just to check if the reduced density matrices are pure. This condition is equivalent to the fact that a bipartite pure
statehasasingleSchmidtcoefficient. Thedeterminationofseparabilityformixedstatesismuchharder,andcurrentlylacksa
completeanswer,evenincompositesystemsofdimensionaslowasC2⊗C4.
To reach a formal definition of separable and entangled states, consider the following preparation procedure of a bipartite
quantumstatebetweenAliceandBob. SupposethatAlicepreparesherphysicalsysteminthestate|e (cid:105)andBobprepareshis
i
physicalsysteminthestate|f (cid:105). Then,thecombinedstateoftheirjointphysicalsystemisgivenby:
i
|e (cid:105)(cid:104)e |⊗|f (cid:105)(cid:104)f |. (2)
i i i i
Wenowassumethattheycancommunicateoveraclassicalchannel(aphoneline,forexample). Then,wheneverAliceprepares
the state |e (cid:105) (i = 1,2,...,K), which she does with probability p , she communicates that to Bob, and correspondingly Bob
i i
(cid:80)
prepareshissysteminthestate|f (cid:105)(i=1,2,...,K). Ofcourse, p =1andp ≥0, ∀i. Thestatethattheyprepareisthen
i i i i
K
(cid:88)
ρ = p |e (cid:105)(cid:104)e |⊗|f (cid:105)(cid:104)f |. (3)
AB i i i i i
i=1
Def. 2Separableandentangledmixedstates:
Aquantumstateρ isseparableifandonlyifitcanberepresentedasaconvexcombinationoftheproductofprojectorson
AB
localstatesasstatedinEq. (3). Otherwise,thestateissaidtobeentangled.
The important point to note here is that the state displayed in Eq. (3) is the most general state that Alice and Bob will be
able to prepare by local quantum operations and classical communication (LOCC) [5]. In LOCC protocols, two parties Alice
and Bob perform local quantum operations separately in their respective Hilbert spaces and they are allowed to communicate
classicalinformationabouttheresultsoftheirlocaloperations. Letusmakethedefinitionsomewhatmoreformal.
Localoperationsandclassicalcommunication(LOCC):SupposeAliceandBobshareaquantumstateρ definedonthe
AB
HilbertspaceH ⊗H . AliceperformsaquantumoperationonherlocalHilbertspaceH ,usingacompletesetofcomplete
A B A
generalquantumoperations{A(1)},satisfying(cid:80) (A(1))†A(1) =I ,andsendshermeasurementresultitoBobviaaclassical
i i i i A
channel. Depending on the measurement result of Alice, Bob operates a complete set of general quantum operations {B(1)},
ij
satisfying (cid:80) (B(1))†B(1) = I on his part belonging to the Hilbert space H . This joint operation along with the classical
j ij ij B B
communication is called one-way LOCC. Furthermore, Bob can send his result j to Alice, and she can choose another set of
localoperations{A(2)},satisfying(cid:80) (A(2))†A(2) =I ,accordingtoBob’soutcome. Theycancontinuethisprocessaslong
ijk k ijk ijk A
asrequired,andtheentireoperationistermedasLOCC,ortwo-wayLOCC.TheoperatorsI andI aretheidentityoperators
A B
onH andH respectively.
A B
EntangledstatescannotbepreparedbytwopartiesifonlyLOCCisallowedbetweenthem.Topreparesuchstates,thephysical
systemsmustbebroughttogethertointeract1.
Thequestionwhetheragivenbipartitestateisseparableornotturnsouttobequitecomplicated. Amongthedifficulties,we
notice that for an arbitrary state ρ , there is no stringent bound on the value of K in Eq. (3), which is only limited by the
AB
CaratheodorytheoremtobeK ≤ (dimH)2 withH = H ⊗H (see[6,7]). Althoughthegeneralanswertotheseparability
A B
problem still eludes us, there has been significant progress in recent years, and we will review some such directions in the
followingsections.
IV. OPERATIONALENTANGLEMENTCRITERIA
Inthissection, wewillintroducesomeoperationalentanglementcriteriaforbipartitequantumstates. Inparticular, wewill
discussthepartialtranspositioncriterion[8,9],themajorizationcriterion[10],thecross-normorrealignmentcriterion[11–13],
1 Becauseofthephenomenonofentanglementswapping[4],onemustsuitablyenlargethenotionofpreparationofentangledstates. So,anentangledstate
betweentwoparticlescanbepreparedifandonlyif,eitherthetwoparticles(callthemAandB)themselvescametogethertointeractatatimeinthepast,or
twootherparticles(callthemCandD)doesthesame,withC(D)havinginteractedbeforehandwithA(B).
4
and the covariance matrix criterion [14, 15]. There exist several other criteria (see e.g. Refs. [16–21]), which will not be
discussed here. However note that, up to now, a necessary and sufficient criterion for detecting entanglement of an arbitrary
givenmixedstateisstilllacking.
A. PartialTransposition
Def. 3Letρ beabipartitedensitymatrix,andletusexpressitas
AB
(cid:88)
ρ = aµν(|i(cid:105)(cid:104)j|) ⊗(|µ(cid:105)(cid:104)ν|) , (4)
AB ij A B
1≤i,j≤dA
1≤µ,ν≤dB
where {|i(cid:105)} ({|µ(cid:105)}) is a set of real orthonormal vectors in H (H ), with d = dimH and d = dimH . The partial
A B A A B B
transposition,ρTA,ofρ withrespecttosubsystemA,isdefinedas
AB AB
(cid:88)
ρTA = aµν(|j(cid:105)(cid:104)i|) ⊗(|µ(cid:105)(cid:104)ν|) . (5)
AB ij A B
1≤i,j≤dA
1≤µ,ν≤dB
Asimilardefinitionexistsforthepartialtranspositionofρ withrespecttoBob’ssubsystem. NoticethatρTB =(ρTA)T.
AB AB AB
Although the partial transposition depends upon the choice of the basis in which ρ is written, its eigenvalues are basis
AB
independent. Wesaythatastatehaspositivepartialtransposition(PPT),wheneverρTA ≥ 0,i.e. theeigenvaluesofρTA are
AB AB
non-negative. Otherwise,thestateissaidtobenon-positiveunderpartialtransposition(NPT).
Theorem2[8]
(cid:16) (cid:17)T
Ifastateρ isseparable,thenρTA ≥ 0andρTB = ρTA ≥ 0.
AB AB AB AB
Proof:
Sinceρ isseparable,itcanbewrittenas
AB
ρ =(cid:80)K p |e (cid:105)(cid:104)e |⊗|f (cid:105)(cid:104)f |≥0. (6)
AB i=1 i i i i i
Nowperformingthepartialtranspositionw.r.t. A,wehave
K
ρTA = (cid:88) p (|e (cid:105)(cid:104)e |)TA ⊗|f (cid:105)(cid:104)f |
AB i i i i i
i=1
K
(cid:88)
= p |e∗(cid:105)(cid:104)e∗|⊗|f (cid:105)(cid:104)f |≥0. (7)
i i i i i
i=1
Notethatinthesecondline,wehaveusedthefactthatA† =(A∗)T. (cid:3)
Thepartialtranspositioncriterion,fordetectingentanglementissimple: Givenabipartitestateρ ,findtheeigenvaluesof
AB
anyofitspartialtranspositions. Anegativeeigenvalueimmediatelyimpliesthatthestateisentangled. Examplesofstatesfor
whichthepartialtranspositionhasnegativeeigenvaluesincludethesingletstate.
ThepartialtranspositioncriterionallowstodetectinastraightforwardmannerallentangledstatesthatareNPTstates. This
isahugeclassofstates. However,itturnsoutthatthereexistPPTstateswhicharenotseparable,aspointedoutinRef. [6](see
also[22]). Moreover,thesetofPPTentangledstatesisnotasetofmeasurezero[23]. Itis,therefore,importanttohavefurther
independent criteria of entanglement detection which permits to detect entangled PPT states. It is worth mentioning here that
PPTstateswhichareentangled,formtheonlyknownexamplesofthe“boundentangledstates”(seeRefs. [22,24]fordetails).
Bound entangled states of bipartite quantum states are the states that cannot be distilled i.e., converted to singlet states under
LOCC[25,26],withotherentangledstatesbeingdistillable.Wewilltalkaboutdistillationofquantumstateslaterinthischapter
inabitmoredetail. Althoughasyetnotfound,itisconjecturedthattherealsoexistNPTboundentangledstates[24]. Notealso
thatbothseparableaswellasPPTstatesformconvexsets. Figure1depictsthestructureofthestatespacewithrespecttothe
partialtranspositioncriteriaanddistillability.
Theorem2isanecessaryconditionofseparabilityinanyarbitrarydimension. However, forsomespecialcases, thepartial
transpositioncriterionisbothanecessaryandasufficientconditionforseparability:
Theorem3[9]
InC2⊗C2orC2⊗C3,astateρisseparableifandonlyifρTA ≥ 0.
Asmentionedabove,PPTboundentangledstatesexist. However,astheTheorem3shows,theycanexistonlyindimensions
higherthanC2⊗C2andC2⊗C3.
5
NPT
PPT
bound
S e p a r a b l e b o u n d Distillable
entangled
s t a t e s e n t a n t a n g l e d states
states
states
?

FIG. 1: The structure of the state space in light of the partial transposition criteria and distillability. Separable and as well as PPT states
form convex sets (while NPT states do not). It also shows the conjectured NPT bound entangled states. It is not clear whether the set of
non-distillablestatesisconvex.
B. Majorization
The partial transposition criterion, although powerful, is not able to detect entanglement in a finite volume of states. It is,
therefore, interesting to discuss other independent criteria. The majorization criterion, to be discussed in this subsection, has
beenshowntobenotmorepowerfulindetectingentanglement. Wechoosetodiscussithere,mainlybecauseithasindependent
roots. Moreover,itrevealsaveryinterestingthermodynamicalpropertyofentanglement.
Beforepresentingthecriterion,wepresentadefinitionofmajorization[27].
Def. 4Letx = (x ,x ,...,x ), andy = (y ,y ,...,y )betwoprobabilitydistributions, arrangedindecreasingorder, i.e.
1 2 d 1 2 d
x ≥x ≥...≥x andy ≥y ≥...≥y . Thenwedefine“xmajorizedbyy”,denotedasx≺y,as
1 2 d 1 2 d
l l
(cid:88) (cid:88)
x ≤ y , (8)
i i
i=1 i=1
wherel=1,2,...d−1,andequalityholdswhenl=d.
Theorem4[10]
Ifastateρ isseparable,then
AB
λ(ρ )≺λ(ρ ), and λ(ρ )≺λ(ρ ), (9)
AB A AB B
whereλ(ρ )isthesetofeigenvaluesofρ ,andλ(ρ )andλ(ρ )arethesetsofeigenvaluesofthereduceddensitymatrices
AB AB A B
ofthestateρ ,andwhereallthesetsarearrangedindecreasingorder.
AB
Themajorizationcriterion: Givenabipartitestate,itisentangledifEq. (9)isviolated. However,itwasshowninRef. [28],
thatastatethatisnotdetectedbythepositivepartialtranspositioncriterion, willnotbedetectedbythemajorizationcriterion
either. Nevertheless,thecriterionhasotherimportantimplications. Wewillnowdiscussonesuch.
Letusreiterateaninterestingfactaboutthesingletstate: Theglobalstateispure,whilethelocalstatesarecompletelymixed.
Inparticular,thisimpliesthatthevonNeumannentropy2ofthesingletislowerthanthevonNeumannentropiesofeitherofthe
local states. Since the von Neumann entropy can be used to quantify disorder in a given state, there exist global states whose
disorderislowerthantheanyofthelocalstates. Thisisanonclassicalfactasfortwoclassicalrandomvariables,theShannon
entropy3 ofthejointdistributioncannotbesmallerthanthatofeither. InRef. [29],itwasshownthatasimilarfactistruefor
separablestates:
Theorem5
Ifastateρ isseparable,
AB
S(ρ )≥S(ρ ), and S(ρ )≥S(ρ ). (10)
AB A AB B
2ThevonNeumannentropyofastateρisS(ρ)=−trρlog ρ.
2
3TheShannonentropyofarandomvariableX,takingupvaluesXi,withprobabilitiespi,isgivenbyH(X)=H({pi})=−(cid:80)ipilog2pi.
6
AlthoughthevonNeumannentropyisanimportantnotionforquantifyingdisorder,thetheoryofmajorizationisamorestringent
quantifier[27]: Fortwoprobabilitydistributionsxandy,x ≺ y ifandonlyifx = Dy,whereD isadoublystochasticmatrix
4. Moreover,x ≺ y impliesthatH({x }) ≥ H({y }). Quantummechanicsthereforeallowstheexistenceofstatesforwhich
i i
globaldisorderisgreaterthanlocaldisordereveninthesenseofmajorization.
AdensitymatrixthatsatisfiesEq.(9),automaticallysatisfiesEq.(10).Inthissense,Theorem4isageneralizationofTheorem
5.
C. Cross-normormatrixrealignment
Thecross-normormatrixrealignmentcriterion[11–13]providesanotherwaytodelineateseparableandentangledstates,and
moreimportantly,cansuccessfullydetectvariousPPTentangledstates. Therearevariouswaystoformulatethiscriterion. Here
wepresentaformulationgiveninRef. [13]asCorollary18.
Adensitymatrixρ onaHilbertspaceH ⊗H ,whered andd arethedimensionsoftheHilbertspacesH andH
AB A B A B A B
respectively,canbewrittenas
ρ = (cid:88) aµν(|i(cid:105)(cid:104)j|) ⊗(|µ(cid:105)(cid:104)ν|) =(cid:88)ξ G˜A⊗G˜B. (11)
AB ij A B kl k l
1≤i,j≤dA k,l
1≤µ,ν≤dB
WehaveusedthesamenotationsasinEq. (4),exceptthatwehaveaddedzerostothetensoraµν,sothattheindicesrununtil
ij
the dimensions of the Hilbert spaces. {G˜A} = {|i(cid:105)(cid:104)j|} and {G˜B} = {|µ(cid:105)(cid:104)ν|} are complete sets of orthonormal Hermitian
k l
operators on the Hilbert spaces H and H respectively, with 1 ≤ k ≤ d2 and 1 ≤ l ≤ d2. Without loss of generality, we
A B A B
assumethatd ≥d . Aftersingularvaluedecompositionofthematrixξ,wehave
A B
ξ =UΣV†, (12)
whereU andV ared2 ×d2 andd2 ×d2 dimensionalunitarymatricesrespectively,andΣisad2 ×d2 dimensionaldiagonal
A A B B A B
matrix. DenotingthekthcolumnvectorofU andV by|u (cid:105)and|v (cid:105),theaboveexpressionbecomes
k k
d2
(cid:88)B
ξ = λ |u (cid:105)(cid:104)v |, (13)
k k k
k=1
whereλ arethediagonalelementsofΣ. So,wehavethematrixelementsofξas
k
ξ =λ δ . (14)
kl k kl
IfGA andGB arethematrixrepresentationsofG˜A andG˜B in{|u (cid:105)}and{|v (cid:105)}basisrespectively,thenusingEqs. (11)and
k l k l k k
(14),weobtain
d2
(cid:88)B
ρ = λ GA⊗GB. (15)
AB k k k
k=1
Eq. (15)canbeinterpretedastheSchmidtdecompositionofthedensitymatrixρ inoperatorspace,wherethesingularvalues
AB
λ arerealandnon-negative. Thecross-normorrealignmentcriterionofseparabilityisgivenbythefollowingtheorem:
k
Theorem6[13]
Ifasharedquantumstateρ isseparable,then
AB
(cid:88)
λ ≤1, (16)
k
k
whereλ arethesingularvaluesofρ asgiveninEq. (15). Iftheinequalityisviolated,onecanconcludethatρ mustbe
k AB AB
anentangledstate.
4AmatrixD=(Dij)issaidtobedoublystochastic,ifDij ≥0,and(cid:80)iDij =(cid:80)jDij =1.
7
D. Covariancematrix
There exist several other operational criteria in the literature to detect whether a quantum state is separable or entangled
[16–21]. We will conclude this section by briefly illustrating one such separability criteria, known as the covariance matrix
criterion[14,15],whichprovidesageneralframeworktolinkandunderstandseveralexistingcriteriaincludingthecross-norm
or realignment criterion. Like the cross-norm or realignment criterion, this method can identify entangled state for which the
partial transposition criterion fails. Before delving into the theory of the covariance matrix criterion, let us first discus the
definitionandpropertiesofthecovariantmatrices.
1. Definitionandproperties
Def. 5 Given a quantum state ρ and a complete set of orthonormal observables {M } on a d-dimensional Hilbert space, the
k
d2×d2covariantmatrixγ =γ(ρ,{M })andsymmetrizedcovariantmatrixγS =γS(ρ,{M })aredefinedas
k k
γ = (cid:104)M M (cid:105)−(cid:104)M (cid:105)(cid:104)M (cid:105), (17)
ij i j i j
(cid:104)M M (cid:105)+(cid:104)M M (cid:105)
γS = i j j i −(cid:104)M (cid:105)(cid:104)M (cid:105), (18)
ij 2 i j
where(cid:104)M(cid:105)=tr(ρM)definestheexpectationoftheoperatorM withrespecttothestateρ.
Thecompletesetoforthonormalobservables{M }hastosatisfytheHilbert-Schmidtorthonormalityconditiontr(M M )=
k i j
δ ,∀i,j =1,2,...,d. OneexampleforsuchasetofobservablesforthecaseofsinglequbitintermsofthePaulimatrices,can
ij
begivenby
I σ σ σ
M = √ , M = √x, M = √y , M = √z . (19)
1 2 3 4
2 2 2 2
Ingeneral,forthed-dimensionalcase,onecanconsiderfollowingmatricestoformthecompletesetoforthonormalobservables:
X = |i(cid:105)(cid:104)i|, i=1,2,...,d, (20)
i
1
Y = √ (|i(cid:105)(cid:104)j|+|j(cid:105)(cid:104)i|), 1≤i<j ≤d, (21)
ij
2
i
Z = √ (|k(cid:105)(cid:104)l|−|l(cid:105)(cid:104)k|), 1≤k <l≤d. (22)
kl
2
Let us now focus on the situation in which the Hilbert space is a tensor product H = H ⊗H of Hilbert spaces of two
A B
subsystems A and B with dimensions d and d respectively. We can consider the complete set of orthonormal observables
A B
in A as {A : k = 1,2...,d2} and in B as {B : k = 1,2...,d2}, and construct a set of d2 +d2 observables as {M } =
k A k B A B k
{A ⊗I,I⊗B }. Althoughthissetisnotcomplete,itcanbeutilizedtodefineaveryusefulformofcovariantmatrices,known
k k
astheblockcovariantmatrices. Theblockcovariantmatrixforagivenbipartitestateρ andorthonormalobservables{M }
AB k
isdefinedasfollows.
Def. 6Letρ beaquantumstateofabipartitesystem,andlet{M }={A ⊗I,I⊗B }beasetoforthonormalobservables
AB k k k
as stated above. Then, the block covariance matrix γ = γ(ρ ,{M }) is given in terms of its matrix elements as γ =
AB k i,j
(cid:104)M M (cid:105)−(cid:104)M (cid:105)(cid:104)M (cid:105),andhastheblockstructure
i j i j
(cid:18) (cid:19)
A X
γ = , (23)
XT B
whereA=γ(ρ ,{A })andB =γ(ρ ,{B })arethecovariantmatricesofthereducedsubsystems,andX =(cid:104)A ⊗B (cid:105)−
A k B k i,j i j
(cid:104)A (cid:105)(cid:104)B (cid:105).
i j
Similarly,wecandefinethesymmetricversionoftheblockcovariancematrixbyreplacingAandBwiththeirsymmetrized
counterparts,whilekeepingX unchanged. Clearly,ifρ =ρ ⊗ρ isaproductstate,thenitsblockcovariantmatrixreduces
AB A B
totheblockdiagonalform,γ(ρ ,{M })=A⊕B,asX becomezero∀i,j.
AB k i,j
Ifρisapurestateonad-dimensionalHilbertspace,thenthecorrespondingcovariancematrixγsatisfiesfollowingproperties:
1. Therankofγ isequaltod−1.
2. Thenonzeroeigenvaluesofγ areequalto1,hencetr(γ)=d−1.
3. γ2 =γ.
8
ThecorrespondingsymmetriccovariancematrixγS satisfiesthefollowing:
1. TherankofγS isequalto2(d−1).
2. ThenonzeroeigenvaluesofγS areequalto1/2,hencetr(γS)=d−1.
Formixedstateρonad-dimensionalHilbertspace,wehavetr[γ(ρ)]=d−(ρ2),andthesameforγS(ρ). Thecovariancematrix
(cid:80)
(symmetricandnon-symmetric)alsosatisfiestheconcavityproperty,i.e.,ifρ= p ρ isaconvexcombinationofstatesρ ,
k k k k
then
(cid:88)
γ(ρ)≥ p γ(ρ ). (24)
k k
k
2. Covariancematrixcriterionforseparability
Theorem7[14,15]
Letρ beaseparablestateonH ⊗H andA andB betheorthonormalobservablesontheHilbertspacesH andH ,
AB A B k k A B
withthelatterhavingdimensionsd andd respectively.Define{M }={A ⊗I,I⊗B }asdiscussedpreviously.Thenthere
A B k k k
(cid:80)
existpurestates|ψ (cid:105)(cid:104)ψ |forA,|φ (cid:105)(cid:104)φ |forB,andconvexweightsp suchthatifwedefineκ = p γ(|ψ (cid:105)(cid:104)ψ |,{A })
(cid:80) k k k k k A k k k k k(cid:48)
andκ = p γ(|φ (cid:105)(cid:104)φ |,{B }),theinequality
B k k k k k(cid:48)
(cid:18) (cid:19) (cid:18) (cid:19)
A X κ 0
γ(ρ ,{M })≥κ ⊕κ ⇔ ≥ A (25)
AB k A B XT B 0 κ
B
holds.
Proof:
Letusassumeρ beaseparablestatewiththefollowingpurestatedecomposition,
AB
(cid:88)
ρ = p (|ψ (cid:105)(cid:104)ψ |⊗|φ (cid:105)(cid:104)φ |). (26)
AB k k k k k
k
Thenusingthepropertiesofcovariancematrices,weget
(cid:88)
γ(ρ ,{M }) = γ( p (|ψ (cid:105)(cid:104)ψ |⊗|φ (cid:105)(cid:104)φ |),{M })
AB k k k k k k k(cid:48)
k
(cid:88)
≥ p γ(|ψ (cid:105)(cid:104)ψ |⊗|φ (cid:105)(cid:104)φ |,{M })
k k k k k k(cid:48)
k
(cid:88)
= p {γ(|ψ (cid:105)(cid:104)ψ |,{A })⊕γ(|φ (cid:105)(cid:104)φ |,{B })}
k k k k(cid:48) k k k(cid:48)(cid:48)
k
= κ ⊕κ , (27)
A B
where we have used concavity property of the covariance matrices in the second line, and in the third line, we have used the
fact that the block covariant matrix of a product state takes the block diagonal form. This theorem can also be proven for the
symmetriccovariantmatricesinthesamemanner. Clearly,ifthereexistnosuchκ andκ ,thestateρ mustbeentangled.(cid:3)
A B AB
Clearly, it is not evident from Theorem 7 that the covariance matrix criterion leads to an efficient and physically plausible
operationalindicationforseparability.Themainproblemistoidentifypossibleκ andκ ,asthisrequiresanoptimizationover
A B
allpurestatedecompositionsofρ . Therefore,wenowfocusonthecases,wheretheabovecriterioncanbeusedefficientlyto
AB
identifyentangledstates,bystatingseveralcorollariesoftheabovetheorem.
Corollary1
(cid:18) (cid:19)
A X
Letγ = betheblockcovariancematrixofabipartitestateρ . Then,ifρ isseparable,wehave
XT B AB AB
||X||2 ≤[1−tr(ρ2)][1−tr(ρ2)], (28)
A B
√
where||A||=tr A†Aisthematrixtracenorm.
Corollary2
Letρ beaquantumstatesharedbetweentwosubsystemsAandBwithdimensionsd andd respectively(withd ≤d ),
AB A B A B
9
(cid:18) (cid:19)
γS = A X bethecorrespondingsymmetriccovariancematrix,andletJ ={j ,...,j }(cid:0)⊂{1,...,d2}(cid:1)beasetofd2
XT B 1 d2A B A
distinctindices. Thenifρ isseparable,wehave
AB
d2
(cid:88)A (cid:88)
2 |X |≤[1−tr(ρ2)]+[1−tr(ρ2)]. (29)
i,j A B
i=1j∈J
NowwewillgiveanotheroperationalentanglementcriterionbasedontheSchmidtdecompositiononoperatorspace,thentry
torelatecovariancematrixcriterionwiththecross-normorrealignmentcriterion. Ageneralbipartitequantumstateρ ona
AB
HilbertspaceH ⊗H ,whered andd arethedimensionsoftheHilbertspacesH andH respectively,canbewrittenas
A B A B A B
d2 d2
ρ =(cid:88)A (cid:88)B ξ G˜A⊗G˜B, (30)
AB kl k l
k=1l=1
whereξ arerealquantities,and{G˜A}and{G˜B}arecompletesetsoforthonormalHermitianoperatorsontheHilbertspaces
kl i j
H andH respectively. Aswehaveseenearlier, ρ inEq. (30), canbewrittenintheSchmidtdecomposed-likeform(in
A B AB
operatorspace)viathesingularvaluedecompositionas
d2
(cid:88)B
ρ = λ GA⊗GB, (31)
AB k k k
k=1
wheresingularvaluesλ arerealandnon-negative,andwehaveassumedthatd ≥d .
k A B
Corollary3
Letρ beaseparablequantumstatesharedbetweentwosubsystemsAandB. Then
AB
(cid:88) (cid:88)
2 |λ −λ2gAgB|≤2− λ2[(gA)2+(gB)2], (32)
i i i i i i i
i i
wheregA(B) =tr[GA(B)].
i i
Proof:
(cid:18) (cid:19)
A X
Let γS = be the symmetric block covariance matrix of the bipartite state ρ . Using the orthogonality of the
XT B AB
matrices{GA}and{GB},onecandeduceX = (cid:0)λ −λ2gAgB(cid:1). Further,onecangettr(ρ2) = (cid:80) λ2(gB)2,andtr(ρ2) =
i i i,j i i i i A i i i B
(cid:80) λ2(gA)2. TogetherwithCorollary2,wecanprovetheclaim. (cid:3)
i i i
Nowusingtherelations|a−b|≥|a|−|b|anda2+b2 ≥2|ab|wecanhave,
(cid:88) (cid:88)
2 |λ −λ2gAgB| ≥ 2 (λ −λ2|gAgB|), (33)
i i i i i i i i
i i
(cid:88) (cid:88)
2(1− λ2|gAgB|) ≥ 2− λ2[(gA)2+(gB)2]. (34)
i i i i i i
i i
Usinginequalities(32)-(34),weget
(cid:88) (cid:88)
2 |λ −λ2gAgB| ≤ 2− λ2[(gA)2+(gB)2]
i i i i i i i
i i
(cid:88)
≤ 2(1− λ2|gAgB|)
i i i
i
(cid:88) (cid:88)
= (2−2 λ )+2 (λ −λ2|gAgB|)
i i i i i
i i
(cid:88) (cid:88)
≤ (2−2 λ )+2 |λ −λ2gAgB|
i i i i i
i i
(cid:88)
⇔ λ ≤ 1. (35)
i
i
Thisisthecross-normorrealignmentcriterionofseparabilitymentionedearlier,whichwegetasacorollaryofthecovariance
matrixcriterion.
Thereareseveralothercorollariesofthecovariancematrixcriterion,whichenableonetoefficientlydetectentangledstatesin
severalcases. Moreover,thecovariancematrixcriterioncanbeimprovedbyusinglocalfilteringoperation[30]. SeeRef. [15]
fordetails.
10
V. NON-OPERATIONALENTANGLEMENTCRITERIA
Inthissection,wewilldiscussthreefurtherentanglementcriteria. WewillshowhowtheHahn-Banachtheoremcanbeused
toobtain“entanglementwitnesses”. Wewillalsointroducethenotionofpositivemaps,andpresenttheentanglementcriterion
basedonit. Andfinallywewillpresenttherangecriterionofseparability. Allthreecriteriaare“non-operational”,inthesense
thattheyarenotstate-independent. Nevertheless,theyprovideimportantinsightintothestructureofthesetofentangledstates.
Moreover,theconceptofentanglementwitnessescanbeusedtodetectentanglementexperimentally,byperformingonlyafew
localmeasurements,assumingsomepriorknowledgeofthedensitymatrix[31,32].
3. TechnicalPreface
Thefollowinglemmaandobservationwillbeusefulforlaterpurposes.
Lemma1
tr(ρTAσ )=tr(ρ σTA).
AB AB AB AB
Observation:
ThespaceoflinearoperatorsactingonH(denotedbyB(H))isitselfaHilbertspace,withthe(Euclidean)scalarproduct
(cid:104)A|B(cid:105)=tr(A†B) A,B ∈B(H). (36)
ThisscalarproductisequivalenttowritingAandBrow-wiseasvectors,andscalar-multiplyingthem:
(dimH)2
(cid:88) (cid:88)
tr(A†B)= A∗B = a∗b . (37)
ij ij k k
ij k=1
A. EntanglementWitnesses
1. EntanglementWitnessfromtheHahn-Banachtheorem
Central to the concept of entanglement witnesses, is the Hahn-Banach theorem, which we will present here limited to our
situationandwithoutproof(seee.g. [33]foraproofofthemoregeneraltheorem):
Theorem8
LetS beaconvexcompactsetinafinitedimensionalBanachspace. Letρbeapointinthespacewithρ(cid:54)∈S. Thenthereexists
ahyperplane5thatseparatesρfromS.
5Ahyperplaneisalinearsubspacewithdimensiononelessthanthedimensionofthespaceitself.