The quantumness of correlations revealed in local measurements exceeds entanglement

0.59 MB·English

by  Marco Piani

Thequantumnessofcorrelationsrevealedinlocalmeasurementsexceedsentanglement
Marco Piani1 and Gerardo Adesso2
1InstituteforQuantumComputingandDepartmentofPhysicsandAstronomy,UniversityofWaterloo,WaterlooONN2L3G1,Canada
2SchoolofMathematicalSciences,TheUniversityofNottingham,UniversityPark,NottinghamNG72RD,UnitedKingdom
(Dated:December15,2011)
Weanalyzeafamilyofmeasuresofgeneralquantumcorrelationsforcompositesystems,definedintermsof
thebipartiteentanglementnecessarilycreatedbetweensystemsandapparatusesduringlocalmeasurements.For
everyentanglementmonotoneE,thisoperationalcorrespondenceprovidesadifferentmeasureQ ofquantum
E
correlations. Examplesofsuchmeasuresaretherelativeentropyofquantumness,thequantumdeficit,andthe
negativity of quantumness. In general, we prove that any so defined quantum correlation measure is always
greater than (or equal to) the corresponding entanglement between the subsystems, Q ≥ E, for arbitrary
E
statesofcompositequantumsystems. Weanalyzequalitativelyandquantitativelytheflowofcorrelationsin
iteratedmeasurements,showingthatgeneralquantumcorrelationsandentanglementcanneverdecreasealong
2 von Neumann chains, and that genuine multipartite entanglement in the initial state of the observed system
1
alwaysgivesrisetogenuinemultipartiteentanglementamongallsubsystemsandallmeasurementapparatuses
0
atanylevelinthechain. Ourresultsprovideacomprehensiveframeworktounderstandandquantifygeneral
2
quantumcorrelationsinmultipartitestates.

n
a PACSnumbers:03.65.Ta,03.65.Ud,03.67.Mn
J

6
1 Introduction.—Thequantumworlddiffersfromourfamiliar S M1 M2 M3 Mn
classicalworldinmany,interrelatedways[1]. Quantumlaws

] forbidbasictaskssuchascloning[2]yetenablecertaininfor-
h
p mationprocessingfeatsotherwiseunfeasiblewithpurelyclas- system apparatus photon retina … consciousness?
- sicalresources[3]. Inparticular, quantumcorrelationsdiffer
t
n from classical ones. Such a difference can assume the strik- FIG.1:(Coloronline)GraphicaldepictionofavonNeumannchain.
a ing traits of entanglement [4–6] and non-locality [7], or the
u
subtlerfeaturesofquantumdiscord[8]. Thelattercapturesa
q
moregeneralsignatureofnon-classicalityofcorrelations(be- ment” on which the “observer” has little or no control (for a
[
ingpresentalsoinalmostallunentangledstates[9])thatstems review on this topic, see [17]). At the same time, it is ex-

2 fromthenon-commutativityofquantumobservables,andcan actly through the establishment of correlations between the
v
be revealed in local measurements; its characterization and observedquantumsystemandameasurementapparatus—and
0
3 applications have recently attracted much attention [10, 11]. finallytheobserver—thatwecandescribethemeasurementto
5 Discord or, in general, quantum correlations different from takeplace.Inthisrespect,themeasurementproblembecomes
2 entanglementhavebeenlinkedtotheadvantage, overclassi- thatoffindinghowthe“quantuminformation”containedina
0. calscenarios,ofquantumalgorithmsforcommunication[12], quantum system becomes correlated to the “classical infor-
1 informationlocking[13],metrology[14],andespeciallycom- mation”inthemindoftheobserver. Inanalyzingsuchanis-
1 putationinthepresenceofnoise[15]. sue,vonNeumann[18]consideredwhatlaterbecameknown
1 as“vonNeumannchain”: asequenceofinteractingphysical
:
v Clarifyingtherelationbetween“classical”and“quantum” systems,startingwiththequantumsystemtobemeasuredand
i is of paramount importance from the practical point of view endingwiththeobserver(seeFig.1).
X
of information processing, as a better understanding of all In this paper we provide quantitative constraints regulat-
r
a genuinelyquantumeffectscanonlyleadtothembeingmore ing the establishment of quantum correlations with appara-
efficiently exploited. Nonetheless, another more fundamen- tusesduringameasurementprocess,andtheirflowalongvon
tal reason to study the non-classicality of correlations is that Neumann chains. Combining and extending the approaches
we do not fully understand the quantum-to-classical transi- recentlyputforwardinRefs.[19,20], wequantifythequan-
tion[17];itisnotclearhowandatwhichscalequantumme- tum correlations Q between subsystems S of a composite
E k
chanics actually leads to an everyday world where (macro- quantumsystemS,intermsoftheminimumentanglementE
scopic) objects appear to be “here” or “there”, but not “here generatedbetweenthewholesystemSandagenerallycom-
and there” as instead allowed by the superposition principle. posed measurement apparatus M which is probing a subset
Thisissueoftheemergenceoftheclassicalfromthequantum of the subsystems of S locally—that is, a local independent
isintimatelyrelatedtothe“measurementproblem”whichhas measurementoneachprobedS isperformedthroughanin-
k
puzzledphysicists,mathematiciansandphilosopherssincethe teraction with a local apparatus M that is part of M. Here
k
birthofquantummechanicsacenturyago. Apotentialsolu- weshow,foranyentanglementmeasureE,thataccordingto
tionofthemeasurementproblemisintermsofdecoherence, such a mapping the key inequality Q (ρ) ≥ E(ρ) holds for
E
that is in terms of correlations established with an “environ- allquantumstatesρ. Thisprovesthatmeasurementprocesses
2
provide a natural and insightful framework for the study of
S
the non-classicality of correlations: for every chosen entan- m
glement monotone E, one obtains a different, operational e ρS
measure QE of general quantum correlations, which incor- syst U US1M1 ρɶS M
U I
poratesandgenerallyexceedstheentanglement(measuredby U
E)betweenthesubsystemsofarbitraryquantumstates.Based a 0 S3M3
at M1
on these results, that definitely elucidate the (so far unclear ar 0
p M
[8,21])interplaybetweenentanglementandgeneralquantum p 3
a
correlationsincompositesystems,wefurthershowthatquan-
tum correlations and entanglement can never decrease along FIG. 2: (Color online) Construction of the pre-measurement state
ρ˜ [Eq. (3)], for S ≡ S = {S ,S ,S ,S },I =
von Neumann chains, and that all the links of the chain ex- SUMI U 1 2 3 4
{1,3},M ={M ,M },andU oftheform(2).
hibitgenuinemultipartiteentanglementifandonlyifthesub- I 1 3 SkMk
systems are genuinely (multipartite) entangled—and not just
quantumlycorrelated—intheinitialstateofthesystemS. (cid:80) |i(cid:105)(cid:104)i| ρ |i(cid:105)(cid:104)i| , for all states ρ of S . The
i Sk Sk Sk Sk k
Preliminaries.— We begin by setting our notation and re- state
calling a number of definitions. Let S be a quantum sys-
ρ˜ =U ρ ⊗|0(cid:105)(cid:104)0| U† , (3)
tem partitioned into n (finite-dimensional) subsystems and SkMk SkMk Sk Mk SkMk
let U = {1,2,...,n}; we denote by S = {S |k ∈ I},
I k beforetracingouttheapparatus,issometimescalledthepre-
with I ⊆ U, a subset of the n subsystems S ,S ,...,S of
1 2 n measurementstate[17]. Thenotionistriviallygeneralizedto
S ≡ S . WedenotebyU\I thecomplementofsetI within
U the case where local projective measurements are performed
U. With respect to the notion of classicality of correlations,
on any subset I of U: in such a case we will refer to a pre-
weadoptthefollowingdefinition(seealso[22]).
measurement state ρ˜ of the whole system plus the set
SUMI
of apparatuses coupled to subsystems S (see Fig. 2). Such
Definition 1 (Classically correlated states). A state ρ is I
SU a pre-measurement state depends of course on the choice of
“classically correlated (CC) with respect to local measure-
local bases in which the local measurements take place. By
ments on subsystems S ”, or, in other words, “classical on
I
consideringtheentanglementpropertiesofρ˜ acrossthe
subsystemsSI”,or,inshort-handnotation,ρSU is“I-CC”,if S :M bipartitionwecanformulatethefollSoUwMinIgtheorem,
thereexistsachoiceoflocalcompletevonNeumannmeasure- U I
straightforwardlygeneralizingRefs.[19,20]:
ments on each subsystem S , k ∈ I such that ρ is left in-
k SU
variantundersuchmeasurements. Equivalently,ρSU isI-CC Theorem1. AstateρSU isI-CCifandonlyifthereexistaset
if there exists a choice of a local orthonormal basis {|i(cid:105)Sk} oflocalcompletevonNeumannmeasurementsonsubsystems
foreachSk,k ∈I,suchthat SI suchthatthepre-measurementstateρ˜SUMI isunentangled
acrossS :M .
(cid:88) U I
ρ = |i (cid:105)(cid:104)i | ⊗|i (cid:105)(cid:104)i | (1)
SU k1 k1 Sk1 k2 k2 Sk2 Theorem1providesaclearoperationalinterpretationofI-
ik1,ik2,...,ik|I| CCstatesastheonlystateswhichdonotnecessarilygiverise
⊗ ···⊗|ik|I|(cid:105)(cid:104)ik|I||Sk|I| ⊗ρiS(cid:126)kU\I , tothecreationofentanglementwithasetofapparatusesdur-
inglocalpre-measurementsofthesubsystemsS . When,on
I
withρi(cid:126)k unnormalizedstatesoftheothersubsystemsS . the other hand, such entanglement is necessarily created, its
SU\I U\I minimum amount (measured by any chosen monotone E),
The above definition provides a finer graining between where the minimization is over the choice of the local mea-
the conventional categories of strictly classically correlated surements,canberegardedasameasureofquantumcorrela-
(i.e., U-CC)andgenuinelyquantumlycorrelated(i.e., ∅-CC) tionsQ intheoriginalstateρ ofthesystem[19,20]. We
E SU
states, including all possible intermediate types of so-called can thus formulate the following quantitative definition for a
classical-quantumstates[23]. generalfamilyofmeasuresofquantumcorrelations.
AcompletevonNeumannmeasurementinabasis{|i(cid:105) }
Sk Definition 2 (Quantumness of correlations). The measure
canberealizedonasubsystemS bylettingS interactwith
k k Q(SI) of quantumness of correlations—or, simply, quantum
a measurement apparatus Mk. Assuming Mk is initially in E
correlations—among all the subsystems of the system S in
somefixedbutotherwisearbitraryinitialpurestate|0(cid:105) ,and U
Mk the state ρ , revealed by local measurements on the (sub-
that Sk and Mk interact through a unitary USkMk, one finds setof)subsSyUstemsS ,andcorrespondingtotheentanglement
thatthelattermustbeofthetype I
measureE,isdefinedby
U |i(cid:105) |0(cid:105) =|i(cid:105) |i(cid:105) , (2)
SkMk Sk Mk Sk Mk Q(SI)(ρ ):= min E (ρ˜ ), (4)
up to a local unitary on M , if we impose that E SU {{|i(cid:105)Sk}|k∈I} SU:MI SUMI
k
the interaction realizes the projective measurement, where the minimum runs over all choices of local bases
that is, Tr (U ρ ⊗ |0(cid:105)(cid:104)0| U† ) = (equivalently, local complete von Neumann measurements)
Mk SkMk Sk Mk SkMk
3
for subsystems S , and the bipartite entanglement measure us,crucially,thataccordingtotheframeworkofDefinition2,
I
E is calculated between the systems S and M in the pre- Q and E always obey a precise ordering relation, with the
U I E
measurementstateρ˜ [Eq.(3)],whichdependsimplicitly latter (however quantified), if present, accounting only for a
SUMI
onthelocalbaseschoice. fraction of the more general quantum correlations (compat-
ibly defined) in arbitrary quantum states. In particular, it is
AccordingtoDefinition2,forabipartitesystemS ≡ S
AB straightforward to check that, thanks to the Schmidt decom-
(n = 2,S ≡ A,S ≡ B), if E is chosen to be, e.g., the
1 2 position,theinequalitiesof(6)aresaturatedforpurestatesof
distillable entanglement E [6, 24], then Q(AB) amounts to
D ED the system, returning that entanglement and quantum corre-
the (two-way) relative entropy of quantumness [20, 25] and
lationsingeneralcoalesceintoauniquesignatureinabsence
Q(A) amounts to the one-way information deficit [19, 26]. of global mixedness [8]. For generally mixed states, Eq. (5)
ED
Other instances of measures falling in the category of Defi- puts the present approach to investigating non-classicality of
nition2are,e.g.,theso-callednegativityofquantumness[20] correlations on firm physical grounds: quantum correlations
(alsoknownasminimumentanglementpotential[27])which trulygobeyondentanglement,inaclearquantitativesense.
correspondstopickingE tobethenegativityN [6], andthe TherightmostsidesofEqs.(5)and(6)involvebipartiteen-
geometric measure of quantumness [28]. All the mentioned tanglementE,butwecanalsoextendtheorderingwehavees-
measures, among all, have been independently proposed in tablishedtothemultipartitecase.Forexample,onecandefine
different contexts (ranging fromthermodynamics to geomet- two quite natural measures of (global) multipartite entangle-
ricapproaches)andhavewellunderstooddefinitionsthatcap- ment starting from any bipartite entanglement monotone E:
turevarious,intertwinedfeaturesofnon-classicalcorrelations E (ρ ) := min(max) E (ρ ), where
in composite systems. Our framework, which builds on and thmeinm(minaxi)mizSaUtion(maximizatio{nP)0i,sP1t}akeSnP0o:vSePr1allSnUon-trivial
generalizestheideasof[19,20],providesuniversalandphysi- partitions of U, that is, P ∩ P = ∅,P ∪ P = U. The
0 1 0 1
callyjustifiedoperationalinterpretationstothoseandawhole twomultipartitemeasuresinherittheLOCCmonotonicity(7)
lot of infinitely-many possible measures QE (one class for fromthebipartitemeasureE,andbothcoincidewiththelat-
each valid E [6]) in terms of the minimum entanglement E terinthebipartitecase. Then,from(5)itfollowsforinstance
necessarilycreatedduringmeasurementprocesses. thatQ(SU)(ρ )≥E (ρ )≥E (ρ ).
E SU max SU min SU
Main result.— We are now ready to state the main result of FlowofquantumcorrelationsandthevonNeumannchain.—
this paper, namely the ordering between entanglement and Wenowmovetocharacterizingthespreadingofentanglement
generalquantumcorrelationsinthestateofanarbitraryquan- amongsystemsandmeasurementapparatuses. Wewillrefer
tumsystemSU. tothefollowingnotionofgenuinemultipartiteentanglement.
Theorem 2. All measures QE of quantum correlations de- Definition3. Apurestate|ψ(cid:105)SU isgenuinelymultipartiteen-
finedbyEq.(4)satisfy tangledif|ψ(cid:105) (cid:54)=|ψ (cid:105) |ψ (cid:105) forallnon-trivialbiparti-
SU 0 SP0 1 SP1
tions{P ,P }ofU. Amixedstateρ isgenuinelymultipar-
Q(ESI)(ρSU)≥Q(ESJ)(ρSU)≥ESK:SU\K(ρSU), (5) tite enta0ngle1d if for any pure-stateSeUnsemble decomposition
(cid:80)
{p ,|ψ (cid:105) } (such that ρ = p |ψ (cid:105)(cid:104)ψ | ) there is at
for all corresponding entanglement measures E, and all i i SU SU i i i i SU
leastonegenuinemultipartiteentangledstateappearingwith
choices of K ⊆ J ⊆ I ⊆ U. In particular, in the bipartite
non-vanishingprobability.
case,onehas
Wegetthefollowingresult,whoseproofisinAppendix.
Q(AB)(ρ )≥max{Q(A)(ρ ), Q(B)(ρ )}
E AB E AB E AB Theorem 3. The state ρ is genuinely multipartite entan-
≥min{Q(A)(ρ ), Q(B)(ρ )} (6) gledifandonlyifanypreS-Umeasurementstateρ˜ ,forany
E AB E AB SUMI
≥E (ρ ). I ⊆ U,isgenuinelymultipartiteentangled,withmultipartite
A:B AB
entanglementsharedamongallsubsystemsofS ∪M .
U I
A proof of Theorem 2 is provided in the Appendix. The
All the above results can be used to analyze qualitatively
theorem is valid for all choices of E as it relies only on the
andquantitativelyavonNeumannchainwhoselinksaregiven
basicmonotonicityproperty
byasequenceofmeasurementapparatuses(seeFig.1). That
E(ρ)≥E(Λ [ρ]), (7) is, suppose that a measurement apparatus M1, initially in
LOCC
a pure state, is used to probe a single system S in a state
satisfied by any entanglement measure by definition [6], for ρ . The measurement is performed by letting S and M
S 1
allstatesρandalltransformationsΛ thatcanberealized interact as in Eq. (2), leading to a pre-measurement state
LOCC
byLocalOperationsandClassicalCommunication(LOCC). ρ˜ . Using the same tools of [19, 20], it is easy to check
SM1
The leftmost inequality in (5) tells us that the more sub- that ρ˜ is S : M entangled if and only if the mea-
SM1 1
systemsaremeasured,themorequantumcorrelationsarere- surement is not performed in the eigenbasis of ρ . We
S
vealedinthestate;thisgeneralizestoarbitrarymeasuresQ can now consider another link in the von Neumann chain,
E
the few known dominance relations involving, e.g., two-way given by another apparatus M brought in to realize a com-
2
versus one-way discord [29]. The rightmost inequality tells plete projective measurement on M . We thus obtain a
1
4
global pre-measurement state ρ˜(cid:48) . Theorem 2 and Def- is trivially the same as the ABM : M entanglement in
inition 2 imply Q(EM2)(ρ˜(cid:48)SM1M2S)M≥1M2ESM1:M2(ρ˜(cid:48)SM1M2) ≥ ρl˜a(cid:48)AttBerMs1tMat2eb=ein(cid:80)goi,bjt|aii(cid:105)n(cid:104)ejd|Aby⊗leρtiBtjin⊗g1M|i(cid:105)(cid:104)ja|nMd21M⊗|iin(cid:105)t(cid:104)ejr|aMct2a,sthine
Q(M1)(ρ˜ ) ≥ E(S : M ) . This argument can be 1 2
E SM1 1 ρ˜SM1 Eq.(2).
reiterated for the next links in the chain, so that we find
Amorerealisticmodelofthemeasurementprocesswould
E ≤ E ≤ ... ≤ E , wherebi-
S:M1 SM1:M2 SM1M2...Mn:Mn+1 involvethesystemS,anumberofapparatusesM,andalarge
partiteentanglementiscalculatedforlargerandlargersystems
set of uncontrollable degrees of freedom—the environment
(that is, longer and longer chains, last link versus the rest).
E—that act by randomly measuring the subsystems and/or
Similarly Q(M1) ≤ ... ≤ Q(Mn) for the quantum correla-
E E theapparatuses[16]. Whilewecanhaveafinecontrolonthe
tionsofthecorrespondingsuccessivepre-measurementstates
measurement apparatuses, tuning them to realize minimally-
of the chain. We thus conclude that bipartite entanglement
disturbing von Neumann measurements on the objects to be
andquantumcorrelationsneverdecreasealongvonNeumann
observed, the non-tailored action of the environment results
chains. Noticethatitfollowsalsothat,whenweconsiderthe
in it getting strongly quantumly correlated with the accessi-
completechainandbreakitatanylevel,namelyinvestigating
ble degrees of freedom through bipartite and genuine multi-
thebipartition(SM ...M ) : (M ...M ),theentangle-
1 j j+1 n partite entanglement (subject to stringent “monogamy” con-
mentandthecorrespondingquantumcorrelationsacrossthis
straints[31]),accordingtothemechanismselucidatedabove.
(global)bipartitionarebothnondecreasingfunctionsofj. In-
Aftertracingovertheunaccessibleenvironmentaldegreesof
deed,successivemeasurementstepscorrespondtoalocalop-
freedom,theobservedsystemsandthemeasurementappara-
eration (actually, a local isometry) with respect to the fixed
tuses are thus typically left in classically correlated “pointer
bipartitionatthelevelj.
states”. Thisistheessenceofdecoherencebyenvironmental-
Theseconsiderationsarereadilygeneralizedtothecaseof
inducedselection[17],andourfindingsallowusinprinciple
S being a composite system. Let us focus for simplicity on
togetaquantitativegriponthenatureandmeasureof(quan-
thebipartitecase,withB beingmeasuredbyM ,whichisin
1 tum)correlationsduringtheprocess. Adetailedstudyofthis
turnbeingmeasuredbyM andsoon. Wehavethen
2 mechanism requires further investigation, possibly consider-
E (ρ ) ≤ Q(B)(ρ )≤E (ρ˜ ) ingalsothecaseof“fuzzymeasurements”inwhichthemea-
A:B AB E AB AB:M1 ABM1 surementapparatusesarethemselvesnoisy,i.e.,initializedin
≤ Q(EM1)(ρ˜ABM1)≤EABM1:M2(ρ˜(cid:48)ABM1M2) mixedratherthanpurestates[32]:inthatcase,itwillbeinter-
≤ Q(M2)(ρ˜(cid:48) )≤... . esting to analyze the role of classical correlations as well as
E ABM1M2 theirinterplaywithentanglementandgeneralquantumones.
The first step of the chain is the most important one, where
Conclusions.— The results presented in this Letter provide
the nature of correlations in the initial state of ρ of the
AB physical justification and demonstrate the insightfulness of
system plays a crucial role. If such a state is B-CC (i.e.,
the operational approach put forward in [19, 20] to address
Q(B)(ρ )=E (ρ )=0),thenthereexistsalocalvon
E AB A:B AB quantitativelythegeneralquantumcorrelationsinacompos-
NeumannmeasurementonB suchthatthepre-measurement
ite system in terms of the entanglement necessarily created
state ρ˜ contains no entanglement and no quantum cor-
ABM1 betweenthesystemandmeasurementapparatusesduringlo-
relationsbetweenthesystemandtheinvolvedapparatus[19].
cal measurements. In this Letter, we proved the key result
Iftheinitialstateofthesystemisunentangledbutquantumly
that,withinsuchaframework,quantumcorrelationsarenever
correlated[30](Q(EB)(ρAB)>0,EA:B(ρAB)=0),thendur- smaller than entanglement, going thus beyond it in general.
ingthepre-measurementtheinitialintra-systemquantumcor- This holds for any entanglement monotone and any corre-
relationsQ(B)(ρ )aretransformed(andpossiblyamplified) spondinglydefinedmeasureofquantumcorrelations.Wepro-
E AB
intoquantumcorrelationsQ(M1)(ρ ) ≥ Q(B)(ρ )be- videdquantitativeandqualitativeresultsonthepresenceand
E ABM1 E AB
tween the composite system and the apparatus, and corre- spreading of entanglement and quantum correlations along
spondingly an amount of entanglement E (ρ ) ≥ von Neumann chains. We hope that, motivated by our find-
AB:M1 ABM1
Q(B)(ρ ) is also created across the AB : M bipartition. ings, coordinated efforts will lead soon to a comprehensive
E AB 1 mathematicalresourcetheoryofgeneralquantumcorrelations
Finally, according to Theorem 3, iff the initial state of the
system is entangled (Q(B)(ρ ) ≥ E (ρ ) > 0), then andtonoveldemonstrationsoftheirpowerfornoise-resilient
E AB A:B AB quantumtechnologies[11].
genuinemultipartiteentanglementisestablishedbetweenthe
twosubsystemsandtheapparatus,andkeepsbeinggenuinely Acknowledgements.— We thank V. P. Belavkin, D. Bruss, E.
shared with all the subsequent apparatuses as well. Notice Chitambar, N. Gisin, T. Nakano, D. Terno and particularly
thataftertheinitialstep,theamountofbipartiteentanglement J.Calsamigliafordiscussions. MPacknowledgessupportby
betweenthelatestapparatusandtherestofthechaincanactu- NSERC, CIFAR, Ontario Centres of Excellence. GA is sup-
allystayconstant(E =...=E )if ported by a Nottingham Early Career Research and Knowl-
AB:M1 ABM1M2...Mn:Mn+1
everyapparatusrealizesanoptimal—inthesenseofEq.(4)— edge Transfer Award. We acknowledge joint support by the
localmeasurement.ThisisseenbynoticingthattheAB :M EPSRC Research Development Fund (Pump Priming grant
1
entanglement in ρ˜ = (cid:80) |i(cid:105)(cid:104)j| ⊗ ρij ⊗ |i(cid:105)(cid:104)j| 0312/09).
ABM1 i,j A B M1
5
Appendix:Proofs ProofofTheorem3
ProofofTheorem2 To prove the theorem we can restrict to measurements on
just one subsystem, say S , since we are considering multi-
n
For the sake of clarity, let us consider the bipartite case, partiteentanglementbetweenallsubsystemsandapparatuses,
with a local measurement applied only to A through an ap- and we can analyze the measurement on each Sk separately.
paratus M. The initial bipartite state of the system can be Following[33],weobservethatpure-statedecompositionsfor
expandedas ρSU andρ˜SUMn areinone-to-onecorrespondence,astheini-
tial state of M is pure (for each |ψ(cid:105) we obtain a pure
ρAB =(cid:88)|i(cid:105)(cid:104)j|A⊗ρiBj, state|ψ˜(cid:105)SUMn =n USnMn|ψ(cid:105)SU|0(cid:105)Mn,aSnUdforeach|ψ˜(cid:105)SUMn
i,j there must be a state |ψ(cid:105) such that U† |ψ˜(cid:105) =
SU SnMn SUMn
|ψ(cid:105) |0(cid:105) ). Onedirectionisthentrivial: ifρ isnotgen-
where {|i(cid:105)} is the basis in which A is measured. The pre- SU Mn SU
uine multipartite entangled, ρ˜ is not genuine multipar-
measurementstateisthen SUMn
titeentangledeither.Indeed,themeasurementinteractionwill
(cid:88)
ρ˜AMB = |i(cid:105)(cid:104)j|A⊗|i(cid:105)(cid:104)j|M ⊗ρiBj. mapastate|ψ0(cid:105)SP0|ψ1(cid:105)SP1,for{P0,P1}apartitionofU,into
i,j astate|ψ0(cid:105)SP0|ψ˜1(cid:105)SP˜1,with{P0,P˜1}anewpartitioninclud-
ingM inS ,ifweassume,withoutlossofgenerality,that
ThekeyobservationisthatthereisasimpleLOCCtransfor- n P˜1
S ∈ P . We now prove also that if ρ˜ is not genuine
mationΛLOCC (withrespecttothetheAB : M bipartitecut) n 1 SUMn
multipartiteentangled,thenρ wasnoteither.
suchthatΛLOCC[ρ˜AMB]=ρMB,whereρMBisthesamestate LetusconsiderapurestateS|Uψ˜(cid:105) =|ψ˜ (cid:105) |ψ˜ (cid:105) in
asρABbutnowsharedbetweenM andB.Dueto(7)wehave SUMn 0 SP˜0 1 SP˜1
the decomposition of ρ˜ . If both S and M are in P˜ ,
E(AB :M) ≥E (ρ )=E (ρ ). SUMn n n 1
ρ˜AMB M:B MB A:B AB then undoing the measurement interaction we obtain a state
|ψ (cid:105) |ψ (cid:105) . Suppose instead S ∈ P˜ , M ∈ P˜ . Im-
It remains to exhibit the operation ΛLOCC on ρ˜AMB, which po0sinSgP0 1 SP1 n 0 n 1
can be constructed as follows. A Fourier transform |i(cid:105) (cid:55)→
√
1/ d(cid:80) e2πiik/d|k(cid:105),withdthedimensionofA(andofM) |ψ˜(cid:105) =|ψ˜ (cid:105) |ψ˜ (cid:105) =U |ψ(cid:105) |0(cid:105) ,
is first apkplied to A, transforming the pre-measurement state SUMn 0 SP˜0 1 SP˜1 SnMn SU Mn
into namely imposing in particular that Tr (|ψ˜(cid:105)(cid:104)ψ˜| ) is
SP˜1 SUMn
d1(cid:88)(cid:88)e2πi(ik−jl)/d|k(cid:105)(cid:104)l|A⊗|i(cid:105)(cid:104)j|M ⊗ρiBj. pouthreer,swuebsfiynsdtemthsatanSdnamctuusatllbyeininaintiaelilgyenusnteantetaonfgtlheedmfreoamsutrhee-
k,l i,j mentperformedbyU . (cid:3)
SnMn
ThensubsystemAismeasuredinthe{|k(cid:105)}basis,anddepend-
ingontheresultaunitaryU =(cid:80) e−2πiik/d|i(cid:105)(cid:104)i|isapplied
k i
toM,obtaining
(cid:88) [1] J.OppenheimandS.Wehner,Science330,1072(2010).
ρ˜ = |i(cid:105)(cid:104)j| ⊗ρij.
MB M B [2] W.K.WoottersandW.H.Zurek,Nature299,802(1982).
i,j [3] M. A. Nielsen and I. L. Chuang, Quantum Computation
andQuantumInformation(CambridgeUniversityPress,Cam-
Theproofcanbestraightforwardlygeneralizedtothemul-
bridge,2000).
tipartitecase. Givenapre-measurementstateρ˜ thereare
SUMI [4] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777
Sk :MkLOCCtransformationsthat“undo”themeasurement (1935).
interaction between Sk and Mk, forany k ∈ I. Indeed, it is [5] E.Schro¨dinger,Naturwissenschaften23,812(1935).
possibletoactviaLOCCtochoosebetweenthefollowingtwo [6] M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007);
options:(i)toreallyjustundothemeasurementinteractionon R.Horodecki,P.Horodecki,M.Horodecki,andK.Horodecki,
Rev.Mod.Phys.81,865(2009).
S (asifnoapparatusM hadbeenintroducedatall)or(ii)to
k k
[7] J.S.Bell,Physics1,195(1964).
transfer coherently the quantum information contained orig-
[8] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901
inally in S into M . Option (i) justifies the first inequality
k k (2001);L.HendersonandV.Vedral,J.Phys.A34,6899(2001).
inEq.(5);option(ii)justifiesthesecondone,asforanysys- [9] A.Ferraro, L.Aolita, D.Cavalcanti, F.M.Cucchietti, andA.
tem Sk interacting with its measuring apparatus (i.e., k ∈ I, Acin,Phys.Rev.A81,052318(2010).
or k ∈ J in Eq. (5)) we can choose to have its information [10] R. Dillenschneider, Phys. Rev. B 78, 224413 (2008); A. Sha-
content transferred to M , on the other side of the S : M bani and D. A. Lidar, Phys. Rev. Lett. 102, 100402 (2009);
k U I
L.Mazzola,J.Piilo,andS.Maniscalco,Phys.Rev.Lett.104,
splitting. The latter possibility means that we can go from
200401(2010);A.BrodutchandD.R.Terno,Phys.Rev.A81,
the the pre-measurement state ρ˜ to ρ , for all
SU:MI SU\K:MK 062103(2010);M.D.LangandC.M.Caves,Phys.Rev.Lett.
K ⊆ I. Thelatterstateisthesameastheoriginalρ ,only
SU 105, 150501(2010); T.Werlang, C.Trippe, G.A.P.Ribeiro,
withtheinformationcontainedinthesystemsSK transferred andG.Rigolin,Phys.Rev.Lett.105,095702(2010);G.Adesso
totheapparatusesMK. (cid:3) andA.Datta, Phys.Rev.Lett.105, 030501(2010); P.Giorda
6
andM.G.A.Paris,Phys.Rev.Lett.105,020503(2010);M.F. [20] M. Piani, S. Gharibian, G. Adesso, J. Calsamiglia, P.
Cornelio,M.C.deOliveira,andF.F.Fanchini,Phys.Rev.Lett. Horodecki,andA.Winter,Phys.Rev.Lett.106,220403(2011).
107020502(2011);G.L.Giorgi,B.Bellomo,F.Galve,andR. [21] F. F. Fanchini, M. F. Cornelio, M. C. de Oliveira, and A. O.
Zambrini,Phys.Rev.Lett.107,190501(2011) Caldeira, Phys.Rev.A84, 012313(2011); A.Al-Qasimiand
[11] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, D.F.V.James,Phys.Rev.A83,032101(2011).
arXiv:1112.6238.(2011)SeealsotheNewsFeaturebyZ.Mer- [22] L.Chen,E.Chitambar,K.Modi,G.Vacanti,Phys.Rev.A83,
ali,Nature474,24(2011). 020101(R)(2011)
[12] D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and [23] M.Piani,P.Horodecki,andR.Horodecki,Phys.Rev.Lett.100,
A. Winter, Phys. Rev. A 83, 032324 (2011); V. Madhok and 090502(2008).
A.Datta,Phys.Rev.A83,032323(2011);V.MadhokandA. [24] C.H.Bennett,D.P.DiVincenzo,J.A.Smolin,andW.K.Woot-
Datta,arXiv:1107.0994(2011). ters,Phys.Rev.A54,3824(1996).
[13] B. M. Terhal, M. Horodecki, D. W. Leung, and D. [25] S.Bravyi,Phys.Rev.A67,012313(2003);K.Modi,T.Paterek,
P.DiVincenzo, J. Math. Phys. 43, 4286 (2002); D. P. DiVin- W. Son, V. Vedral, and M. Williamson, Phys. Rev. Lett. 104,
cenzo,M.Horodecki,D.Leung,J.Smolin,andB.M.Terhal, 080501(2010).
Phys. Rev. Lett. 92, 067902 (2004); S. Boixo, L. Aolita, D. [26] J.Oppenheim,M.Horodecki,P.Horodecki,andR.Horodecki,
Cavalcanti,K.Modi,andA.Winter,Int.J.Quant.Inf.9,1643 Phys.Rev.Lett.89,180402(2002).
(2011). [27] R.ChavesandF.deMelo,Phys.Rev.A84,022324(2011)
[14] K.Modi,M.Williamson,H.Cable,andV.Vedral,Phys.Rev. [28] A. Streltsov, H. Kampermann, and D. Bruss, Phys. Rev. Lett.
X1,021022(2011), 107,170502(2011).
[15] A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett 100, [29] M. D. Lang, C. M. Caves, and A. Shaji, Int. J. Quant. Inf. 9,
050502(2008);B.P.Lanyon,M.Barbieri,M.P.Almeida,and 1553(2011).
A.G.White,Phys.Rev.Lett.101,200501(2008);A.Brodutch [30] For the sake of the discussion we consider here a faithful en-
and D. R. Terno, Phys. Rev. A 83, 010301 (2011); B. Eastin, tanglementmeasureE suchthatE(ρ ) = 0iffρ isnot
AB AB
arXiv:1006.4402(2010); G.Passante,O.Moussa,D.A.Trot- entangled.
tier,andR.Laflamme,Phys.Rev.A84,044302(2011);R.Auc- [31] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A
caise,J.Maziero,L.Celeri,D.Soares-Pinto,E.deAzevedo,T. 61, 052306 (2000); B. M. Terhal, IBM J. Res. & Dev. 48,
Bonagamba,R.Sarthour,I.Oliveira,andR.Serra,Phys.Rev. 71 (2004); A. Streltsov, G. Adesso, M. Piani, and D. Bruss,
Lett.107,070501(2011). arXiv:1112.3967(2011).
[16] W.H.Zurek,PhysicsToday44,36(1991). [32] V.Vedral,Phys.Rev.Lett.90,050401(2003).
[17] W.H.Zurek,Rev.Mod.Phys.75,715(2003). [33] J.K.Asbo´th,J.Calsamiglia,andH.Ritsch,Phys.Rev.Lett.94,
[18] J. Von Neumann, Mathematical Foundation of Quantum Me- 173602(2005).
chanics(PrincetonUniversityPress,Princeton,NJ,1955). [34] S. Gharibian, M. Piani, G. Adesso, J. Calsamiglia, and P.
[19] A. Streltsov, H. Kampermann, and D. Bruss, Phys. Rev. Lett. Horodecki,Int.J.Quant.Inf.9,1701(2011).
106,160401(2011).