Steering Bell-diagonal states

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by  Quan Quan

Steering Bell-diagonal states
Quan Quan1, Huangjun Zhu2,3*, Si-Yuan Liu1‡, Shao-Ming Fei4,5, Heng Fan6,7,1, and
Wen-Li Yang1,8
1InstituteofModernPhysics,NorthwestUniversity,Xi’an710069,China
2InstituteforTheoreticalPhysics,UniversityofCologne,Cologne50937,Germany
3PerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2L2Y5,Canada
4SchoolofMathematicalSciences,CapitalNormalUniversity,Beijing100048,China
6
5Max-Planck-InstituteforMathematicsintheSciences,Leipzig04103,Germany
1
6InstituteofPhysics,ChineseAcademyofSciences,Beijing 100190,China
0
2 7CollaborativeInnovationCenterofQuantumMatter,Beijing100190,China
8CenterforMathematicsandInformationInterdisciplinarySciences,Beijing,100048,China
b
e *Correspondingauthor: [email protected],[email protected]
F ‡[email protected]

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2
ABSTRACT

]
h
p We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In
- thesimplestnontrivialscenariooftwoprojectivemeasurements,wesolvethisproblemcompletelybyvirtueoftheconnection
t
n betweenthesteeringproblemandthejoint-measurementproblem.Anecessaryandsufficientcriterionisderivedtogetherwith
a asimplegeometricalinterpretation. OurstudyshowsthataBell-diagonalstateissteerablebytwoprojectivemeasurements
u iffitviolatestheClauser-Horne-Shimony-Holt(CHSH)inequality,insharpcontrastwiththestricthierarchyexpectedbetween
q steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the
[
volumeofthesteeringellipsoid. Inparticular,wedeterminethemaximalconcurrenceandellipsoidvolumeofBell-diagonal

2 statesthatarenotsteerablebytwoprojectivemeasurements.Finally,weexplorethesteerabilityofBell-diagonalstatesunder
v threeprojectivemeasurements. Asimplesufficientcriterionisderived,whichcandetectthesteerabilityofmanystatesthat
3 are not steerableby two projectivemeasurements. Our studyoffersvaluableinsighton steeringof Bell-diagonalstates as
1 wellastheconnectionsbetweenentanglement,steering,andBellnonlocality.
1
0
0
. Introduction
1
0
Einstein-Podolsky-Rosen(EPR) steering,1 as noticed bySchro¨dinger,2 is an intermediatetypeof nonlocalcorrelationsthat
6
1 sitsbetweenentanglementandBellnonlocality.Intheframeworkofmodernquantuminformationtheory,this“spookyaction”
: canbedescribedasataskofentanglementverificationwithanuntrustedparty,asexplainedbyWisemanetal. 3,4 Ithinges
v
onthequestionofwhetherAlicecanconvinceBobthattheyshareanentangledstate,despitethefactthatBobdoesnottrust
i
X
Alice. In order to achieve this task, Alice needs to change Bob’s state remotely in a way that would be impossible if they
r sharedclassicalcorrelationsonly. ContrarytoentanglementandBellnonlocality,steeringfeaturesafundamentalasymmetry
a
becausethetwoobserversplaydifferentrolesinthesteeringtest.3–5 Recently,growingattentionhasbeendirectedtosteering
because of its potential applications in quantum information processing, such as quantum key distribution (QKD),6 secure
teleportation,7andentanglementassistedsubchanneldiscrimination.8
Two basic questions concerning steering are its detection and quantification. One approach for detecting steering is to
provetheimpossibilityofconstructinganynon-steeringmodel.3,4 Apracticalalternativeistodemonstratetheviolationsof
varioussteeringinequalities.9–14 ThefirststeeringinequalitywasderivedbyReidin1989,9whichisapplicabletocontinuous
variablesystems,asconsideredinEPR’soriginalargument. Generaltheoryofexperimentalsteeringcriteriaweredeveloped
in Ref.,10 followed by many other works.11–14 In line with theoretical development, a loophole-free steering experiment
wasreportedinRef.,15 andone-waysteeringwasdemonstratedinRef.16 Inaddition,steeringdetectionbasedonall-versus-
nothingargumentwasproposedinRefs.,17,18alongwithanexperimentaldemonstration.19 Meanwhile,steeringquantification
hasreceivedincreasingattentioninthepastfewyears,8,20,21whichleadstoseveralusefulsteeringmeasures,suchassteerable
weight20andsteeringrobustness.8
Despitethesefruitfulachievements,steeringdetectionandquantificationhaveremainedchallengingtasks,andmanybasic
questionsarepoorlyunderstood.Forexample,noconclusivecriterionisknownfordeterminingthesteerabilityofgenerictwo-
qubitstatesexceptforWernerstates.3,4 EvenforBell-diagonalstates, onlya fewpartialresultsareknownconcerningtheir
steerability,includingseveralnecessarycriteriaandseveralsufficientcriteria;22–24furtherprogressesarethushighlydesirable.
Inaddition,manyresultsintheliteraturerelyheavilyonnumericalcalculationandlackintuitivepictures. Analyticalresults
arequiteraresincedifficultoptimizationproblemsareofteninvolvedinsolvingsteeringproblems.
Inthiswork,weinvestigatethesteerabilityoftwo-qubitBell-diagonalstatesunderprojectivemeasurementsbythesteering
party.Thesestatesareappealingtoboththeoreticalandexperimentalstudiessincetheyhavearelativelysimplestructureand
are particularly suitable for illustrating ideas and cultivating intuition. In addition, generic two-qubit states can be turned
intoBell-diagonalstatesbyinvertiblestochasticlocaloperationandclassicalcommunication(SLOCC),25soanyprogresson
Bell-diagonalstatesmaypotentiallyhelpunderstandtwo-qubitstatesingeneral.
WefirstconsiderthesteerabilityofBell-diagonalstatesunderthesimplestnontrivialmeasurementsettingonthesteering
party, that is, two projective measurements. We solve this problem completely by virtue of the connection between the
steering problem and the joint-measurement problem.14,26–28 In particular, we derive a necessary and sufficient steering
criterion analytically and provide a simple geometricalinterpretation. Such analyticalresults are valuable but quite rare in
the literature on steering. Our study leads to a measure of steering, which turns out to equal the maximal violation of the
Clauser-Horne-Shimony-Holt(CHSH)inequality.29,30 Asanimplication,aBell-diagonalstateissteerablebytwoprojective
measurementsiffitviolatestheCHSHinequality.Thisconclusionpresentsasharpcontrastwiththeobservationthatsteering
isnecessarybutusuallynotsufficientforBellnonlocality.3,4,31 Ontheotherhand,inthespecialcaseofrank-2Bell-diagonal
states,entanglementissufficienttoguaranteesteeringandBellnonlocality,inlinewiththespiritofGisin’stheorem.32,33 The
relationsbetweenoursteeringmeasureandconcurrenceaswellasthevolumeofthesteeringellipsoidarethenclarified.Quite
surprisingly,thesteeringmeasureandthevolumeofthesteeringellipsoidseemtodisplayoppositebehaviorsforstateswith
givenconcurrence.
Finally,weexplorethesteerabilityofBell-diagonalstatesunderthreeprojectivemeasurements. Althoughsuchproblems
are generallyvery difficultto address, we derivea nontrivialsufficientcriterion, which also hasa simple geometricalinter-
pretation. Thiscriterioncandetectthesteerabilityofmanystatesthatarenotsteerablebytwoprojectivemeasurements. The
relationbetweenentanglementandsteeringinthisscenarioisalsoclarified.
Setting up the stage
Consider two remote parties, Alice and Bob, who share a bipartite quantum state r with reduced states r and r for the
A B
two parties, respectively. Alice canperforma collectionoflocalmeasurementsascharacterizedbya collectionofpositive-
operator-valued measures (POVMs) A , where x labels the POVM and a labels the outcome in each POVM. Recall
ax a,x
thata POVM A is composedof{a s|e}t ofpositiveoperatorsthatsum up to the identity,thatis, (cid:229) A =I. The whole
ax a a ax
collection of P{OV|M}s A is called a measurement assemblage. If Alice performsthe measurement|x and obtains the
ax a,x
outcomea,thenBob’s{sub|n}ormalizedreducedstateisgivenbyr =tr [(A I)r ]. Notethat(cid:229) r =r isindependent
ax A ax a ax B
ofthemeasurementchosenbyAlice,asrequiredbythenosignali|ngprinciple|.⊗Thesetofsubnormaliz|edstates r fora
ax a
givenmeasurementxisanensembleforr ,andthewholecollectionofensembles r isastateassemblag{e.12| }
B ax a,x
Thestateassemblage r isunsteerableifthereexistsalocalhiddenstate({LH|S})model:3,4,14,26–28
ax a,x
{ | }
r ax=(cid:229) pr (ax,l )s l , (1)
| l |
where pr (ax,l ) 0,(cid:229) apr (ax,l )=1,ands l areacollectionofsubnormalizedstatesthatsumuptor B andthusforman
ensemblefo|rr .≥Thismodelm| eansthatBobcan interprethisconditionalstates r as comingfromthe preexistingstates
B ax
s l ,whereonlytheprobabilitiesarechangedduetotheknowledgeofAlice’smeasur|ementsandoutcomes.
Thesteeringproblemiscloselyrelatedtothejoint-measurementproblem. A measurementassemblage A iscom-
ax a,x
patibleorjointlymeasurable26–28,34,35 ifthereexistaPOVM Gl andprobabilitiespA(ax,l )with(cid:229) apA({ax,|l})=1such
{ } | |
that
Aax=(cid:229) pA(ax,l )Gl . (2)
| l |
Physically,thismeansthatallthemeasurementsintheassemblagecanbemeasuredjointlybyperformingthemeasurement
Gl l and post processing the measurement data. According to the above discussion, determining the compatibility of a
{ }
measurement assemblage is mathematically equivalent to determining the unsteerability of a state assemblage. Therefore,
manycompatibilityproblemscanbetranslatedintosteeringproblems,andviceversa.14,26–28 Thisobservationwillplayan
importantroleinthepresentstudy.
When r is of full rank, the state assemblage r for Bob can be turned into a measurement assemblage as fol-
B ax a,x
lows,14,28 { | }
Bax=r B−1/2r axr B−1/2. (3)
| |
2/12
Notethatthesetofoperators B foragivenxformsaPOVM,whichisreferredtoasBob’ssteering-equivalentobservable
ax a
(or POVM).28 The measurem{en|t}assemblage B is compatible iff the state assemblage r is unsteerable. For
ax a,x ax a,x
erexaasmonpilne,gi.fTrhai|xs=ob(cid:229)selrvpa(taio|xn,lsu)gsgle,stthseanfBruai|xtf=ul{(cid:229)aplpp|ro(}aa|cxh,lfo)rGulndweirtshtaGnldi=ngrsB−te1e/r2isnlgrvB−ia1/s2t;eethriencgo-{neqvue|rivs}ealfeonltloowbssefrrvoambltehse.same
Results
SteerBell-diagonalstatesbyprojectivemeasurements
Anytwo-qubitstatecanbewritteninthefollowingform
r = 1(I I+aaa sss I+I bbb sss + (cid:229) 3 t s s ), (4)
ij i j
4 ⊗ · ⊗ ⊗ · ⊗
i,j=1
wheres for j=1,2,3arethreePaulimatrices,sss isthevectorcomposedofthesePaulimatrices,aaaandbbbaretheBlochvectors
j
associatedwiththereducedstatesofAliceandBob,respectively,andT =(t )isthecorrelationmatrix. Thetwo-qubitstate
ij
isaBell-diagonalstateiffaaa=bbb=000,36inwhichcasewehave
r = 1(I I+ (cid:229) 3 t s s ), (5)
ij i j
4 ⊗ ⊗
i,j=1
withtwocompletelymixedmarginals,thatis, r =r =I/2. Bell-diagonalstatesareofspecialinterestbecausetheyhave
A B
a simple structure and are thus a good starting pointfor understandingstates with more complexstructure. In addition, all
two-qubitstatesexceptforasetofmeasurezerocanbeturnedintoBell-diagonalstatesbyinvertibleSLOCC.25
Withasuitablelocalunitarytransformation,thecorrelationmatrixT in(5)canbeturnedintodiagonalform,sothat
r = 1(I I+(cid:229) 3 t s s ). (6)
j j j
4 ⊗ ⊗
j=1
As an implicationofthisobservation,a Bell-diagonalstate is steerable byonepartyiff it issteerable bythe otherparty, so
thereisnoone-waysteering5forBell-diagonalstates. Itdoesnotmatterwhichpartyservesasthesteeringpartyinthepresent
study.
Inthecaseofaqubit,anyprojectivemeasurement A withtwooutcomes isuniquelydeterminedbyaunitvector
x
eee on the Bloch sphere as A =(I eee sss )/2. If A{lic±e| }and Bob share the Be±ll-diagonal state (5) and Alice performs
x x x
the projectivemeasurementd±e|termine±dby·eee , then the two outcomeswill occur with the same probability of 1/2, and the
x
subnormalizedreducedstatesofBobaregivenbyr =[I (TTeee ) sss ]/4.Accordingly,Bob’ssteering-equivalentobservable
x x
takesontheform ±| ± ·
1
B = (I rrr sss ), rrr =TTeee . (7)
±|x 2 ± x· x x
Notethatthisobservableisuniquelycharacterizedbythesubnormalizedvectorrrr , whichdeterminesanunbiasednoisy(or
x
unsharp)vonNeumannobservable.Here“unbiased”meansthattr(B )=tr(B )=1. Inthisway,thecorrelationmatrixT
+x x
inducesamapfromprojectivemeasurementsofAlicetonoisyprojecti|vemeasu−re|mentsofBob.AlicecansteerBob’ssystem
usingthemeasurementassemblage A iffthesetofnoisyprojectivemeasurements B isincompatible.
x x x x
Toseethegeometricmeaningof{the±m| }apinducedbyT,notethattheendpointofrrr lie{so±n|a}nellipsoidE centeredatorigin
x
and characterizedby thesymmetricmatrixTTT: the threeeigenvaluesof TTT are the squaresofthe threesemiaxes(some
ofwhichmayvanish),andtheeigenvectorsdeterminetheorientationsofthesesemiaxes;see Fig.1. Thisellipsoidencodes
thesetofpotentialnoisyprojectivemeasurementsofBobinducedbyprojectivemeasurementsofAlice. Geometrically,this
ellipsoid is identicalto the steeringellipsoid introducedin Refs.,23,37,38 whichencodesthe setof states to whichAlice can
steerBob’ssystem. Itisalsoreferredtoasthesteeringellipsoidherealthoughthemeaningisslightlydifferentfromthatin
Ref.23,37,38 Sinceitsdiscovery,thesteeringellipsoidhasplayedanimportantroleinunderstandingvariousfeaturespertinent
to entanglement and steering.23,24,37–39 To appreciate its significance in the current context, note that the steerability of a
Bell-diagonalstatebythemeasurementassemblage A iscompletelydeterminedbythesetofvectorsrrr onthesteering
x x x
ellipsoid.Moreover,inseveralcasesofprimaryinter{est±to| }us,thesteerabilitycanbedeterminedbypurelygeometricalmeans,
asweshallseeshortly.
3/12
Figure1. ThesteeringellipsoidsofthreeBell-diagonalstates. TheellipsoidofaBellstate(left)coincideswiththeBloch
sphere;theellipsoidofarank-2Bell-diagonalstateoranedgestate(middle)isrotationallysymmetricwiththelargest
semiaxisequaltotheradiusoftheBlochsphere;theellipsoidofaWernerstate(right)isaspherecontainedintheBloch
sphere.
Steeringbytwoprojectivemeasurements
InthissectionwederiveanecessaryandsufficientcriteriononthesteerabilityofaBell-diagonalstateundertwoprojective
measurements. We also introduce a steering measure and illustrate its geometrical meaning. Our study shows that a Bell-
diagonalstate is steerable by two projectivemeasurementsiff it violatesthe CHSH inequality. Furthermore,we clarifythe
relations between entanglement, steering, and Bell nonlocality by deriving tight inequalities between the following three
measures:theconcurrence,thesteeringmeasure,andthevolumeofthesteeringellipsoid.
Theorem 1. A Bell-diagonal state with correlation matrix T is steerable by two projective measurements iff l +l >1,
1 2
wherel ,l arethetwolargesteigenvaluesofTTT.
1 2
Proof. SupposeAliceperformstwoprojectivemeasurements A 2 = (I eee sss )/2 2 .ThenBob’ssteeringequivalent
observablesaregivenby B 2 = (I rrr sss )/2 2 ,wh{ere±rrr|x}=x=T1 Teee{,a±sspxe·cified}inx=(17). AccordingtoRef.40 (seealso
Refs.35,41–43),thetwoob{ser±va|xb}lxe=s1are{com±paxti·bleiff}x=1 x x
rrr +rrr + rrr rrr 2. (8)
1 2 1 2
| | | − |≤
Notethatrrr andrrr aretwovectorsonthesteeringellipsoid,andthelefthandsideoftheinequalityishalfoftheperimeter
1 2
ofaparallelograminscribedonthesteeringellipsoid,withtheplanespannedbytheparallelogrampassingthecentreofthe
ellipsoid.SotheBell-diagonalstateissteerableiffthemaximalperimeterofsuchparallelogramsislargerthan4.Interestingly,
themaximumcanbederivedwithasimilarmethodusedforderivingthemaximalviolationoftheCHSHinequality,30,44
max rrr1+rrr2 + rrr1 rrr2 =max TT(eee111+eee222) + TT(eee111 eee222) = max 2cosc TTccc +2sinc TTccc⊥
eee111,eee222{| | | − |} eee111,eee222{| | | − |} c ,ccc,ccc⊥{ | | | |}
=2mccc,cccaxq|TTccc|2+|TTccc⊥|2=2mccc,cccaxpcccTTTTccc+ccc⊥TTTTccc⊥=2pl 1+l 2, (9)
⊥ ⊥
where2c istheanglespannedbyeee andeee ;cccandccc arethedirectionvectorsof(eee +eee )and(eee eee ),respectively,which
1 2 ⊥ 111 222 111 222
−
arealwaysorthogonal. Herethemaximuminthelaststepisattainedwhencccandccc spanthesamespaceasthatspannedby
⊥
thetwoeigenvectorsassociatedwiththetwolargesteigenvaluesofTTT. Themaximumovereee andeee canbeattainedwhen
1 2
thetwovectorsareeigenvectorscorrespondingtothetwolargesteigenvaluesofTTT. TheBell-diagonalstateissteerableby
twoprojectivemeasurementsiff2√l +l >2,thatis,l +l >1.
1 2 1 2
The choicesof ccc andccc thatmaximize (9) are highlynotunique. Therefore,the optimalprojectivemeasurementsthat
⊥
Aliceneedstoperformarealsonotunique.Althoughtheoptimalmeasurementscanalwaysbechosentobemutuallyunbiased
asshownintheaboveproof,itisusuallynotnecessarytodoso. Asanexample,considertheBell-diagonalstatecharacterized
bythecorrelationmatrixT =diag(t ,t ,t )witht t t . Onechoiceofcccandccc readsccc=(1,0,0)andccc =(0,1,0),
1 2 3 1 2 3 ⊥ ⊥
≥ ≥| |
whichleadstotheoptimalmeasurementdirectionseee =(t ,t ,0)/ t2+t2 andeee =(t , t ,0)/ t2+t2. Notethateee and
1 1 2 q1 2 2 1 −2 q1 2 1
eee arenotorthogonalingeneral,sothecorrespondingprojectivemeasurementsarenotmutuallyunbiased.
2
The proof of Theorem 1 also suggests a steering measure of a Bell-diagonal state under two projective measurements,
namely,S:=2√l +l . Thismeasurehasasimplegeometricalmeaning: (S/2)2 isequaltothesumofsquaresofthetwo
1 2
largestsemiaxesofthesteeringellipsoid.ABell-diagonalstateissteerableinthisscenarioiffS>2. Themaximum2√2ofS
isattainedwhenl =l =1,whichcorrespondstoaBellstate. Toobtainanormalizedmeasureofsteering,wemayoptfor
1 2
4/12
max 0,(S 2)/(2√2 2) . AccordingtoRef.,30themaximalviolationoftheCHSHinequalitybytheBell-diagonalstateis
equa{lto2√−l +l (c−f.Re}f.45 forageometricalinterpretation),whichcoincideswiththesteeringmeasureSintroducedhere.
1 2
ThisobservationhasanimportantimplicationfortherelationbetweensteeringandBellnonlocality.
Corollary1. ABell-diagonalstateissteerablebytwoprojectivemeasurementsiffitviolatestheCHSHinequality.
ToclarifythegeometricmeaningofTheorem1andthesteeringmeasureS,itisconvenienttochooseaconcreteBellbasis.
Hereweshalladoptthefollowingchoice,46
1
b mn = (0,n +( 1)m 1,1 n ), m ,n =0,1. (10)
| i √2 | i − | ⊕ i
Notethat b isthesinglet.ThankstothechoiceoftheBellbasis,thecorrelationmatricesofthefourBellstatesarediagonal
11
asgivenb|ydiiag(( 1)m , ( 1)m +n ,( 1)n ). Uptoalocalunitarytransformation,anyBell-diagonalstateisamixtureofthe
− − − −
fourBellstates. Withoutlossofgenerality,wecanfocusonBell-diagonalstatesofthisform,whosecorrelationmatricesare
alsodiagonal,asin(6).
Figure2. GeometricillustrationofBell-diagonalstatessteerablebytwoprojectivemeasurements.(left)Theregular
tetrahedronrepresentsthesetofBell-diagonalstates. Theoctahedroningreenrepresentsthesetofseparablestates. Theblue
regionsrepresentthosestatesthataresteerablebytwoprojectivemeasurements.(right)Afaceoftheregulartetrahedron
whichrepresentsthesetofrank-3Bell-diagonalstates.
Geometrically, the set of Bell-diagonal states forms a regular tetrahedron, whose vertices correspond to the four Bell
states.36,46 ThesetofseparableBell-diagonalstatesformsanoctahedroninsidethetetrahedron.36,46 Thetetrahedroncanbe
embeddedintoacubewhosesidesarealignedwiththethreeaxeslabelledbyt ,t ,t ,asshowninFig.2. Inthisway,aBell-
1 2 3
diagonalstateisuniquelyspecifiedbyitsthreecoordinates(t ,t ,t ). ThehalfsteeringmeasureS/2ofthisBell-diagonalstate
1 2 3
isequaltothemaximumover t2+t2, t2+t2,and t2+t2,whichisequaltothemaximallengthofthethreeprojections
q1 2 q2 3 q3 1
of (t ,t ,t ) onto the three coordinateplanes. Note that S is convexint ,t ,t and definesa normin the three-dimensional
1 2 3 1 2 3
vector space that accommodates Bell-diagonal states. Each level surface of this norm is determined by three orthogonal
cylindersofequalradius.Inparticular,thesetofunsteerableBell-diagonalstates(determinedbythelevelsurfacewithS=2)
iscontainedintheintersectionofthethreesolidcylindersspecifiedbythefollowingthreeinequalities,respectively,
t2+t2 1, t2+t2 1, t2+t2 1. (11)
1 2 ≤ 2 3 ≤ 3 1 ≤
In the rest of this section we clarify the relations between the following three measures: the concurrence, the steering
measureS,andthe volumeof thesteeringellipsoid. SinceS isequalto themaximalviolationof theCHSH inequality,our
discussionisalsoofinteresttostudyingBellnonlocality.
Recallthatatwo-qubitstateisentanglediffithasnonzeroconcurrenceandthattheconcurrenceofaBell-diagonalstate
is givenbyC=max 0,2p 1 , where p is the maximaleigenvalueof the state.47 Givena Bell-diagonalstate with
max max
correlationmatrixT,{the norm−aliz}edvolumeV of the steeringellipsoid isdefinedasV := det(T).23 IfT isdiagonal, say
| |
T =diag(t ,t ,t ), thenV = t t t . Theconstraints t 1for j=1,2,3implythat0 V 1,wheretheupperboundis
1 2 3 12 3 j
| | | |≤ ≤ ≤
saturatedonlyforBellstates.
5/12
Figure3. RelationsbetweenthreeentanglementandsteeringmeasuresforBell-diagonalstates. HereCistheconcurrence,
Sisthesteeringmeasure,andV isthenormalizedvolumeofthesteeringellipsoid. Theorangeregionineachplotindicates
therangeofvalues.ThedashedlinesrepresentedgestatesandthesolidlinesrepresentWernerstates.
CalculationshowsthatC,S,V satisfythefollowinginequalities(seeMethodssectionformoredetails):
2√2
(1+2C) S 2 1+C2, (12)
3 ≤ ≤ p
1+2C 3
C2 V , (13)
≤ ≤(cid:16) 3 (cid:17)
2√2√3V S 2√1+V. (14)
≤ ≤
Here the lower boundin (12) is applicable to entangledBell-diagonalstates, while the other five boundsin (12), (13), and
(14) are applicable to all Bell-diagonal states. The inequality C2 V was also derived in Ref.39 As an implication of
≤
the aboveinequalities, anyBell-diagonalstate with concurrencelargerthan (3 √2)/(2√2) is steerable by two projective
−
measurements. ThenormalizedvolumeofthesteeringellipsoidofanyseparableBell-diagonalstateisboundedfromabove
by1/27,inagreementwiththeresultinRef.,23 whilethatofanyunsteerableBell-diagonalstate isboundedfromaboveby
1/(2√2).
TwotypesofBell-diagonalstatesdeservespecialattentionastheysaturatecertaininequalitiesin(12),(13),and(14). A
Wernerstatehastheform
1 f
W = f b b + − (I b b ), (15)
f 11 11 11 11
| ih | 3 −| ih |
where 0 f 1. Note that f is equal to the singlet fraction when f 1/4. Geometrically, the Werner state lies on a
≤ ≤ ≥
diagonalofthecubeinFig. 2; conversely,anyBell-diagonalstatelyingona diagonalofthecubeisequivalenttoa Werner
state undera localunitarytransformation. The correlationmatrix forthe Werner state has the formT =diag(t ,t ,t ) with
1 2 3
t =t =t =(1 4f)/3. Therefore,thesteeringellipsoidreducestoaspherewithradiust =t =t = 4f 1/3;seethe
1 2 3 1 2 3
− | − |
rightplotinFig.1. Inaddition,
2√2 4f 13
C=max 0,2f 1 , S= 4f 1, V = | − | . (16)
{ − } 3 | − | 27
TheWernerstateissteerablebytwoprojectivemeasurementsiff(3√2+2)/8< f 1. Itsaturatesthelowerboundin(14)
and,when f 1,alsothelowerboundin(12)andtheupperboundin(13). ≤
≥ 2
ThosestateslyingonanedgeofthetetrahedroninFig.2arecallededgestates(orrank-2Bell-diagonalstates). Ifanedge
statehastwononzeroeigenvalues pand1 pwith p 1/2,thent2 =1andt2 =t2 =(2p 1)2(assumingt t t ).
− ≥ 11 22 33 − 1≥ 2≥|3|
Therefore, the steering ellipsoid is rotationally symmetric with the largest semiaxis equal to 1 and the other two semiaxes
equalto2p 1;seethemiddleplotinFig.1. Inaddition,
−
C=2p 1, S=2q1+(2p 1)2, V =(2p 1)2. (17)
− − −
6/12
Theedgestateissteerablebytwoprojectivemeasurementswheneverp=1/2,thatis,whenitisentangled.Soentanglementis
sufficienttoguaranteesteeringandBellnonlocalityinthisspecialcase,6 whichcomplementsGisin’stheorem.32,33 Inaddition,
theedgestatesaturatestheupperboundsin(12)and(14)aswellasthelowerboundin(13).
Fig.3illustratestherelationsbetweenC,S,V. WhentheconcurrenceCislarge,thethreemeasuresarecloselycorrelated
toeachother,whiletheytendtobemoreindependentintheoppositescenario. Quitesurprisingly,thenormalizedvolumeV
of the steering ellipsoidseems to have a closer relationwith concurrenceC rather thanthe steering measureS. In addition,
forgivenconcurrenceC>0,thevolumeV attainsthemaximumwhenthesteeringmeasureSattainstheminimum,andvice
versa.
Steeringbythreeprojectivemeasurements
InthissectionweexplorethesteerabilityofBell-diagonalstatesunderthreeprojectivemeasurementsbythesteeringparty.To
thisend,weneedacriterionfordeterminingthecompatibilityofthreeunbiasednoisyprojectivemeasurements. Fortunately,
thisproblemhasbeensolvedinRefs.,48,49accordingtowhich,threenoisybinaryobservables B 3 = (I rrr sss )/2 3
arecompatibleiff { ±|x}x=1 { ± x· }x=1
3
(cid:229) LLL LLL 4. (18)
x FT
| − |≤
x=0
HereLLL =rrr +rrr +rrr ,LLL =2rrr LLL forx=1,2,3,andLLL denotestheFermat-Toricelli(FT)vectorof LLL 3 ,whichis
0 1 2 3 x x− 0 FT { x}x=0
thevectorLLL thatminimizesthetotaldistance(cid:229) 3 LLL LLL . Ingeneral,LLL hasnoanalyticalexpression.48,49
GivenaBell-diagonalstatewithcorrelationmx=a0tr|ixxxxT−,su|pposeAliceperFfTormsthreeprojectivemeasurements A 3 =
(I eee sss )/2 3 . ThenBob’ssteeringequivalentobservablesaregivenby B 3 = (I rrr sss )/2 3 ,{w±he|xr}ex=rrr1=
T{Teee±foxr·x=1,}2x,=31.Define { ±|x}x=1 { ± x· }x=1 x
x
S = 1 max (cid:229) 3 LLL LLL (19)
3 x FT
2rrr1,rrr2,rrr3∈Ex=0| − |
asasteeringmeasureoftheBell-diagonalstateunderthreeprojectivemeasurements,whereE isthesteeringellipsoid. Then
theBell-diagonalstateissteerablebythreeprojectivemeasurementsiffS >2. Ingeneral,itisnoteasytocomputeS . Here
3 3
we shall derive a nontriviallower bound, which is very useful for understandingthe steerability of Bell-diagonalstates by
threeprojectivemeasurements.
Whenrrr rrr ,theFTvectorcanbedeterminedexplicitly49(notethatthereisatypoinRef.49 aboutthesign),
3 1,2
⊥
rrr rrr rrr +rrr
LLL FT= |rrr1−rrr2|−+|rrr1+rrr2|rrr3, (20)
1 2 1 2
| − | | |
whichimplythat
3
(cid:229) |LLL x−LLL FT|=2q(|rrr1−rrr2|+|rrr1+rrr2|)2+4rrr23. (21)
x=0
Theorem2. AnyBell-diagonalstatewith T >1issteerablebythreeprojectivemeasurements,where T = tr(TTT)=
F F
k k k k p
tr(TTT)istheFrobeniusnormofthecorrelationmatrixT.
p
Proof. Letl ,l ,l betheeigenvaluesofTTT innonincreasingorderandeee ,eee ,eee theassociatedorthonormaleigenvectors.
1 2 3 1 2 3
Letrrr =TTeee forx=1,2,3. Thenrrr ,rrr ,rrr aremutuallyorthogonaland
x x 1 2 3
S3≥q(|rrr1−rrr2|+|rrr1+rrr2|)2+4rrr23=2pl 1+l 2+l 3=2kTkF. (22)
IftheBell-diagonalstateisnotsteerablebythreeprojectivemeasurements,thenS 2,so T 1.
3 F
≤ k k ≤
TheFrobeniusnorm T happenstobetheEuclideannormofthevector(t ,t ,t )thatrepresentstheBell-diagonalstate
F 1 2 3
inFigs.2and4;itssquarkeiskequaltothesumofsquaresofthethreesemiaxesofthesteeringellipsoid(cf.Ref.50). Thesetof
Bell-diagonalstateswiththesamenorm T liesonasphere. ItisclearfromtheabovediscussionthatS 2 T S,so
F 3 F
k k ≥ k k ≥
anyBell-diagonalstatethatissteerablebytwoprojectivemeasurementsisalsosteerablebythreeprojectivemeasurements,as
expected. Theconverseisnottrueingeneral,asillustratedinFig.4. ConsidertheWernerstatein(15)forexample,wehave
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Figure4. IllustrationofBell-diagonalstatessteerablebythreeprojectivemeasurements(cf.Fig.2). (left)Theregular
tetrahedronrepresentsthesetofBell-diagonalstates. Theoctahedroningreenrepresentsthesetofseparablestates. Theblue
regionsrepresentthosestatesthataresteerablebytwoprojectivemeasurements,andtheredregionsrepresentthosestates
thatarenotsteerablebytwoprojectivemeasurementsbutsteerablebythreeprojectivemeasurementsasspecifiedinthe
proofofTheorem2. (right)Afaceoftheregulartetrahedronwhichrepresentsthesetofrank-3Bell-diagonalstates.
T = 4f 1/√3,sotheWernerstateissteerablebythreeprojectivemeasurementsif1 f >(√3+1)/4.Bycontrast,it
F
k k | − | ≥
issteerablebytwoprojectivemeasurementsonlyif1 f >(3√2+2)/8.
≥
Therelationsbetween T andC,V canbederivedwithsimilarmethodsusedinderiving(12)and(14),withtheresults
F
k k
1
(1+2C) T 1+2C2, (23)
√3 ≤k kF≤p
√3V1/3 T √1+2V. (24)
F
≤k k ≤
Herethelowerboundin(23)isapplicabletoentangledBell-diagonalstates,whiletheotherthreeboundsareapplicabletoall
Bell-diagonalstates. Asin(12)and(14), thetwo lowerboundsaresaturatedbyWernerstates, whilethetwoupperbounds
aresaturatedbyedgestates;seeFig.5. TheseinequalitiesarequiteinstructivetounderstandingthesteeringofBell-diagonal
statesbythreeprojectivemeasurementsgiventhatS 2 T .Asanimplication,anyunsteerableBell-diagonalstatesatisfies
3 F
≥ k k
C (√3 1)/2andV 1/(3√3).
≤ − ≤
Discussion
In summary, we studied systematically the steerability of Bell-diagonal states by projective measurements on the steering
party. In the simplest nontrivialscenarioof two projectivemeasurements, we solvedthe problemcompletelybyderivinga
necessaryandsufficientcriterion,whichhasasimplegeometricalinterpretation. Wealsointroducedasteeringmeasureand
provedthatit is equalto the maximalviolation of the CHSH inequality. This conclusionimplies thata Bell-diagonalstate
issteerablebytwoprojectivemeasurementsiffitviolatestheCHSHinequality. Inthespecialcaseofedgestates,ourstudy
showsthatentanglementissufficienttoguaranteesteeringandBellnonlocality.Inaddition,weclarifiedtherelationsbetween
entanglementand steeringbyderivingtightinequalitiessatisfied by theconcurrence,our steeringmeasure, andthe volume
ofthesteeringellipsoid. Finally,weexploredthesteerabilityofBell-diagonalstatesunderthreeprojectivemeasurements. A
simplesufficientcriterionwasderived,whichcandetectthesteerabilityofmanystatesthatarenotsteerablebytwoprojective
measurements.
Our study provided a number of instructive analytical results on steering, which are quite rare in the literature. These
results notonly furnisha simple geometricpicture aboutsteering of Bell-diagonalstates, butalso offervaluableinsight on
the relations between entanglement, steering, and Bell nonlocality. They may serve as a starting point for exploring more
complicated steering scenarios. In addition, our work prompts several interesting questions, which deserve further studies.
Forexample,isthesteeringcriterioninTheorem2bothnecessaryandsufficient? Isthereanyupperboundonthenumberof
measurementsthatare sufficientto inducesteeringforall steerableBell-diagonalstates? We hopethatthese questionswill
stimulatefurtherprogressonthestudyofsteering.
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Figure5. Rangesofvaluesof2 T versustheconcurrenceC(left)andthenormalizedvolumeV ofthesteeringellipsoid
F
k k
(right)forBell-diagonalstates. Here T istheFrobeniusnormofthecorrelationmatrixT,and2 T isalowerboundfor
F F
k k k k
thesteeringmeasureS ,whichdeterminesthesteerabilityofBell-diagonalstatesunderthreeprojectivemeasurements.The
3
dashedlinesrepresentedgestatesandthesolidlinesrepresentWernerstates.
Methods
Concurrenceandsteeringmeasure
Herewederivetheinequalitiesin(12), (13), and(14)inthemaintext,whichcharacterizetherelationsbetweentheconcur-
renceC, the steering measure S (undertwo projectivemeasurements), and the volumeV of the steering ellipsoid. We also
determinethoseBell-diagonalstatesthatsaturatetheseinequalities. Similarapproachcanbeappliedtoderive(23)and(24),
whicharepertinenttosteeringofBell-diagonalstatesbythreeprojectivemeasurements.
Withoutlossofgenerality,wemayassumethatr hastheformin(6)with t t t 1. Thenthespectrumofr is
3 2 1
| |≤ ≤ ≤
givenby
1
1 t t t ,1 t +t +t ,1+t t +t ,1+t +t t , (25)
1 2 3 1 2 3 1 2 3 1 2 3
4(cid:8) − − − − − − (cid:9)
wheretheeigenvaluesarearrangedinnondecreasingorder. Theminimalandthemaximaleigenvaluesarerespectivelygiven
by p =(1 t t t )/4 0and p =(1+t +t t )/4.
min 1 2 3 max 1 2 3
IftheBell−-dia−gona−lstatei≥sseparable,thatisC=0,th−en0 p p 1/2,36whichimpliesthat
min max
≤ ≤ ≤
t +t + t 1, t2+t2 1, t t t 1/27. (26)
1 2 |3|≤ 1 2 ≤ |1 23|≤
SotheinequalitiesS 2√1+C2,V (1+2C)3/27,andS 2√1+V in(12),(13),and(14)holdforseparableBell-diagonal
states. The inequalit≤yS 2√1+C≤2 is saturated ifft =1≤, t =t =0, in which case r is an edge state with two nonzero
1 2 3
≤
eigenvaluesequalto1/2.TheinequalityS 2√1+V issaturatedunderthesamecondition.TheinequalityV (1+2C)3/27
issaturatedifft =t = t =1/3,inwhic≤hcaser isaWernerstatewhicheitherhassingletfraction1/2oris≤proportionalto
1 2 3
| |
aprojectorofrank3. HerestatesthatareequivalenttoW in(15)underlocalunitarytransformationsarealsocalledWerner
f
states. TheinequalityC2 V in(13)istrivialforseparablestates;itissaturatediffV =C=0,thatis,t =0,inwhichcase
3
≤
theBell-diagonalstateliesonacoordinateplaneinFig.2. Theinequality2√2√3V SfollowsfromthedefinitionsofSand
V andisapplicabletobothseparableandentangledstates. Itissaturatedifft =t =≤t ,inwhichcaser isaWernerstate.
1 2 3
| |
If the Bell-diagonalstate is entangled,then p >1/2,C=2p 1=(t +t t 1)/2, andt =t +t 1 2C.
max max 1 2 3 3 1 2
Thepositivityofr andtherequirement t t t leadtothefollowing−setofinequa−lities−, − −
3 2 1
| |≤ ≤
t t t +t 1+C, t +2t 1+2C. (27)
2 1 1 2 1 2
≤ ≤ ≥
Theseinequalitiesdetermineatriangularregionintheparameterspaceoft ,t withthefollowingthreevertices:
1 2
1 1
(1,C), (1+C,1+C), (1+2C,1+2C). (28)
2 3
The maximum1+C2 oft2+t2 under these constraintsis attained ifft =1,t = t =C, in which case the state has two
nonzeroeigenvaluesequal1to(12 C)/2andisthusanedgestate. Them1inimum2 2(−1+32C)2/9isattainedifft =t = t =
1 2 3
± −
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(1+2C)/3, in whichcase the state hasone eigenvalueequalto (1+C)/2 andthree eigenvaluesequalto (1 C)/6, andis
thusaWernerstate. Bycontrast,themaximum(1+2C)3/27of t t t isattainedexactlywhent2+t2 attains−theminimum,
|1 23| 1 2
and the minimumC2 of t t t is attained whent2+t2 attains the maximum. Therefore, (12) and (13) hold for entangled
|1 2 3| 1 2
Bell-diagonalstates. Asanimmediatecorollary,(14)alsoholdsinthiscase.
Insummary,thelowerboundin(12)isapplicabletoentangledBell-diagonalstates, whiletheotherfiveboundsin(12),
(13),(14)areapplicabletoallBell-diagonalstates. ThetwoinequalitiesS 2√1+C2 andS 2√1+V aresaturatedonly
for edge states. The inequalityC2 V is saturated only for edge states an≤d those states with≤V =0. The two inequalities
≤
2√2(1+2C)/3 S andV (1+2C)3/27 are saturated only for Werner states that have singlet fractions at least 1/2 or
≤ ≤
Wernerstates thatare proportionaltorank-3projectors. Theinequality2√2√3V S is saturatedonlyforWernerstates. In
≤
particular,amongentangledBell-diagonalstates,onlyedgestatesandWernerstateswithsingletfractionslargerthan1/2can
saturatetheseinequalities.
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