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Einstein-Podolsky-RosenSteering inQuantum PhaseTransition ChunfengWu,1 Jing-LingChen,1,2,∗ Dong-LingDeng,3 Hong-YiSu,1,2 X.X.Yi,1,4 andC.H.Oh1,5,† 1CentreforQuantumTechnologies, NationalUniversityofSingapore, 3ScienceDrive2,Singapore117543 2Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China 3Department of PhysicsandMCTP,UniversityofMichigan, AnnArbor, Michigan 48109, USA 4School ofPhysicsandOptoelectronicTechnology, DalianUniversityofTechnology, Dalian116024, China 5Department of Physics, NationalUniversityof Singapore, 2Science Drive3, Singapore 117542 (Dated:January31,2012) WeinvestigatetheEinstein-Podolsky-Rosen(EPR)steeringanditscriticalityinquantumphasetransition. It isfoundthattheEPRsteerabilityfunctionofthegroundstateofXYspinchainexhibitsnonanalyticfeaturein thevicinityofaquantumphasetransitionbyshowingthatitsderivativewithrespecttoanisotropyparameter divergesatthecriticalpoint. WethenverifytheuniversalityofthecriticalphenomenaoftheEPRsteerability function in the system. Wealso use two-qubit EPR-steering inequality to explore the relation between EPR steeringandquantumphasetransition. 2 PACSnumbers:03.65.Ud,75.10.Jm,64.70.Tg 1 0 2 Einstein et al. presented the so-called Einstein-Podolsky- maynotviolateBell inequalityandhenceonedoesnothave n Rosen (EPR) paradoxto questionthe completenessof quan- Bellnonlocality,buttheBellfunctionvaluestillcanbeusedto a tum mechanics based on locality and realism in 1935 [1]. captureQPTs.EPRsteeringliesstrictlyintermediatebetween J Soon after, Schro¨dinger[2] introducedthe term of entangle- Bell nonlocality and entanglement[15, 16] and in principle, 8 mentto describethe correlationsbetweentwo particles. En- EPR steering should be easier observed than Bell nonlocal- 2 tanglementthataquantumstatewhichcannotbeseparated,is ity due to the asymmetry descriptionbetween two observers ] the first type of nonlocal effect identified. In 1964, Bell [3] Alice and Bob. It has been shown experimentallythat some h presentedamethodintheformofBellinequalitytodescribe Bell-local states exhibit EPR steering and the EPR-steering p - quantumnonlocalpropertybasedontheassumptionsoflocal- inequality allowing for multi-setting measurements for two t n ityandrealism. TheviolationofBellinequalityrulesoutlo- partieshasbeenpresented[11]. Althoughmanyeffortshave a calhiddenvariabletheoriestodescribequantummechanicsas beendevotedtotheinvestigationsofEPRsteering,theEPR- u wellasimpliestheexistenceoftheso-calledBellnonlocality steering inequalities in the literatures are not strong enough q [4–7],thatisthesecondtypeofnonlcoalphenomenonarising fortwo-qubitsystems. Therefore,itisnotpossibletoobserve [ fromtheEPRparadox. ForBellnonlocality,entanglementis theEPRsteeringforsomestates,especiallyformixedstates. 2 necessarybutnotsufficient. Recently,EPRsteeringhasbeen Very recently, a criterion for EPR steering of two qubitshas v showntobethethirdtypeofnonlocalproperty[8–14]. EPR presented and it has been numerically proved to be a strong 3 steering,likeentanglement,wasoriginatedfromShro¨dinger’s condition to witness steerability [19]. This offers an effec- 4 2 replytotheEPRparadoxtoreflecttheinconsistencybetween tivewaytodetectEPRsteeringfortwoqubits. Aninteresting 0 quantum mechanicsand local realism. For a pure entangled question is whether EPR steering can be used to investigate . state held by two separated observers Alice and Bob, Bob’s thebehaviorofcondensedmattersystems. 1 0 qubitcanbe“steered”intodifferentstatesalthoughAlicehas 2 no access to the qubit. The EPR steering has been experi- 1 mentally observed by violation of EPR-steering inequalities Inthiswork,weinvestigatetheXYspinchaintoestablish v: and the violation demonstratesthe impossibility of using lo- the relation between EPR steering and QPTs. Utilizing the i calhiddenstatetheoriestocompletequantummechanics[11]. EPR-steering criterionproposedin our recentwork [19], we X Within the hierarchy of nonlocality, Bell nonlocality is the find the EPR steerability function as defined in Eq. (6) of ar strongest, followed by EPR steering, while entanglement is reduced density matrix of the grouSnd state. The function S theweakest[15,16]. indicatestheexistenceofEPRsteeringwhenitissmallerthan 0. Weanalyzethefunction anditsnonanalyticbehaviorat Inspiteoftheessentialroleplayedbyquantumnonlocality thetransitionpoint. OurresSultsshowthatthequantumcriti- inquantuminformationandcomputation,Literatures[17,18] calityintheXYmodelcanbecapturedbytheEPRsteerabil- have shownthat bothentanglementandBell nonlocalitycan ityfunctionofthegroundstate,andthisenablesconveniently beusedasatooltorevealquantumphasetransition(QPT)in testable quantum nonlocalityto signal the QPT. We also ex- many-body quantum systems. For entanglement, one needs ploretherelationbetweenEPRsteeringandQPTbyutilizing tofindouttheexactformofthereduceddensitymatrixofthe two-qubitEPR-steeringinequality. groundstateandutilizeentanglementmeasuretosignalQPTs. WhileforBellnonlocality,whichisthemoststringenttotest experimentally,thereduceddensitymatrixofthegroundstate Considerone-dimensionalXYspinchainwithHamiltonian 2 givenby M H = [(1+γ)σxσx +(1 γ)σyσy +hσz], (1) i i+1 − i i+1 i X −M where M = (N 1)/2 for the spin number N odd, γ is − anisotropy paramter, h is the strength of magnetic field, and (a) (b) σx,y,z are Pauli operators associated with local spin at site i i. The system undergos a QPT at the critical point h = 1 FIG.1: (Coloronline)ThevarianceofS of2-qubitreduceddensity differingXX-likephaseγ =0fromIsing-likephase0<γ ≤ matrixρij anditsderivativedS/dhversushwhenγ isfixedtobe 1. 0.6.ThecurvesarefordifferentspinsizesN Thetwo-spinreduceddensitymatrixatsitesiandj ofthe groundstateofthespinchainisoftheform, strengthhwhenγ = 0.6fordifferentvaluesofN. Fig.1(a) ρij = 41(I+hσiziσiz ⊗1+hσjzi1⊗σjz +hσizσjziσiz ⊗σjz sbheo0w.s6.thIetviasroiabnvcieouosfSthawtithiisncsrmeaasllienrgthhawnh0enfoγritshfiexreadngtoe S σXσY σX σY). (2) of h > 0.8, and this tells us that ρij exhibits EPR steering. X h i j i i ⊗ j Weplotd /dhversushinFig.1(b)fordifferentspinsizeN. X,Y=x,y S The nonanalyticpropertyof the EPR steerability function in Thenonzerocorrelations σz and σzσz fortheXYmodel the XY model is clearly shown at the critical point h = 1. h i,ji h i ji c aregivenby[20] Thus, ofthegroundstateisawitnessofQPT. S We next explore the scaling behavior of by the finite hσizi=−G0, size scaling approach [21] to further understaSnd the relation σzσz =G2 G G , (3) between EPR steering and QPT. In QPT, the critical feature h i ji 0− i−j j−i can be characterizedby a universalquantity whose behavior where at criticality is entirely described by a critical exponentν in theformofξ λ λ −ν. Tostudyquantumcriticalityin 1 M 2πk 2πk ∼ | − c| XY model, one needsto distinguish two universality classes G = ( cos )cos(r )/Λ r k M X1 − N N depending on the anisotropy parameter γ. For any value of γ, quantum critical behavior occurs at the transition point M γ 2πk 2πk + sin sin(r )/Λk, (4) hc = 1. Forγ = 0theXYmodelbelongstoXXuniversality M X1 N N classandcriticalexponentisν =1/2,andfor0<γ 1the ≤ modelbelongstoIsinguniversalityclassandcriticalexponent with Λ = (γsin2πk)2+cos2 2πk. We need to explore isν =1[22]. k q N N Letus considerthe derivativeof steerabilityfunctionwith quantumnonanalytic propertyin thermodynamiclimit when respecttohasafunctionhfordifferentspinsizeN asshown N ,thenthesumsintheexpectationvaluesarereplaced →∞ in Fig. 1 (b). For finite spin size, the curvesdo nothavedi- byintegrals, vergence,butshowobvioushumpsnearcriticalpointh =1. c 1 π WithincreasingspinsizeN,thepeakofeachcurvebecomes G = dφ( cosφ)cos(φr)/Λ r π Z − φ sharper. Eachcurveapproachestoitsmaximalvalueatpseu- 0 γ π docriticalpoint hm which changestowards the critical point +π Z dφsinφsin(φr)/Λφ, (5) hc when N increases. In the thermodynamic limit, when 0 N ,thesingularbehaviorofd /dhisclearinthevicin- →∞ S andΛ = (γsinφ)2+cos2φ. ityofthequantumcriticalpoint,anditcanbeanalyzedas, φ The veryprecent proposed criterion for EPR steering [19] d is obtainedfromthe constraintsontheeigenvaluesofpartial S κ1ln h hc +const, (7) dh ≈ | − | transposeof2-qubitdensityoperatorρij. Let λ1,λ2,λ3,λ4 befoureigenvaluesofρTijj listedinascending{order,theexpe}- wvehresuresκsp1in=si0z.e2l3n5N6. iWneFitgh.en2pwlohticthheshvoawlusetohfedreSla/tdiohna,thm riencedconditionforρ bearingEPRsteeringis ij d S =λ1+λ2−(λ1−λ2)2 <0. (6) dSh|hm ≈κ2lnN +const, (8) If the above eigenvalue combination is negative, the EPR with κ2 = 0.2355. According to the scaling ansatz in the S steeringofρAB canbecertified. Todemonstratetherelation case of logarithmic divergence [21], the ratio κ1/κ2 gives | | between EPR steering and QPTs, we plot EPR steerability the exponent ν of . By our numerical results, ν = 1 is S function of ρ and its derivative with respect to the field approximately obtained for the XY model when γ = 0.6. ij S 3 é é 1.5 é é ééééé é hm 1.0 é é h é ∆ é S(cid:144) é ∆ é Γ=0.6 0.5 éé é é (a) (b) é 0.0 FIG. 3: Quantum predictions of S10 and dS10/dh versus h when 2 3 4 5 6 7 8 γ =0.6inthermodynamiclimitwhenN →∞. lnN FIG. 2: The maximum value of dS/dhat the pseudocritical point inequalityfor EPRsteering. We expectmoreeffectiveEPR- hmversuslatticesizeslnN. steeringinequalitieswhichcanenabletodetectEPRsteering in QPT, and this will make it convenientto demonstrate ex- perimentallytheconnectionbetweenEPRsteeringandQPT. Therefore,ourresultsshowthattheEPRsteerabilityfunction of the ground state can signal the quantum criticality in the To summarize, we have investigated the relation between XYmodel. EPRsteeringandQPTintheanisotropicspin-1/2XYmodel The known EPR-steering inequalities can also be used to by using the 2-qubitEPR steering criterion. The EPR steer- demonstratethefactthatEPRsteeringisabletocapturequan- ability function shows the existence of the EPR steering tumcriticalityinXYmodel. HereweconsidertheN-setting of the ground staSte of the model. As the spin number goes EPR-steering inequalities proposed in Ref. [11] which is toinfinity,thesystemundergoesQPT betweenthespin-fluid based on the assumption that observer Alice’s measurement and the Ising-like phases, which can be captured by . By result is described by the random variable Ak = 1 (k = studyingthenonanalyticbehaviorof inthevicinityoSftran- 1,...N) and Bob’s kth measurement is defined by±Pauli ob- sition point h = 1, we find that Sis a universal quantity c servables ~σkB along some axis nˆk, and the two qubit EPR- todescribeQPTintheXYmodel,aSndthismakesitpossible steeringinequalityisoftheform to demonstrate experimentally the connection between EPR steeringandQPT.TheresultthatEPRsteerabilityfunctionis N = 1 A ~σB C , (9) able to signalQPT can be understoodas follows. The func- SN N Xh k k i≤ N tion is the combinationof eigenvaluesof partial transpose k=1 S ofρ whichchangesdramaticallyatthetransitionpoint,and ij where CN is the limit imposed by local hidden state theo- the informationof the critical changeis obviouslycontained reties.WhenN =2,C2 =1/√2;whenN =3,C3 =1/√3; in the eigenvalues of ρ or its partial transpose. Thus, the ij and when N = 10, C10 = 0.5236. It is obvious that the EPR steerability function can reflect the critical feature in more the number of measurement settings, the stronger the QPT. We believe that theSresult is applicable to other quan- two-qubit EPR-steering inequality is. We utilize 10-setting tummany-bodysystems.Wealsodiscusstherelationbetween EPR-steeringinequalitytoinvestigatetheEPRsteeringofXY EPRsteeringandQPTbyresortingto10-settingEPR-steering spinchainandplotquantumpredictionof 10 anditsderiva- inequality. Although the EPR-steering inequality is not vio- S tivewithrespecttohinthermodynamiclimitwhenN latednearthecriticalpoint,quantumpredictionofleft-hand- → ∞ in Fig. 3. From Fig. 3 (a), we find that for some values side of the inequality still exhibits singular behavior. This ofh,thequantumpredictionsdonotexceedC10 andthe10- suggeststwoparticularpoints:(1)Quantumpredictionofleft- settingEPR-steeringinequalitycannotidentifytheEPRsteer- hand-sideofEPR-steeringinequalitiesplaysainterestingrole ingofthegroundstateinthevicinityofcriticalpoint.Evenif to capture QPT no matter whether the inequalities are vio- noviolationisfound,thederivativeofquantumpredictionof lated or not, just like the case for Bell’s inequalities in QPT 10 stillexhibitssingularpropertyatthe criticalpointhc, as [18];(2)Thepresenttwo-qubitEPR-steeringinequalitiesare S showninFig. 3(b).Inaword, 10isabletosignalthenonan- not strong enough to detect EPR steering in the XY model. S alyticfeaturesintheXYspinchainwhenquantumprediction OurresultsfromEPRsteeringcriterionshowthatthereduced of 10issmallerthanC10,ornoEPRsteeringidentifiedbythe densitymatrixofthegroundstateofXYspinchainbearsEPR S inequality. TheresultissimilartothatforBell’sinequalityin steeringanditindeedsignalstheQPT.Weexpectmoreeffec- QPT[18]thatBellfunctionvaluecancaptureQPTsalthough tiveEPR-steeringinequalitieswhichcanenabletodetectEPR Bell’sinequalityisnotviolated.Ontheotherhand,according steeringinQPT. totheEPRsteeringcriterion ,itisfoundthatthedensityma- S trix ofthe groundstate hasEPRsteering. Thissuggeststhat J.L.C. is supported by National Basic Research Program the10-settingEPR-steeringinequalitiesarenotstrongenough (973Program)ofChinaunderGrantNo. 2012CB921900and todetectEPRsteeringintheXYmodelandsoitisnotatight NSF of China (Grant Nos. 10975075 and 11175089). This 4 work is also partly supported by National Research Founda- [9] Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. tionandMinistryofEducation,Singapore(GrantNo. WBS: Rev.Lett.68, 3663(1992); J.C.Howell, R.S.Bennink, S.J. R-710-000-008-271). Bentley,andR.W.Boyd,Phys.Rev.Lett.92,210403(2004). 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