Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering

Transcription

1 PHYSICAL REVIEW A 76, Entanglement, Einstein-Poolsky-Rosen correlations, Bell nonlocality, an steering S. J. Jones, H. M. Wiseman, an A. C. Doherty 2 Centre for Quantum Computer Technology, Centre for Quantum Dynamics, Griffith University, Brisbane, Ql. 4, Australia 2 School of Physical Sciences, University of Queenslan, Brisbane Ql. 4072, Australia Receive 4 September 2007; publishe 28 November 2007 In a recent work Phys. Rev. Lett. 98, we efine steering, a type of quantum nonlocality that is logically istinct from both nonseparability an Bell nonlocality. In the bipartite setting, it hinges on the question of whether Alice can affect Bob s state at a istance through her choice of measurement. More precisely an operationally, it hinges on the question of whether Alice, with classical communication, can convince Bob that they share an entangle state uner the circumstances that Bob trusts nothing that Alice says. We argue that if she can, then this emonstrates the nonlocal effect first ientifie in the famous Einstein- Poolsky-Rosen paper Phys. Rev. 47, as a universal effect for pure entangle states. This ability of Alice to remotely prepare Bob s state was subsequently calle steering by Schröinger, whose terminology we aopt. The phenomenon of steering has been largely overlooke, an prior to our work ha not even been given a rigorous efinition that is applicable to mixe states as well as pure states. Arme with our rigorous efinition, we prove that steerable states are a strict subset of the entangle states, an a strict superset of the states that can exhibit Bell nonlocality. In this work we expan on these results an provie further examples of steerable states. We also elaborate on the connection with the original EPR paraox. DOI: 0.03/PhysRevA PACS numbers: U, Mn I. INTRODUCTION Entanglement is arguably the central concept in the fiel of quantum information. However, there is an unresolve tension between ifferent notions of what entanglement is, even in the bipartite setting. On the one han, entangle states are efine as those that cannot be create from factorizable states using local operations an classical communication LOCC. On the other han, entanglement is regare as a resource that enables the two parties to perform interesting or in more recent times useful nonlocal protocols. For pure states, which were the only states consiere in this context for many ecaes, these notions coincie, an the wor entangle introuce by Schröinger is ientical with factorizable. The first authors to ientify an interesting nonlocal effect associate with unfactorizable states were Einstein, Poolsky, an Rosen EPR in They consiere a general unfactorizable pure state of two systems, hel by two istant parties say Alice an Bob, = c n u n n = n v n n. n= n=. All we have change from EPR s presentaton is to use Dirac s notation rather than wave functions Here u n an v n are two ifferent orthonormal bases for Alice s system. If such states exist, then if Alice chose to measure in the u n respectively, v n basis, then she woul instantaneously collapse Bob s system into one of the states n respectively, n. That is, as a consequence of two ifferent measurements performe upon the first system, the secon system may be left in states with two ifferent wave functions 2. Now comes the paraox: the two systems no longer interact, so no real change can take place in Bob s system in consequence of anything that may be one to Alice s system 2. That is, if Bob s quantum state is the real state of his system, then Alice cannot choose to make it collapse into either one of the n or one of the n because that woul violate local causality. Note that it is crucial to consier more than one sort of measurement for Alice; if Alice were restricte to measuring in one basis say the u n basis, then it woul be impossible to emonstrate any real change in Bob s system, because she might know beforehan which of the n is the real state of his system. That is, the paraox exists only if there is not a local hien state LHS moel for Bob s system, in which the real state n is hien from Bob but may be known to Alice. As the above quotations show, EPR assume local causality to be a true feature of the worl; inee, they say that no reasonable theory coul be expecte to permit otherwise. They thus conclue that the wave function cannot escribe reality; that is, the quantum mechanical QM escription must be incomplete. Their intuition was thus that local causality coul be maintaine by completing QM. This intuition was supporte by the famous example that they then presente as a special case of Eq.., involving a bipartite entangle state with perfect correlations in position an momentum. The EPR paraox in this case is trivially resolve by consiering local hien variables LHVs for position an momentum. Although the argument of EPR against the completeness of QM was correct, their intuition was not. As proven by Bell 3,4, local causality cannot be maintaine even if one allows QM to be complete by hien variables. That is, assuming as always an with goo justification 5 that QM is correct, Bell s theorem proves that local causality is not a true /2007/765/ The American Physical Society

2 JONES, WISEMAN, AND DOHERTY feature of the worl. 2 Interestingly, any unfactorizable pure state can be use not only to emonstrate the EPR paraox 2, but also to emonstrate Bell nonlocality that is, the violation of local causality. This fact was perhaps first state in 989 by Werner 6; the first etaile proof was given in 99 by Gisin 7; see also 8. With the rise of quantum information experiment, the iealization of consiering only pure states has become untenable. The question of which mixe states were Bell nonlocal that is, allowe a emonstration of Bell nonlocality was first aresse by Werner 6, in a founational paper pregnant with implications for, an applications in, quantum information science. Revealing the first hint of the complexity of mixe-state entanglement, still being uncovere 9, Werner showe that not all mixe entangle states can emonstrate Bell nonlocality. Here, for mixtures, an entangle state is efine as one which cannot be written as a mixture of factorizable pure states. Inee Werner s paper is often cite as that which introuce this efinition. That is, he is creite with introucing the ichotomy of entangle states versus separable i.e., locally preparable states. However, it is interesting to note that he use neither the term entangle nor the term separable. For a iscussion of the history of terms use in this context, an their relation to the present work, see Appenix A. In a recent paper, the present authors also consiere the issue of mixe states an nonlocality 0. We rigorously efine the class of states that can be use to emonstrate the nonlocal effect which EPR ientifie in 935. We propose the term steerable for this class of states for reasons given in Appenix A, an prove that the set of Bell-nonlocal states is a strict subset of the set of steerable states, which in turn is a strict subset of the set of nonseparable states. This was our main result. Like entangle, steering is a term introuce by Schröinger in the aftermath of the EPR paper. Specifically, he creits EPR with calling attention to the obvious but very isconcerting fact that for a pure entangle state like Eq.., Bob s system can be steere or pilote into one or the other type of state at Alice s mercy in spite of her having no access to it. He referre to this as a paraox, because if such states can exist, an if the QM escription is complete, then local causality must be violate. In Ref. 0 we first supplie an operational efinition of steering in the style of a quantum informational task involving two parties in contrast to emonstrating Bell nonlocality, which can be efine as a task involving three parties. Next we turne this operational efinition into a mathematical efinition. Applying this to the case of 22 imensional Werner states enable us to establish our main result, quote 2 For both Bell an EPR, there is an escape, by enying inepenent real situations as such to things which are spatially separate from each other, as state by Einstein in That is, Alice, for example, can refuse to amit the reality of Bob s measurement results until she observes them, by talking only about the outcomes of her own future measurements. However, Einstein state that in his opinion this antirealism was equally unacceptable as violating local causality; see Ref. 42 for a iscussion. PHYSICAL REVIEW A 76, above. We then completely characterize steerability for -imensional Werner states an isotropic states. Finally, we completely characterize the Gaussian states that are steerable by Gaussian measurements, an relate this to the Rei criterion 2 for the EPR paraox. In the present paper we expan an exten the material in Ref. 0. In Sec. II we present the operational efinitions of Bell nonlocality an steering as before, an also that for emonstrating nonseparability. In aition we use these operational efinitions to show that they lea to a hierarchy of states: Bell nonlocal within steerable within nonseparable. In Sec. III we turn our operational efinitions into mathematical efinitions, an in aition we explain how our efinition of steering conforms to Schröinger s use of the term. In Sec. IV we erive conitions for steerability for four families of states. As before, we consier Werner states, isotropic states, an Gaussian states, but here we expan the proofs for the benefit of the reaer. In aition, we consier another class of states: the inept states of Ref. 3. We also consier a subclass of Gaussian states in more etail: the symmetric two-moe states prouce in parametric own conversion. We conclue with a summary an iscussion in Sec. V. II. OPERATIONAL DEFINITIONS It is useful to begin with some operational efinitions for the ifferent properties of quantum states that we wish to consier. This is useful for a number of reasons. First, it presents the ieas that we wish to iscuss in an accessible format for those familiar with concepts in moern quantum information. Secon, it allows us to present an elementary proof of the hierarchy of the concepts we will present a more etaile proof of this hierarchy in subsequent sections. First, let us efine the familiar concept of Bell nonlocality 3 as a task, in this case with three parties; Alice, Bob, an Charlie. Alice an Bob can prepare a share bipartite state, an repeat this any number of times. Each time, they measure their respective parts. Except for the preparation step, communication between them is forbien this prevents them from colluing in an attempt to fool Charlie. Their task is to convince Charlie with whom they can communicate that the state they can prepare is entangle. Charlie accepts QM as correct, but trusts neither Alice nor Bob. If the correlations between the results they report can be explaine by a LHV moel, then Charlie will not be convince that the state is entangle; the results coul have been fabricate from share classical ranomness. Conversely, if the correlations cannot be so explaine then the state must be entangle. Therefore they will succee in their task iff if an only if they can emonstrate Bell nonlocality. This task can be thus consiere as an operational efinition of violating a Bell inequality. The analogous efinition for steering uses a task with only two parties. Alice can prepare a bipartite quantum state an sen one part to Bob, an repeat this any number of times. Each time, they measure their respective parts, an communicate classically. Alice s task is to convince Bob that the state she can prepare is entangle. Bob like Schröinger accepts that QM escribes the results of the measurements he

3 ENTANGLEMENT, EINSTEIN-PODOLSKY-ROSEN PHYSICAL REVIEW A 76, makes which, we assume, allow him to o local state tomography. However, Bob oes not trust Alice. In this case Bob must etermine whether the correlations between his local state an Alice s reporte results are proof of entanglement. How he shoul etermine this is explaine in etail in Sec. III, but the basic iea is that he shoul not accept the correlations as proof of entanglement if they can be explaine by a LHS moel for Bob. If the correlations between Bob s measurement results an the results Alice reports can be so explaine then Alice s results coul have been fabricate from her knowlege of Bob s LHS in each run. Conversely, if the correlations cannot be so explaine then the bipartite state must be entangle. Therefore we say that Alice will succee in her task iff she can steer Bob s state. Finally, the simplest task is for Alice an Bob to etermine iff a bipartite quantum state that they share is nonseparable. In this case they can communicate results to one another, they trust each other, an they can repeat the experiment sufficiently many times to perform state tomography. By analyzing the reconstructe bipartite state, they coul etermine whether it is nonseparable. That is, whether it can be escribe by correlate LHSs for Alice an Bob. Because Alice an Bob trust each other an can freely communicate, this is really a one party task. Using these operational efinitions we can show that Bell nonlocality is a stronger concept than steerability. That is, that Bell-nonlocal states are a subset of the steerable states. The operational efinition of Bell nonlocality is base on three parties an requires a completely istrustful Charlie. If we weaken this conition by allowing Charlie to trust Bob completely, we arrive at the following situation. Charlie can now, in principle, o state tomography for Bob s local state as he believes everything tol to him by Bob, an he only istrusts the measurement results reporte by Alice. In this case, he will only concee that the state prepare by Alice an Bob is entangle if the state is steerable. Thus it is possible to arrive at the operational efinition for steering by weakening the operational efinition for Bell nonlocality. Thus, the Bell-nonlocal states are a subset of the steerable states. Similarly, if we weaken the conition for steerability we arrive at the conition for nonseparability as follows. In this case we weaken the conition by allowing for Bob to trust Alice completely. Since Bob now has access to the measurement information for both subsystems as he believes everything tol to him by Alice he can, in principle, perform state tomography. Clearly, in this situation Bob will only concee that they share an entangle state if the state that Alice prepares really is entangle. Thus, the steerable states are a subset of the entangle states. We illustrate these relations graphically in Fig.. While these operational efinitions give a goo insight into the relationships between the three classes of states it is also esirable to have a strict mathematical way to efine the classes. We present such efinitions in the following section. III. MATHEMATICAL DEFINITIONS First, we efine some terms. Let the set of all observables on the Hilbert space for Alice s system be enote D.We FIG.. Color online Operational efinitions for classes of entangle states. Bell-nonlocal states a can be efine via a threeparty task involving Alice A, Bob B, an Charlie C. Steerable states b may be efine using a two-party task. Defining an entangle state c essentially requires only one party. In all cases shaing inicates the skeptical party, otte arrows inicate twoway communication, an soli arrows inicate trust an two-way communication. enote an element of D by Â, an the set of eigenvalues a of  by Â. ByPaÂ;W we mean the probability that Alice will obtain the result a when she measures  on a system with state matrix W. We enote the measurements that Alice is able to perform by the set M D. Note that, following Werner 6, we are restricting to projective measurements. The corresponing notations for Bob, an for Alice an Bob jointly, are obvious. Thus, for example, Pa,bÂ,Bˆ ;W =Trˆ a A ˆ b B W, 3. where ˆ a A is the projector satisfying ˆ a A =aˆ a A. The strongest sort of nonlocality in QM is Bell nonlocality 3. This is a property of entangle states which violate a Bell inequality. This is exhibite in an experiment on state W if the correlations between a an b cannot be explaine by a LHV moel. That is, if it is not the case that for all a Â, bbˆ, for all ÂM, Bˆ M, we have Pa,bÂ,Bˆ ;W = aâ,bbˆ,. 3.2 Here, an below, aâ,, bbˆ,, an enote some positive, normalize probability istributions, involving the LHV. We say that a state is Bell nonlocal if there exists a measurement set M M that allows Bell nonlocality to be emonstrate. If Eq. 3.2 is always satisfie we say W is Bell local. A strictly weaker 6 concept is that of nonseparability or entanglement. A nonseparable state is one that cannot be written as

4 JONES, WISEMAN, AND DOHERTY W =. 3.3 Here, an below, D an D are some positive, normalize quantum states. We can also give an operational efinition, by allowing Alice an Bob the ability to measure a quorum of local observables, so that they can reconstruct the state W by tomography 4. Since the complete set of observables D is obviously a quorum, we can say that a state W is nonseparable if it is not the case that for all aâ, bbˆ, for all ÂD, Bˆ D, we have Pa,bÂ,Bˆ ;W = PaÂ; PbBˆ ;. 3.4 Bell nonlocality an nonseparability are both concepts that are symmetric between Alice an Bob. However steering, Schröinger s term for the EPR effect, is inherently asymmetric. It is about whether Alice, by her choice of measurement Â, can collapse Bob s system into ifferent types of states in the ifferent ensembles E A aa :aâ. Here aa Tr Wˆ a A ID is Bob s state conitione on Alice measuring  with result a. The tile enotes that this state is unnormalize its norm is the probability of its realization. Of course Alice cannot affect Bob s unconitione state =Tr W= a aa that woul allow superluminal signaling. Despite this, steering is clearly nonlocal if one believes that the state of a quantum system is a physical property of the system, as i Schröinger. This is apparent from his statement that It is rather iscomforting that the theory shoul allow a system to be steere or pilote into one or the other type of state at the experimenter s mercy in spite of his having no access to it. As this quote also shows, Schröinger was not wee to the terminology steering. He also use the term control for this phenomenon, an the wor riving in the context of his 936 result that a sophisticate experimenter can prouce a non-vanishing probability of riving the system into any state he chooses. By this he means that if a bipartite system is in a pure entangle state, then one party Alice can, by making a suitable measurement on her subsystem, create any pure quantum state for Bob s subsystem with probability, whenever this is well efine. He regare steering or riving as a necessary an inispensable feature of quantum mechanics, but foun it repugnant, an oubte whether it was really true. That is, he was not satisfie about there being enough experimental evience for its existence in Nature. What experimental evience woul have convince Schröinger? The pure entangle states he iscusse are an iealization, so we cannot expect ever to observe precisely the phenomenon he introuce. On the other han, Schröinger was quite explicit that a separable but correlate state, which allows etermining the state of the first system by suitable his emphasis measurement of the secon or vice versa coul never exhibit steering. Of this situation, he says that it woul utterly eliminate the experimenter s influence on the state of that system which he oes not touch. Thus it PHYSICAL REVIEW A 76, is apparent that by steering Schröinger meant something that coul not be explaine by Alice simply fining out which state Bob s system is in, out of some preefine ensemble of states. In other wors, the experimental evience Schröinger sought is precisely the evience that woul convince Bob that Alice has prepare an entangle state uner the conitions escribe in our first operational efinition of steering. To reiterate, we assume that the experiment can be repeate at will, an that Bob can o state tomography. Prior to all experiments, Bob emans that Alice announce the possible ensembles E A :ÂM she can steer Bob s state into. In any given run after he has receive his state, Bob shoul ranomly pick an ensemble E A, an ask Alice to prepare it. 3 Alice shoul then o so, by measuring  on her system, an announce to Bob the particular member A a she has prepare. Over many runs, Bob can verify that each state announce is inee prouce, an is announce with the correct frequency Tr a A. If Bob s system i have a preexisting LHS as Schröinger thought, then Alice coul attempt to fool Bob, using her knowlege of. This state woul be rawn at ranom from some prior ensemble of LHSs F= with =. Alice woul then have to announce a LHS aa base on her knowlege of, accoring to some stochastic map from to a. Alice will have faile to convince Bob that she can steer his system if, for all ÂM, an for all a Â, there exists an ensemble F an a stochastic map aâ, from to a such that aa = aâ,. 3.5 That is, if there exists a coarse-graining of ensemble F to ensemble E A then Alice may simply know Bob s preexisting state. Conversely, if Bob cannot fin any ensemble F an map aâ, satisfying Eq. 3.5 then Bob must amit that Alice can steer his system. We can recast this efinition as a hybri of Eqs. 3.2 an 3.4: Alice s measurement strategy M on state W exhibits steering if it is not the case that for all aâ,b Bˆ, for all ÂM, Bˆ D, we can write Pa,bÂ,Bˆ ;W = aâ,pbbˆ ;. 3.6 That is, if the joint probabilities for Alice an Bob s measurements can be explaine using a LHS moel for Bob an a LHV moel for Alice correlate with this state, then we have faile to emonstrate steering. Iff there exists a measurement strategy M that exhibits steering, we say that the state W is steerable by Alice. 3 This ensures that Bob nee not trust Alice that they share the same state W in each run, because Alice gains nothing by preparing ifferent states in ifferent runs, because she never knows what ensemble Bob is going to ask for

5 ENTANGLEMENT, EINSTEIN-PODOLSKY-ROSEN It is straightforwar to see that the conition for no steering implies the conition for Bell locality, since if there is a moel with PbBˆ, satisfying Eq. 3.6, then there is a moel with bbˆ, that satisfies Eq. 3.2; simply make bbˆ,= PbBˆ ; for all Bˆ,. Since no steering implies no Bell nonlocality, we see that if a state is Bell nonlocal, then it implies that it is also steerable. Hence Bell nonlocality is a stronger concept than steerability. Similarly, the conition for separability implies the conition for no steering. If there is a moel with PaÂ; satisfying Eq. 3.4, then there is a moel with aâ, that satisfies Eq. 3.6; simply make aâ,= PaÂ; for all Â,. Thus, steerability is also a stronger concept than nonseparability. At least one of these relations must be strictly stronger than, because Bell nonlocality is strictly stronger than nonseparability 6. In the following sections we prove that in fact steerability is strictly stronger than nonseparability, an strictly weaker than Bell nonlocality. IV. CONDITIONS FOR STEERABILITY Below we erive conitions for steerability for four families of states W. In each example we parametrize the family of states in terms of a mixing parameter R, an a secon parameter that may be iscrete. In each case, the upper boun for W to be a state is =, an W is a prouct state if =0, an except in the last case W is linear in. For the first two examples Werner an isotropic states the conitions erive are both necessary an sufficient for steerability. For the other examples inept states an Gaussian states the conitions erive are merely sufficient for steerability. In terms of the parameter we can efine bounaries between ifferent classes of states. For example, we will make use of Bell, efine by W being Bell nonlocal iff Bell. Similarly a state W is entangle iff ent. Our goal is then to etermine or at least boun the steerability bounaries for the above classes of states, efine by W being steerable iff steer. Crucial to the erivations of the conitions for steerability of these states is the concept of an optimal ensemble F = ; that is, an ensemble such that iff it cannot satisfy Eq. 3.5 then no ensemble can satisfy it. In fining an optimal ensemble F we use the symmetries of W an M : Lemma. Consier a group G with a unitary representation Û g=û g Û g on the Hilbert space for Alice an Bob. Say that ÂM, aâ, gg, we have Û gâû gm an Û gâû a g = Û g aa Û g. 4. Then there exists a G-covariant optimal ensemble: g G, =Û g Û g. Proof. For specificity, consier a iscrete group with orer G. Say there exists an ensemble F= satisfying Eq. 3.5 for some map aâ,. Then uner the conitions of Lemma, aa can be rewritten as G gg PHYSICAL REVIEW A 76, Û g Û g aû gâû g,. Thus the G-covariant ensemble F = g, g,, with g, =Û g Û g an g,= /G, satisfies Eq. 3.5 with the choice aâ,g, = aû gâû g,. 4.2 The analogous formulas for the case of continuous groups are elementary. Once we have etermine the optimal ensemble for a given class of states an a given measurement strategy it remains to etermine if there exists a stochastic map aâ, such that Eq. 3.5 is true. In each steering experiment we assume that Alice really oes sen Bob an entangle state. To etermine if the state is steerable, we take the perspective of a skeptical Bob an imagine that in each case Alice is attempting to cheat; that is, that she sens Bob a ranom state from the optimal ensemble F an oes not perform her measurements. She simply announces her allege measurement results base on aâ, which efines her cheating strategy. We compare the states that Bob woul obtain if Alice really i sen half of an entangle state an perform a measurement with those that coul be prepare using an optimal ensemble an cheating strategy. There are two possible reasons why Bob coul fin that his measurement results are consistent with results reporte by Alice. First, Alice coul really be sening Bob half of an entangle state an steering his system via her measurements. Or, as the skeptical Bob believes, Alice coul really just be sening him ifferent pure states in each run an announcing her results base on her knowlege of this state. Now if the optimal ensemble which we are assuming Bob is clever enough to etermine can explain the correlations between Alice s announce results an Bob s results then the state sent by Alice is not steerable. However, if the best cheating strategy that Alice coul possibly use is insufficient to explain the correlations then Bob must amit that Alice has sent him part of an entangle state. Furthermore, if he makes this amission, the state must be steerable. A. Werner states This family of states in C C was introuce by Werner in Ref. 6. As mentione above, we parametrize it by R such that W is linear in, it is a prouct state for =0, an is a state at all only for. W = + I 2 V. 4.3 Here I is the ientity an V is the flip operator efine by V. Defining = +/ allows one to reprouce Werner s notation 6 for these states. Werner states are nonseparable iff ent =/+6. For = 2, the Werner states violate the Clauser-Horne-Shimony- Holt CHSH inequality iff / 2 5. This places an

6 JONES, WISEMAN, AND DOHERTY upper boun on Bell. For 2 only the trivial upper boun 4 of is known. However, Werner foun a lower boun on Bell of / 6, which is strictly greater than ent. Now let us consier the possibility of steering Werner states. We allow Alice all possible measurement strategies: M =D, an without loss of generality take the projectors to be rank one: ˆ a A =aa. For Werner states, the conitions of Lemma are then satisfie for the -imensional unitary group U. Specifically, g Û, an Û g Û Û 6. Again without loss of generality we can take the optimal ensemble to consist of pure states, in which case there is a unique covariant optimal ensemble, F = Haar, where Haar is the Haar measure over U. If Alice were to make any projective measurement of her half of a Werner state an obtain the result a, Bob s unnormalize conitione state woul be given by aa =Tr A ˆ a A IW = aw a = + I aa. 4.4 This is a state proportional to the completely mixe state minus a term proportional to the state Alice s system is projecte into by her measurement. We now etermine if it is possible for Alice to simulate this conitione state using the optimal ensemble F an an optimal cheating strategy efine by aâ,. That is, we imagine that in each run of the experiment Alice simply sens Bob a state = rawn at ranom from F = Haar. When aske to perform a measurement  an announce her result, she uses aâ, which is base on her knowlege of = to etermine her answer. In testing whether this is actually what Alice coul be oing, we only nee to consier the quantity a aa a = This is ue to the form of aa note above in Eq If on average the strategy use by Alice with the ensemble F prouces the correct overlap with the state aa then Eq. 3.5 will hol an steering is not possible. Thus Alice makes use of the overlap with aa of the ranom states in etermining the optimal aâ,. Since Alice s goal is to simulate aa, as efine in Eq. 4.4, she will etermine which of the eigenstates of  has the least overlap with in each run of the experiment an announce the eigenvalue associate with that eigenstate as her result. On average Bob woul then fin that his conitione state has the least possible overlap with aa. Writing this explicitly, the optimal istribution is given by 4 This is because no Bell inequality has been foun that the Werner states violate for 2. It is only an upper boun because this is not a test of all possible Bell inequalities. PHYSICAL REVIEW A 76, if ˆ a aâ, = A ˆ a A a a 0 otherwise. 4.6 It is straightforwar to see that this ensemble is normalize, that is, Â,, aâ, =. 4.7 a Clearly the optimal istribution aa, is the istribution that will preict the same overlap with aa as that given by Eq This occurs at precisely the steering bounary steer. When steer steering cannot be emonstrate, as it is possible that Alice is using a cheating strategy to simulate Bob s conitione state. This means that Alice s optimal cheating strategy coul actually make Bob believe that his conitione state has a smaller overlap with aa than woul be expecte from Eq In this case Alice coul correctly simulate aa simply by introucing the appropriate amount of ranomness to her responses i.e., increase the overlap to the correct size by choosing a ifferent aâ,. To reiterate, when steer it is possible that Alice is performing a classical strategy which is consistent with Bob s results, so he will not believe that the state is genuinely steerable. To fin the form of steer we compare with Werner s result 6 for the lower boun on ent. We fin that he actually use the construction outline above. His LHVs for Bob s system were in fact the LHSs use in the optimal ensemble F. Werner shows that for any positive normalize istribution aâ,, a Haar aâ,a / The equality is attaine for the optimal aâ, specifie by Eq. 4.6 this prouces the smallest possible preicte overlap with aa. Now to etermine when Eq. 3.5 is satisfie by F an thus to etermine steer we simply compare Eq. 4.8 with Eq We fin that Alice cannot simulate the correct overlap with aa iff / 2 / 3. Hence we see that for Werner states steer = Recently a new lower boun for Bell was foun for =2 by Acìn et al. 6, greater than steer, as shown in Fig. 2. Reference 6 makes use of a connection with Grothenieck s constant a mathematical constant from Banach space theory to evelop a local hien variable moel for projective measurements when =2. Acìn et al. show that for twoqubit Werner states

7 ENTANGLEMENT, EINSTEIN-PODOLSKY-ROSEN PHYSICAL REVIEW A 76, η η (a) // (b) // (c) B. Isotropic states The isotropic states, which were introuce in 7, can be parametrize ientically to the Werner states; that is, in terms of their imension an a mixing parameter, W = I/ 2 + P Here P + = + +, where + = i= ii/ is a maximally entangle state. In fact, for =2 it is straightforwar to verify that the isotropic states are ientical to Werner states up to local unitaries. Isotropic states are nonseparable iff ent =/+7. A nontrivial upper boun on Bell for all is known; in Ref. 8 it is shown that a Bell inequality is certainly violate by a -imensional isotropic state if 2 I QM Bell, where I QM is efine as 4.3 η η ε n FIG. 2. Color online Bounaries between classes of entangle states for Werner a an isotropic b states W, inept states W c, an two-moe symmetric Gaussian states W n. The bottom blue line is ent, above which states are entangle. The next re line is steer, above which states are steerable. In cases c an the own arrows inicate that we have only an upper boun on steer. The top green line with own arrows is an upper boun on Bell, above which states are Bell nonlocal. The up arrows in cases a an b are lower bouns on Bell for =2. This lower boun establishes that the classes are strictly istinct. In cases a an b, ots join values at finite with those at =. The separate point in c is explaine at the en of Sec. IV C / 2 Bell /K g , 4. where K g is Grothenieck s constant of orer 3. Bouns on K g 3 ensure that for =2 Werner states Bell Using Eq. 4.0, we see that when =2, steer =/ 2. This proves that steerability is strictly weaker than Bell nonlocality as steer Bell. It is also well known that for =2, ent =/3, which is strictly less than steer. Thus using the =2 Werner states as an example we also see that steerability is strictly stronger than nonseparability. This clear istinction between the three classes can be seen on the left-han axis of Fig. 2a. () I QM =4 /2 2k k q k+, 4.4 k=0 q an q k =/2 3 sin 2 k+/4/. Collins et al. 8 go on to show that in the limit as the limiting value this upper boun on Bell approaches 2 /6Catalan , where Catalan0.959 is Catalan s constant. In etermining steerability we again allow Alice all possible measurement strategies: M =D, an take the projectors to be rank one: ˆ a A =aa. The isotropic states have the symmetry property that they are invariant uner transformations of the form Û * Û, hence the conitions of Lemma are again satisfie for the -imensional unitary group U. In this case, g Û an Û g Û * Û. Thus we can again take the optimal ensemble to be F = Haar. Now consier the conitione state that Bob woul obtain if Alice were to make a measurement  on her half of W, aa =Tr A ˆ a A IW = I + aa. 4.5 This is a state proportional to the completely mixe state plus a term proportional to the state Alice s system woul be projecte into by her measurement. Note the similarity with the Werner state example, where the conitione state was proportional to the completely mixe state minus a term proportional to aa. This ifference arises because the isotropic states are symmetrically correlate rather than antisymmetrically correlate as in the Werner state example. Again we wish to etermine if it is possible for Alice to simulate the conitione state aa using the optimal ensemble F an a cheating strategy efine by an optimal istribution aâ,. Imagine that in each run of a steering experiment Alice simply sens Bob a state rawn at ranom from F = G,m. When aske to perform a measurement  an announce her result, she uses aâ, to etermine her answer. In testing whether this is actually what Alice

8 JONES, WISEMAN, AND DOHERTY coul be oing, we again only nee to consier the quantity a aa a = In this case Alice s strategy is similar to the Werner state example, except now she wants to simulate the maximum possible overlap with aa ue to the form of aa. Therefore, Bob will only concee that W is steerable if the maximum overlap with aa preicte using the ensemble F an the optimal cheating strategy aa, is less than that preicte by Eq In this case there woul be no possible classical strategy that Alice coul possibly be using to simulate the correlations with Bob s results. Ientical preictions for the overlap with aa will again occur precisely at the steering bounary steer, which occurs when aâ, is use. The optimal aâ, is efine in a similar manner to the Werner state example. However, in each run of the experiment Alice now etermines which of the eigenstates of  is closest to an announces the eigenvalue associate with that eigenstate as her result. That is, if ˆ a aâ, = A ˆ a A a a 0 otherwise. 4.7 To test if Eq. 3.5 hols, Alice an Bob woul nee to run the experiment many times an compare a aa a with the quantity a Haar aa,a. 4.8 This can be written as Haar a Haar a aa =a 2, 4.9 where the subscript a on the integral means that in the integral only those states with a greater than all others will contribute. As shown in Appenix B, a ranom state from the ensemble F can be escribe by the unnormalize state = m = j= z j j, 4.20 where the z j are mutually inepenent complex Gaussian ranom variables with zero mean an zero secon moments except for z j * zk = jk. That is, we can replace the Haar measure Haar by G,m= G m Haar. In terms of the variables z j, this can be expresse as PHYSICAL REVIEW A 76, Now using the Gaussian measure G,m to escribe the ensemble F, we can rewrite Eq. 4.9 as G mm 2 Haar a a 2 Haar a =a 2 G mm 2 G,ma 2 =a G mm 2 It is straightforwar to show that the enominator equals one see Appenix B 2, an hence we can evaluate the numerator left to Appenix B 3 to fin that G,ma a 2 = H 2, 4.23 where H =+/2+/3+ +/ is the harmonic series. Thus we fin that for any positive normalize istribution aâ, we must have a Haar aâ,a H 2, 4.24 with the equality obtaine for the optimal aâ, as efine in Eq Comparing this with Eq. 4.6 we see that steering can be emonstrate iff + 2 H 2. Thus for isotropic states steer = H large 4.25 ln For =2 the isotropic states are equivalent up to local unitaries to the Werner states, an we again fin that steer =/2, which is strictly less than Bell an strictly greater than ent. For 2, steer is greater than ent an significantly less than an upper boun on Bell. This is shown in Fig. 2b. For large we see that both steer an ent ten to zero, however, steer approaches zero more slowly; it is larger than ent by a factor of ln 9. C. Inept states We now consier a family of states with less symmetry than the previous examples. This makes the analysis more ifficult, meaning that we cannot fin steer exactly. However, making use of the symmetry properties of the states allows us to fin an upper boun on steer. We efine a family of two-qubit states by G,m exp z i 2 2 z 2 z. i= 4.2 where W = +,

9 ENTANGLEMENT, EINSTEIN-PODOLSKY-ROSEN = 0 0 +, 4.28 an the reuce states are foun by partial tracing with respect to Bob Alice. That is, =Tr As in the previous examples, this is a two-parameter family of states; the parameter is again a mixing parameter, an the parameter etermines how much entanglement is present in the state. Note that when =/2 these states are equivalent to the two-imensional Werner an isotropic states. This family of states was stuie in Ref. 3 in the context of istributing entanglement. The authors consiere an inept company attempting to istribute pure entangle states to many pairs of parties. However, they mixe up the aresses some fraction of the time, meaning that on average the company woul actually istribute mixe entangle states of the form of Eq Hence we will refer to this family of states as inept states. As note above, the inept states are a family of two-qubit states, which means that it is possible to evaluate ent analytically. This was one in Ref. 3 leaing to the following conition for nonseparability of inept states: ent = Reference 3 also consiers Bell nonlocality of the state matrix W by testing if a violation of the CHSH inequality 20 occurs. This was one using the metho of 5 for etermining the optimal violation of the CHSH inequality for two-qubit states. One fins that the state W violates the CHSH inequality if an only if Bell Now, in orer to emonstrate steering we must specify a measurement strategy. In the two previous examples we have use the complete set of projective measurements, M =D. This woul be a suitable measurement strategy to allow us to efine an optimal ensemble, however, in orer to make our task simpler we will consier a more restricte set of measurements. We note that states efine by Eq have the symmetry property that they are invariant uner simultaneous contrary rotations about the z axes. This immeiately suggests a restricte measurement scheme; we allow all measurements in the xy plane but only allow a single measurement along the z axis. That is, Alice s measurement scheme is given by M =ˆ zˆ :0,2, where ˆ = ˆ x cos + ˆ y sin In this case the conitions for Lemma are satisfie for the Lie group G generate by /2ˆ z I /2I ˆ z see Appenix C. This is a more restricte scheme than we have consiere so far, but will be sufficient to emonstrate steerability if Eq. 3.5 oes not hol since it must hol for all measurements to preclue steering. Thus we are only consiering an upper PHYSICAL REVIEW A 76, boun on steer the bounary between steerable an nonsteerable states using all projective measurements. We now consier the optimal ensemble for this restricte set of measurements. We use an ensemble of pure states F =, where = 2 I + z 2 cosˆ x + z 2 sinˆ y zˆ z, 4.33 an =/2zz. It is straightforwar to show that this ensemble is of the form of the optimal ensemble since the conitions for Lemma hol see Appenix C. While this ensemble has the form of the optimal, it is not completely specifie as z is still general. Thus to fin the optimal ensemble we nee to etermine the optimal probability istribution z. First consier the reuce states that Bob woul obtain if Alice really were to measure ˆ z on her half of W. If she i so, an obtaine the + result then Bob s state woul be given by z + = 2 I z +ˆ z. Similarly, for the result, Bob woul obtain z = 2 I z ˆ z, where the constants z + an z are efine as z + = 2 2, z = Now we wish to etermine if Alice coul simulate these conitione states using the ensemble F an a suitable strategy (±ˆ z,z,). Due to the form of z ± the best strategy for Alice is to split the ensemble F into two subensembles, one to simulate z + an the other to simulate z. Thus we can separate z into two positive istributions z = + z + z We imagine that Alice will attempt to simulate measuring ˆ z by ranomly generating states using the istribution z an sening them to Bob. If in a particular run of the experiment the state she sent Bob was from the subensemble etermine by + z then she will announce the result +. Similarly, if she sent Bob a state from z then she will announce. Now if Alice uses this strategy, Bob will fin on average that + z ± = z ± z ˆ z ± zz I z Comparing with Eqs an 4.35 we fin that in orer for the ensemble F to be able to simulate Alice measuring ˆ z we have the following constraints on z:

10 JONES, WISEMAN, AND DOHERTY PHYSICAL REVIEW A 76, z + z =, z z =, z + zz = z +, z zz = z Now consier the following conitione states that Bob woul obtain if Alice were to measure ˆ : ± = 2 I ± cosˆ x ± sinˆ y 2ˆ z How well coul Alice simulate the above state using the ensemble F an a cheating strategy efine by (±ˆ,z,)? We know that the ensemble F is symmetric uner rotations about the z axis. So in this case Alice woul use her knowlege of to etermine the outcome to announce when aske to measure ˆ. That is, if the state that she sent Bob is closer to the positive axis efine by ˆ then she will announce the + result. Similarly, if is closer to the negative measurement axis then she announces. This correspons to if 2 ±ˆ,z, 2 =, ± 0 if ± 2, From the symmetry uner rotations about the z axis we can see that Alice will be able to o equally well using this strategy to simulate states prepare by any measurement ˆ in the xy plane. Thus without loss of generality we set =0 an consier the specific case where Alice allegely measures ˆ x. Uner these conitions Eq reuces to x ± = 2 I ± ˆ x 2ˆ z If Alice ranomly sens Bob states from F an uses Eq to etermine her responses, Bob will fin on average the state + + ± 2I z z 2 zˆ x zzzˆ z We know that when F is optimal, Eq will exactly simulate Eq In etermining the optimal F we must fin the optimal z, however, we are constraine in etermining z by the fact that the ensemble must also simulate the states that Bob woul obtain if Alice were to measure ˆ z. These constraints are enforce by Eqs Note that Eqs an 4.40 ensure that the ˆ z term in Eq an Eq will be the same. Therefore, to etermine how well Alice s strategy can simulate Eq we only nee to consier the coefficient of the ˆ x term. If the coefficient of this term preicte by Eq is as large as in Eq then Alice s strategy simulates Bob s conitione state perfectly. Thus Bob woul not believe that the state W is genuinely steerable. Hence we nee to fin the istribution z, which maximizes the ˆ x coefficient in Eq to etermine if steering is possible. That is, we wish to fin the z that gives the maximum value of / + z z 2 z. This is equivalent to maximizing + z z 2 + z + z, 4.47 subject to the constraints given by Eqs Writing ± z= f ± 2 z for real functions f ± z we can use Lagrange multiplier techniques to perform the optimization. We fin that the optimal z has the unsurprising form z = z z + + z z, 4.48 where the constants z ± are efine in Eq an z z is the Dirac elta function. We see now why the choice of splitting the ensemble into two istributions was the best choice for Alice. The optimal ensemble F is compose of pure states in two rings aroun the z axis of the Bloch sphere; one in the +z hemisphere efine by z +, which on average may be use to simulate z +, an the other in the z hemisphere efine by z, which may simulate z +. These comments apply to the case 0/2. Using z to evaluate Eq we fin that x ± = 2I ± 2 z+ + 2 z ˆ x 2ˆ z Finally, comparing this with x ± given by Eq we fin that Alice s optimal cheating strategy fails to simulate measurements of ˆ x when Thus uner these conitions we know that steering is possible using the measurement scheme M. Note that we have not etermine steer as we have not consiere all possible projective measurements. However, we can make Eq an equality to provie an equation for, which is an upper boun on steer. This bounary is plotte in Fig. 2c. For =/2 we know explicitly that Bell steer ent 4.5 since these states are equivalent to the =2 Werner states see Appenix C 2. This special case yiels the isolate points at =/2 in Fig. 2c. For the remaining range of we

11 ENTANGLEMENT, EINSTEIN-PODOLSKY-ROSEN fin that our upper boun on steer is significantly lower than the upper boun on Bell an significantly higher than ent. This fact, taken with the known bounary values for =/ 2 gives us goo reason to conjecture that the three bounaries are strictly istinct for all 0,. D. Gaussian states Finally, we investigate a general multimoe bipartite Gaussian state W 2. The moe operators are efine as qˆ i=â i +â i an pˆ i= iâ i â i for the position an momentum, respectively. Here â i an â i are the annihilation an creation operators for the ith moe. For an n-moe state one may efine a vector Rˆ =qˆ, pˆ,...,qˆ n, pˆ n, which allows the commutation relations for the moe operators to be compactly expresse as ij Here n J i, where = i= ij R i,r j =2i are matrix elements of the symplectic matrix J i = A Gaussian state is efine by the mean of the vector of phase-space variables Rˆ, as well as the covariance matrix CM V for these variables. The mean vector can be arbitrarily altere by local unitary operations an hence cannot etermine the entanglement properties of W. Thus for our purposes a Gaussian state is characterize by the CM. In Alice, Bob block form it appears as CMW = V = V C C T V This represents a vali state if the linear matrix inequality LMI V + i is satisfie 2. Rather than aressing steerability in general, we consier the case where Alice can only make Gaussian measurements 2,22, the set of which will be enote by G. Thus, as for the previous section, since we are consiering a restricte class of measurements, if we emonstrate steerability with this measurement scheme it will provie an upper boun on steer. A measurement AG is escribe by a Gaussian positive operator with a CM T A satisfying T A +i 0 2,22. When Alice makes such a measurement, Bob s conitione state A a is Gaussian with a CM 23, PHYSICAL REVIEW A 76, CM A a = V A = V C T V + T A C, 4.56 which is actually inepenent of Alice s outcome a. Our goal is to etermine a sufficient conition for steerability of Gaussian states. We o this by etermining the necessary an sufficient conition for steerability with Gaussian measurements. In the previous examples after specifying a measurement scheme we consiere Bob s conitione state if Alice were to perform a measurement an etermine when it was possible for this state to be simulate by a cheating strategy. In the following we are working towar the same goal. If Alice were to perform a Gaussian measurement on half of the state W an sen the other part to Bob, then Bob s conitione state woul have a covariance matrix efine by Eq. 4.56; however, this is inepenent of Alice s result a. Thus we o not nee to consier a strategy for Alice to announce correctly correlate results to Bob. We simply nee to etermine when Alice coul simulate Bob s conitione state by sening Bob states from a pure state ensemble rather than actually sening part of W. We will show that there exists an optimal ensemble of Gaussian states istinguishe by their mean vectors but sharing the same covariance matrix, which we will label U which Alice coul use for this task. If there exists a vali ensemble of Gaussian states efine by U which can simulate V A then Bob will not believe that W is entangle, an hence the state is not steerable. Before moving to the presentation of our main result consier the following result from linear algebra theory relating to Schur complements of block matrices. The Schur complements of P an Q in a general block matrix B = P R R Q T 4.57 are efine as P =Q R T P R an Q = P RQ R T, respectively. The matrix B is positive semiefinite PSD, iff both P an its Schur complement are PSD an likewise for Q an its Schur complement. The proof of our main theorem is base on the following inequality: V + 0 i 0, 4.58 an relies on the following facts: Lemma 2. IfEq.4.58 is true then there exists an ensemble efine by covariance matrix U such that U + i 0, 4.59 V A U 0, 4.60 which implies that the state W is not steerable. Proof. See Appenix D for proof of this lemma. Lemma 3. If the Gaussian state W efine in Eq is not steerable by Alice s Gaussian measurements then there exists a Gaussian ensemble efine by covariance matrix U such that Eqs an 4.60 hol. Proof. See Appenix D 2 for proof of this lemma. Lemma 4. If AG there exists U such that Eqs an 4.60 hol, then V + T A CV + i C T 0, 4.6 must also hol. Proof. See Appenix D 3 for proof of this lemma. We are now in a position to present our main theorem. Theorem 5. The Gaussian state W efine in Eq is not steerable by Alice s Gaussian measurements iff Eq is true