Some remarks on Bell non-locality and Einstein-Podolsky-Rosen steering of bipartite states
aa r X i v : . [ qu a n t - ph ] J a n Some remarks on Bell non-locality andEinstein-Podolsky-Rosen steering ofbipartite states
Huaixin Cao, Zhihua Guo
School of Mathematics and Information Science, Shaanxi Normal UniversityXi’an 710119, ChinaEmail: [email protected], [email protected]
Abstract . Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering are ev-ery important quantum correlations of a composite quantum system. Bell nonlocalityof a bipartite state is a quantum correlation demonstrated by some local quantummeasurements, while EPR steering is another form of quantum correlations, observedfirstly by Schrodinger in the context of famous EPR paradox. In this paper, wegive some remarks on Bell nonlocality and EPR steering of bipartite states, includ-ing mathematical definitions and characterizations of these two quantum correlations,the convexity and closedness of the set of all Bell local states and the set of all EPRunsteerable states. We also derive a EPR-steering criteria, with which the EPR steer-ability of the maximally entangled states are checked.
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Generally, quantum correlations means the correlations between subsystems of a com-posite quantum system, including Bell nonlocality, steerability, entanglement andquantum discord.Bell nonlocality of a bipartite state is a quantum correlation demonstrated by somelocal quantum measurements whose statistics of the measurement outcomes cannotbe explained by a local hidden variable (LHV) model [1, 2]. Such a nonclassical fea-ture of quantum mechanics can be used in device-independent quantum informationprocessing [2]. For more works on Bell nonlocality, please refer to Clauser and Shi-mony [3], Home and Selleri [4], Khalfin and Tsirelson [5], Tsirelson [6], Zeilinger [7],Werner and Wolf [8], Genovese [9], and Buhrman et al. [10], and references therein.Einstein-Podolsky-Rosen (EPR) steering as a form of quantum correlations, wasfirst observed by Schrodinger [11] in the context of famous Einstein-Podolsky-Rosen(EPR) paradox [12–15]. EPR steering arises in the scenario where some local quan-tum measurements on one part of a bipartite system are used to steer the other part.This scenario demonstrates EPR steering if the obtained ensembles cannot be ex-plained by a local hidden state (LHS) model [16]. Followed in close analogy withcriteria for other forms of quantum nonlocality (Bell nonlocality and entanglement),Cavalcanti et al. [17] developed a general theory of experimental EPR-steering criteriaand derived a number of criteria applicable to discrete as well as continuous-variableobservables. Saunders et al. [18] contributed experimental EPR-steering by using Belllocal states. Bennet et al. [19] derived arbitrarily loss-tolerant tests, which enable us1o perform a detection-loophole-free demonstration of Einstein-Podolsky-Rosen steer-ing with parties separated by a coiled 1-km-long optical fiber. H¨andchen et al. [20]presented an experimental realization of two entangled Gaussian modes of light thatin fact shows the steering effect in one direction but not in the other. The generatedone-way steering gives a new insight into quantum physics and may open a new fieldof applications in quantum information.EPR steering, as a form of bipartite quantum correlation that is intermediate be-tween entanglement and Bell nonlocality, allows for entanglement certification whenthe measurements performed by one of the parties are not characterized (or are un-trusted) and has applications in quantum key distribution. Branciard et al. [21] an-alyzed the security and feasibility of a protocol for quantum key distribution (QKD)in a context where only one of the two parties trusts his measurement apparatus andclarified the link between the security of this one-sided DI-QKD scenario and thedemonstration of quantum steering, in analogy to the link between DI-QKD and theviolation of Bell inequalities. Wittmann et al. [22] presented the first loophole-freedemonstration of EPR-steering by violating three-setting quadratic steering inequal-ity in light of polarization entangled photons shared between two distant laboratories.Steinlechner et al. [23] achieved an unprecedented low conditional variance productof about 0 . <
1, where 1 is the upper bound below which steering is present, andobserved the steering effect on an unconditional two-mode-squeezed entangled statethat contained a total vacuum state contribution of less than 8%. Reid [24] provedthat EPR paradox can be used to verify that the quantum benchmark for qubitteleportation has been reached, without postselection and EPR steering inequalitiesinvolving m measurement settings can also be used to confirm quantum teleporta-tion if one assumes trusted detectors for Charlie and Alice. Skrzypczyk et al. [25]proposed a way of quantifying this phenomenon and use it to study the steerabilityof several quantum states and shown that every pure entangled state is maximallysteerable and the projector onto the antisymmetric subspace is maximally steerablefor all dimensions.Piani et al. [26] provided a necessary and sufficient characterization of steering,based on a quantum information processing task: the discrimination of branches ina quantum evolution, which we dub subchannel discrimination. They also provedthat, for any bipartite steerable state, there are instances of the quantum subchan-nel discrimination problem for which this state allows a correct discrimination withstrictly higher probability than in absence of entanglement, even when measurementsare restricted to local measurements aided by one-way communication. Many of thestandard Bell inequalities (e.g. CHSH ) are not effective for detection of quantumcorrelations which allow for steering, because for a wide range of such correlationsthey are not violated. Zukowski et al. [27] presented some Bell like inequalities whichhave lower bounds for non-steering correlations than for local causal models. Theseinequalities involve all possible measurement settings at each side. Geometric Belllike inequalities for steering.By definition, it is easy to check that every separable state is unsteerable state andany unsteerable state is Bell local. Thus, quantum states that demonstrate Bell non-locality form a subset of EPR steerable states which, in turn, form a subset of entan-gled states. Furthermore, Quintino et al. proved in [28] that entanglement, one-waysteering, two-way steering, and Bell nonlocality are genuinely different. Specifically,considering general POV measurements, they proved the existence of (i) entangledstates that cannot lead to steering, (ii) states that can lead to steering but not to Bellnonlocality, and (iii) states which are one-way steerable but not two-way steerable.Zhu et al. [29] proposed a general framework for constructing universal steeringcriteria that are applicable to arbitrary bipartite states and measurement settings of2he steering party. The same framework is also useful for studying the joint mea-surement problem. Based on the data-processing inequality for an extended R´enyirelative entropy, they also introduced a family of steering inequalities, which detectsteering much more efficiently than those inequalities known before. Sun et al. [30]experimentally demonstrated asymmetric EPR steering for a class of two-qubit statesin the case of two measurement settings and proposed a practical method to quan-tify the steerability. They also provided a necessary and sufficient condition for EPRsteering and clearly demonstrate one-way EPR steering.Recently, Cavalcanti et al. [31] contributed a review on quantum steering withfocus on semidefinite programming. Moreover, based on decomposing the measure-ment correlations in terms of extremal boxes of the steering scenario, Das et al. [32]presented a method to check EPR steering in the scenario where the steering partyperforms two black-box measurements and the trusted party performs two mutuallyunbiased projective qubit measurements. In this context, they proposed a measure ofsteerability called steering cost and proved that their steering cost is a convex steeringmonotone.In this paper, we will give some remarks on Bell nonlocality and EPR steering ofbipartite states, including mathematical definitions and characterizations of these twoquantum correlations, the convexity and closedness of the set of all Bell local statesand the set of all EPR unsteerable states. We also derive a EPR-steering criteria,with which the EPR steerability of the maximally entangled states are checked. Theother parts of this note are divided as follows. In Section 2, we will give the definitionof Bell locality and Bell nonlocality of bipartite states, and establish some equivalentcharacterizations of Bell locality. Moreover, we will prove that the closedness andconvexity of the set of all Bell local states. In Section 3, we will give the definitionsof PER unsteerability and PER steerability of bipartite states, and establish someequivalent characterizations of PER unsteerability. Moreover, we will prove that theclosedness and convexity of the set of all unsteerable states. In Section 4, we willestablish a EPR steering criteria and prove the EPR steerability of the maximallyentangled states. In what follows, we use H A and H A to denote two finite dimensional complex Hilbertspaces, which describe two quantum systems A and B , respectively. We also use D X to denote the set D ( H X ) of all quantum states of the system X described by a Hilbertspace H X .A standard nonlocality scenario (SNLS) consists of two distant systems on whichtwo observers, Alice and Bob, perform respectively m A and m B different measure-ments of o A and o B possible outcomes. More explicitly, when the outcomes of Aliceand Bob are labeled a and b , respectively, while their POV measurement choices are M x = { M a | x : a = 1 , . . . , o A } ( x = 1 , . . . , m A ) ,N y = { N b | y : b = 1 , . . . , o B } ( y = 1 , . . . , m B ) , respectively, the family M AB ≡ M A ⊗ N B := { M x ⊗ N y : x = 1 , . . . , m A , y = 1 , . . . , m B } is said to be a standard nonlocality scenario (SNLS) for system AB , where M A = { M x : x = 1 , , . . . , m B } , N B = { N y : y = 1 , , . . . , m B } , measurement assemblages of A and B , respectively, and M x ⊗ N y = { M a | x ⊗ N b | y : a = 1 , . . . , o A , b = 1 , . . . , o B } . Definition 2.1.
Let ρ AB be a state of the system AB , M A = { M x } m A x =1 and N B = { N y } m B y =1 be two sets of some POV measurements (POVMs) of A and B ,respectively.(1) A state ρ AB is said to be Bell local for M A ⊗ N B if there exist a probabilitydistribution (PD) { π λ } dλ =1 such that for each ( λ, x ) and each ( λ, y ), there exist PDs { P A ( a | x, λ ) } o A a =1 and { P B ( b | y, λ ) } o B b =1 , respectively, such thattr[( M a | x ⊗ N b | y ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) P B ( b | y, λ ) , ∀ a, b, x, y. (2 . local hidden variable (LHV) model of ρ AB withrespect to M AB and λ is said to be a local hidden variable. Denote by BL ( M A , N B )the set of all states ρ AB that are Bell local for M A ⊗ N B .(2) A state ρ AB is said to be Bell nonlocal for M A ⊗ N B if it is not Bell local for M A ⊗ N B . Denote by BN L ( M A , N B ) the set of all states ρ AB that are Bell nonlocalfor M A ⊗ N B .(3) A state ρ AB is said to be Bell local if for every M A ⊗ N B , there exists a PD { π λ } dλ =1 such that Eq. (2.1) holds. Denote by BL ( AB ) the set of all Bell local states ρ AB of AB .(4) A state ρ AB is said to be Bell nonlocal if it is not Bell local, i.e. there existsan M A ⊗ N B such that ρ AB is not Bell local for M A ⊗ N B . Denote by BN L ( AB )the set of all states ρ AB that are Bell nonlocal. Remark 2.1.
By definition above, we see that when a state ρ AB is Bell local for M A ⊗N B , it has an LHV model (2.1). Finding the sums of two sides for b = 1 , , . . . , o b yields that tr[( M a | x ⊗ I B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) , ∀ a, x. This shows that the measurement results of Alice with M A are independent of themeasurements of Bob. Similarly, we havetr[( I A ⊗ N b | y ) ρ AB ] = d X λ =1 π λ P B ( b | y, λ ) , ∀ b, y, implying that the measurement results of Bob with N B are independent of the mea-surements of Alice. Moreover, we see from definition that Bell local states: BL ( AB ) = T M A , N B BL ( M A , N B ); Bell nonlocal states:
BN L ( AB ) = S M A , N B BN L ( M A , N B ) . By Definition 2.1, we know that
Remark 2.2.
Every separable state is Bell local. Equivalently, Bell nonlocal statemust be entanglement.To see this, let ρ AB = P dλ =1 c λ ρ Aλ ⊗ ρ Bλ be separable. Then for every M A ⊗ N B ,we have tr[( M a | x ⊗ N b | y ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) P B ( b | y, λ ) , ∀ a, b, x, y, where P A ( a | x, λ ) = tr( M a | x ρ Aλ ) , P B ( b | y, λ ) = tr( N b | y ρ Bλ ) .
4y Definition 2.1, ρ AB is Bell local. Note that in this case, response functions P A ( a | x, λ ) and P B ( b | y, λ ) are “quantum”, i.e. they are induced by quantum states. Remark 2.3.
In the definition of locality of a state, the probability distribution { π λ } dλ =1 of the hidden variable λ is necessary. Generally, the dimension d of hiddenvariable space depends on not only the measurement assemblage M A ⊗ N B but alsothe state ρ AB .An expectation is to find the same dimension of hidden variable spaces for allBell local states for a given M A ⊗ N B . To do this, let us consider the set Ω A of allpossible maps from S m = { , , . . . , m A } into S o = { , , . . . , o A } . Clearly, Ω A hasjust N A := o m A A elements and so can be written asΩ A = { J , J , . . . , J N A } . Each element J of Ω denotes a “measurement scenario”, which assigns an outcomevalue a for each POVM M x , that is, J ( x ) = a . We use p A ( k, λ ) to denote theprobability of a measurement scenario J k to be used when Alice receives a classicalmessage λ in Λ. Thus, { p A ( k, λ ) } N A k =1 is a PD and depending only on the number m A of measurement operators and the number o A of the common outcomes. Let P ( a, x, λ )be the probability of obtaining the outcome a when Alice receives a classical message λ in Λ and uses M x . Then the total probability formula yields that P ( a, x, λ ) = N A X k =1 p A ( k, λ ) δ a,J k ( x ) , ∀ a ∈ S o . (2 . P ( b, y, λ ) be the probability of obtaining the outcome b when Bobreceives a classical message λ in Λ, and Ω B the set of all possible maps from T m = { , , . . . , m B } into T o = { , , . . . , o B } . Clearly, Ω B has just N B := o m B B elementsand so can be written as Ω B = { K , K , . . . , K N B } . Then P ( b, y, λ ) = N B X j =1 p B ( j, λ ) δ a,K j ( y ) , ∀ b ∈ T o , (2 . N B := o m B B is the number of elements K j ’s of Ω B .When a state ρ AB is Bell local for M A ⊗ N B , it has an LHV model (2.1). Thus,for every k we have X a P A ( a | x, k ) = 1(1 ≤ x ≤ m A ) , X b P B ( b | y, k ) = 1(1 ≤ y ≤ m B ) . By finding the sums of two sides of (2.1) for b ∈ T o , we get thattr[( M a | x ⊗ I B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) , ∀ a, x. (2 . I A ⊗ N b | y ) ρ AB ] = d X λ =1 π λ P B ( b | y, λ ) , ∀ b, y. (2 . a when the mea-surement M x is used. The quantity π λ can be viewed as the probability of Alice5eceiving a message λ , and the quantity P A ( a | x, λ ) should be the probability of ob-taining outcome a when the measurement M x is used and a message λ is received byAlice. From Eqs. (2.2) and (2.3), we know that P A ( a | x, λ ) = N A X k =1 p A ( k, λ ) δ a,J k ( x ) , ∀ a ∈ S o , (2 . P N A k =1 p A ( k, λ ) = 1 for all λ . Similarly, P B ( b | y, λ ) = N B X j =1 p B ( j, λ ) δ b,K j ( y ) , ∀ b ∈ T o , (2 . P N B j =1 p B ( j, λ ) = 1 for all λ . It follows from (2.6), (2.7) and (2.1) thattr[( M a | x ⊗ N b | y ) ρ AB ] = N A X k =1 N B X j =1 q k,j δ a,J k ( x ) δ b,K j ( y ) , (2 . q k,j = P dλ =1 π λ p A ( k, λ ) p B ( j, λ ) ≥ k, j satisfying P N A k =1 P N B j =1 q k,j = 1 . Conversely, if there exist a probability distribution { q k,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } := { π , π , . . . , π N A N B } satisfying (2.8), then Eq. (2.1) holds for P A ( a | x, λ ) = δ a,J k ( x ) and P B ( b | y ) = δ b,K j ( y ) if π λ = q k,j and then a state ρ AB is Bell local for M A and M B .As a result, we have the following conclusion. Theorem 2.1.
A state ρ AB is Bell local for M A ⊗ N B if and only if there existsa probability distribution { q k,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } satisfying Eq. (2.8). This characterization of Bell locality is very useful due to the sum in (2.8) wastaken for a fixed number N A N B of terms, the PDs { δ a,J k ( x ) } o A a =1 depending only on M a | x and { δ b,K j ( y ) } o B b =1 depending only on N b | y are independent of ρ AB , while the PD { q k,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } depends only on ρ AB . For instance, we can provethe following conclusion by using this characterization. Corollary 2.1.
The set BL ( M A , N B ) is a compact convex subset of D AB . Fur-thermore, BL ( AB ) is a compact convex set. Proof.
Let { ρ n } ∞ n =1 ⊂ BL ( M A , N B ) with ρ n → ρ as n → ∞ . We see fromTheorem 2.1 that for each n , there exists a PD { q nk,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } suchthat tr[( M a | x ⊗ N b | y ) ρ n ] = N A X k =1 N B X j =1 q nk,j δ a,J k ( x ) δ b,K j ( y ) , ∀ a, x, b, y, (2 . n = 1 , , . . . . By choosing subsequence, we may assume that for each ( k, j ), thesequence { q nk,j } ∞ n =1 is convergent, sat q nk,j → q k,j as n → ∞ . Clearly, { q k,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } is a PD. Letting n → ∞ in Eq. (2.9) yields thattr[( M a | x ⊗ N b | y ) ρ ] = N A X k =1 N B X j =1 q k,j δ a,J k ( x ) δ b,K j ( y ) , ∀ a, x, b, y. By Theorem 2.1, we conclude that ρ ∈ BL ( M A , N B ). This shows that BL ( M A , N B )is closed and then compact due to the compactness of D AB .
6o check the convexity of BL ( M A , N B ), we let ρ , ρ ∈ BL ( M A , N B ) and 0 1. We see from Theorem 2.1 that for n = 1 , 2, there exists a PD { q nk,j : 1 ≤ k ≤ N A , ≤ j ≤ N B } such thattr[( M a | x ⊗ N b | y ) ρ n ] = N A X k =1 N B X j =1 q nk,j δ a,J k ( x ) δ b,K j ( y ) , ∀ a, x, b, y. (2 . ∀ a, x, b, y, tr[( M a | x ⊗ N b | y )( tρ + (1 − t ) ρ ] = N A X k =1 N B X j =1 [ tq k,j + (1 − t ) q k,j ] δ a,J k ( x ) δ b,K j ( y ) = N A X k =1 N B X j =1 tq k,j δ a,J k ( x ) δ b,K j ( y ) , where q k,j = tq k,j + (1 − t ) q k,j for all k, j . Clearly, P k,j q k,j = 1. By using Theorem2.1 again, we see that tρ + (1 − t ) ρ ∈ BL ( M A , N B ).Lastly, by using the fact that BL ( AB ) = \ M A , N B BL ( M A , N B ) , we see that BL ( AB ) is a compact convex set. The proof is completed. Definition 3.1. (Steerability) Let ρ AB be a state of the system AB , and let M A = {{ M a | x } o A a =1 : x = 1 , , . . . , m A } be any measurement assemblage of A .(1) A state ρ AB of the system AB is said to be unsteerable from A to B withrespect to M A if there exists a PD { π λ } dλ =1 and a set of states { σ λ } dλ =1 ⊂ D B suchthat ρ a | x := tr A [( M a | x ⊗ B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) σ λ , ∀ x, a, (3 . { P A ( a | x, λ ) } o A a =1 is a PD for each ( a, x ). In this case, we also say that Eq. (3.1)is an LHS model of ρ AB with respect to M A (2) A state ρ AB is said to be steerable from A to B with respect to M A if it isnot unsteerable from A to B with respect to M A . In this case, we also say that ρ AB exhibits quantum steering with respect to M A .(3) A state ρ AB is said to be unsteerable from A to B if for any M A , ρ AB is unsteerable from A to B with respect to M A .(4) A state ρ AB is said to be steerable from A to B if ∃ an M A such that it issteerable from A to B with respect to M A , i.e. it is not unsteerable from A to B with respect to M A .Symmetrically, we define unsteerability and steerability of a state from B to A .(5) A state ρ AB is said to be is steerable if it is steerable from A to B or B to A .(6) A state ρ AB is said to be unsteerable if it is not steerable, i.e. it is unsteerableboth from A to B , and B to A .Here are some remarks to the definitions above.7 emark 3.1. Denote by US ( A → B, M A ) the set of all states which are unsteer-able from A to B with respect to M A , by US ( A → B ) the set of all states which areussteerable from A to B , and denote by S ( A → B, M A ) the set of all states whichare steerable from A to B with respect to M A , by S ( A ∨ B ) the set of all states whichare steerable from either A to B , or B to A . From definition above, we have US ( A → B ) = \ M A US ( A → B, M A ); US ( B → A ) = \ M B US ( B → A, M B ); US ( A ∧ B ) = US ( A → B ) ∩ US ( B → A ); S ( A ∨ B ) = S ( A → B ) ∪ S ( B → A ) . Remark 3.2. When ρ AB ∈ US ( A → B, M A ), Eq. (3.1) holds. Thus, we have ρ B = tr A " o A X a =1 ( M a | x ⊗ B ) ρ AB = d X λ =1 π λ o A X a =1 P A ( a | x, λ ) σ λ . Since P o A a =1 P A ( a | x, λ ) = 1 for all λ and x , we get ρ B = d X λ =1 π λ σ λ , (3 . M x . This means that thechoice of Alice’s measurements can not change (steer) Bob’s state ρ B , which is alwaysgiven by Eq. (3.2).Generally, the PD { π λ } dλ =1 and the states { σ λ } dλ =1 depend on the state ρ AB andthe measurement assemblage M A .The physical interpretation is the following: when a state ρ AB is unsteerablewith respect M A , Eq. (3.1) enables that Bob can interpret his conditional states ρ a | x := tr A [( M a | x ⊗ B ) ρ AB ] as coming from the pre-existing states { σ λ } and the PD { π λ } , where only the probabilities are changed due to the knowledge { P A ( a | x, λ ) } ofAlice’s measurement and result. Also, he can obtain his state ρ B from the pre-existingstates { σ λ } and the PD { π λ } in light of Eq. (3.2). Contrarily, when a state ρ AB issteerable with respect to M A , Bob must believe that Alice can remotely steer thestates in his lab by making measurements M A on her side. Example 3.1. Let us now assume that Alice’s measurements in M A are com-patible, in the sense of being jointly measurable [31]. This means that there exists asingle ‘parent’ POV measurement N = { N λ } dλ =1 such that ∀ M x = { M a | x } o A a =1 ∈ M A ,there is d PDs { P A ( a | x, λ ) } o A a =1 ( λ = 1 , , . . . , d ), such that M a | x = d X λ =1 P A ( a | x, λ ) N λ ( a = 1 , , . . . , o A ) . Thus, for any state ρ AB of the system AB , we have for each ( a, x ),tr A [( M a | x ⊗ ρ AB ] = d X λ =1 P A ( a | x, λ )tr A [( N λ ⊗ ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) σ λ , where π λ = tr[( N λ ⊗ ρ AB ] , σ λ = 1 π λ tr A [( N λ ⊗ ρ AB ] . ρ AB is unsteerable from A to B with respect to a com-patible measurement assemblage M A . Especially, when Alice has just one POVmeasurement M = { M a } o A a =1 , i.e. M A = { M } , any state ρ AB of the system AB isunsteerable from A to B with respect to M A . Explicitly,tr A [( M a ⊗ ρ AB ] = o A X λ =1 π λ P A ( a | M, λ ) σ λ , where π λ = tr[( M λ ⊗ ρ AB ] , P A ( a | M, λ ) = δ λ,a , σ λ = 1 π λ tr A [( M λ ⊗ ρ AB ] . In a word, it is not possible that Alice wants to steer Bob with just one POVM. Theorem 3.1. A state ρ AB of the system AB is unsteerable from A to B withrespect to M A if and only if there exists a PD { π λ } dλ =1 , a group of states { σ λ } dλ =1 ⊂D B , and dm A PDs { P A ( a | x, λ ) } o A a =1 (1 ≤ x ≤ m A , ≤ λ ≤ d ) such that every localPOVM { N b } o B b =1 of B , it holds that tr[( M a | x ⊗ N b ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ )tr( N b σ λ ) , ∀ x, a, b. (3 . Proof. Necessity. Let ρ AB be unsteerable from A to B with respect to M A .Then by definition, there exists a PD { π k } dk =1 and a group of states { σ k } dk =1 ⊂ D B such that Eq. (3.1) holds for all x, a. For any POVM { N b } o B b =1 of B , we see from Eq.(3.1) that ∀ x, a, b, tr[( M a | x ⊗ N b ) ρ AB ] = tr (cid:0) N b tr A [( M a | x ⊗ B ) ρ AB ] (cid:1) = tr d X λ =1 π λ P A ( a | x, λ )( N b σ λ ) ! = d X λ =1 π λ P A ( a | x, λ )tr( N b σ λ ) . Sufficiency. Suppose that Eq. (3.3) holds for every POVM { N j } o B j =1 of B . Thenfor every M x = { M a | x } o A a =1 ∈ M A and for every projection P on H B , using Eq. (3.3)for N = P, N = I B − P yields that for every ( x, a ),tr (cid:0) P tr A [( M a | x ⊗ B ) ρ AB ] (cid:1) = tr[( M a | x ⊗ P ) ρ AB ]= d X λ =1 π λ P A ( a | x, λ )tr( P σ λ ) ! = tr P d X λ =1 π λ P A ( a | x, λ ) σ λ ! . Thus, for every ( x, a ), (cid:10) P, tr A [( M a | x ⊗ B ) ρ AB ] (cid:11) HS = * P, d X λ =1 π λ P A ( a | x, λ ) σ λ + HS , where h X, Y i HS := tr( X † Y ) denotes the Hilbert-Schmidt inner product on the oper-ator space B ( H B ). Hence, for every ( x, a ),tr A [( M a | x ⊗ B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) σ λ . ρ AB is unsteerable from A to B with respect to M A .The proof is completed.We see from Corollary 3.1 that the steering of Alice to Bob needs to get a helpfrom BoB.Similarly, one can prove the following. Theorem 3.2. A state ρ AB of the system AB is unsteerable from A to B if andonly if for every M A , there exists a PD { π λ } dλ =1 , a set of states { σ λ } dλ =1 ⊂ D B and dm A PDs { P A ( a | x, λ ) } o A a =1 (1 ≤ x ≤ m A , ≤ λ ≤ d ) such that for every POVM { N b } o B b =1 of B , it holds that tr[( M a | x ⊗ N b ) ρ AB ] = d X λ =1 π λ P A ( a | x, k )tr( N b σ λ ) , ∀ x, a, b, (3 . A to B with respect to M A . To do this, we consider the set Ω ofall possible maps from S m = { , , . . . , m A } into S o = { , , . . . , o A } . Clearly, Ω hasjust N := o m A A elements and so can be written asΩ = { J , J , . . . , J N } . Each element J of Ω denotes a “measurement scenario”, which assigns an outcomevalue a for each POVM x ≡ M x , that is, J ( x ) = a . We use p ( k, λ ) to denote theprobability of a measurement scenario J k to be used when Alice receives a classicalmessage λ in Λ, and P ( a, x, λ ) to denote the probability of obtaining the outcome a under the condition that Alice receives a classical message λ in Λ and chooses M x .Then the Law of Total Probability yields that P ( a, x, λ ) = N X k =1 p ( k, λ ) δ a,J k ( x ) , ∀ a ∈ S o , ∀ x ∈ S m , (3 . N X k =1 p ( k, λ ) = 1( ∀ λ ∈ Λ) , o A X a =1 δ a,J k ( x ) = 1( ∀ k, x ) , o A X a =1 P ( a, x, λ ) = 1( ∀ a, x ) . Please refer to [ ? ] and [31] for Eq. (3.5).Suppose that ρ AB is unsteerable from A to B with respect to M A . Then bydefinition, there exists a PD { π λ } dλ =1 and a set of states { ρ Bλ } dλ =1 ⊂ D B such thattr A [( M a | x ⊗ B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) ρ Bλ , ∀ a ∈ S o , ∀ x ∈ S m . (3 . M a | x ⊗ B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) , ∀ a ∈ S o , ∀ x ∈ S m . (3 . a when measurement x is performed and P A ( a | x, λ ) is the probability of obtaining the outcome a underthe condition that Alice receives a classical message λ and chooses M x . Thus, P A ( a | x, λ ) = P ( a, x, λ ) and so Eq. (3.5) yields that P A ( a | x, λ ) = N X k =1 p ( k, λ ) δ a,J k ( x ) , ∀ a ∈ S o , ∀ x ∈ S m . 10t follows from (3.6) thattr A [( M a | x ⊗ B ) ρ AB ] = N X k =1 δ a,J k ( x ) d X λ =1 π λ p ( k, λ ) ρ Bλ , ∀ a ∈ S o , ∀ x ∈ S m . By putting τ k = P dλ =1 π λ p ( k, λ ) ρ Bλ , we obtain thattr A [( M a | x ⊗ B ) ρ AB ] = N X k =1 δ a,J k ( x ) τ k , ∀ a ∈ S o , ∀ x ∈ S m , (3 . τ k ≥ k and P Nk =1 tr( τ k ) = 1. See [31, 32].Conversely, we suppose that there exists there exists positive operators τ k ( k =1 , , . . . , N ) on H B satisfying P Nk =1 tr( τ k ) = 1 and such that (3.8) holds. Let π k =tr( τ k ) , σ k = π k τ k . Then (3.8) becomestr A [( M a | x ⊗ ρ AB ] = N X k =1 π k δ a,J k ( x ) σ k , ∀ x ∈ S m , ∀ a ∈ S o . (3 . P o A a =1 δ a,J k ( x ) = 1 for all x ∈ S m and all k = 1 , , . . . , N , by taking P A ( a | x, k ) = δ a,J k ( x ) we see by definition that ρ AB is unsteerable from A to B with respect to M A .As a conclusion, we have established the following theorem. Theorem 3.3. A state ρ AB of the system AB is unsteerable from A to B withrespect to M A if and only if there exists a family { τ k } Nk =1 of positive operators τ k on H B with P Nk =1 tr( τ k ) = 1 such that (3.8) holds. It is remarkable to point out that positive operators τ k in Eq. (3.8) depend onlyon the state ρ AB and are independent of the measurement operators M a | x , while thedeterministic PDs { δ a,J k ( x ) } a depend only on the measurement operators { M a | x } ,independent of the state ρ AB . Also, the number N = o m A A of terms of summation isfixed whenever the measurement assemblage M A is given. This enables us to provethe following important properties of unsteerable states. Corollary 3.1. US ( A → B, M A ) is a compact convex subset of D AB . Proof. Let ρ , ρ ∈ US ( A → B, M A ) and 0 < t < 1. Then by Theorem 3.3,there exist there exist families { τ k } Nk =1 and { τ k } Nk =1 of positive operators τ ik on H B with P Nk =1 τ ik ) = 1( i = 1 , , k = 1 , , . . . , N ) such thattr A [( M a | x ⊗ ρ i ] = N X k =1 δ a,J k ( x ) τ ik ( i = 1 , , ∀ x ∈ S m , ∀ a ∈ S o . Thus, ∀ x ∈ S m , ∀ a ∈ S o , we havetr A [( M a | x ⊗ tρ + (1 − t ) ρ ] = N X k =1 δ a,J k ( x ) τ k , where τ k = tτ k + (1 − t ) τ k ≥ k and P Nk =1 tr( τ k ) = 1 . Thus, Theorem 3.3implies that tρ + (1 − t ) ρ is unsteerable from A to B with respect to M A and then US ( A → B, M A ) is convex.Next, let { ρ m } ∞ m =1 be a sequence in US ( A → B, M A ) such that ρ m → ρ as m → + ∞ . By Theorem 3.3, there are positive operators τ mk ( k = 1 , , . . . , N, m = 1 , , . . . )on H B such that P Nk =1 tr( τ mk ) = 1 andtr A [( M a | x ⊗ B ) ρ m ] = N X k =1 δ a,J k ( x ) τ mk , ∀ n ∈ S m , ∀ n ∈ S o , (3 . m . By the compactness of D B , we may assume that for each k = 1 , , . . . , N , { τ mk } ∞ m =1 is convergent and let τ mk → τ k as m → + ∞ . Then by letting m → + ∞ in(3.10), we get tr A [( M a | x ⊗ B ) ρ ] = N X k =1 δ a,J k ( x ) τ k , ∀ n ∈ S m , ∀ a ∈ S o . (3 . P Nk =1 tr( τ mk ) = 1( m = 1 , , . . . ), we see P Nk =1 tr( τ k ) = 1. Clearly, τ k ≥ k . Now, Theorem 3.3 shows that is unsteerable from A to B withrespect to M A and therefore US ( A → B, M A ) is closed and then compact. Theproof is completed. Corollary 3.2. The set US ( A → B ) is a compact convex subset of D AB and S ( A → B ) is open. Proof. From Remark 3.1, we know that US ( A → B ) = \ M A US ( A → B, M A ) , where the intersection was taken over all measurement assemblages M A of A . Itfollows from Corollary 3.1 that US ( A → B ) is compact and convex. The proof iscompleted.As the end of this section, let us discuss some relationships among steerability,nonlocality, entanglement and quantum correlations. From Theorem 3.1 and Theorem3.2, we see the following remarks.(1) When ρ AB is unsteerable from A to B with respect to M A , we see fromDefinition 3.1 that there exists a PD { π λ } dλ =1 , a set of states { σ λ } dλ =1 ⊂ D B , and dm A PDs { P A ( a | x, λ ) } o A a =1 (1 ≤ x ≤ m A , ≤ λ ≤ d ) such thattr A [( M a | x ⊗ B ) ρ AB ] = d X λ =1 π λ P A ( a | x, λ ) σ λ , ∀ x, a. Thus, for any M B ,tr[( M a | x ⊗ N b | y ) ρ AB ] = tr( N b | y tr A [( M a | x ⊗ B ) ρ AB ]) = X k π k P A ( a | x, k ) P B ( b | y, k ) , where P B ( b | y, k ) = tr( N b | y σ k ) . Thus, ρ AB is Bell local for M A ⊗ M B .(2) When ρ AB is unsteerable either from A to B , or from B to A , it is Bell local.This shows that an unsteerable state must be Bell local, i.e. US ( A ∧ B ) = US ( A → B ) ∩ US ( B → A ) ⊂ US ( A → B ) ∪ US ( B → A ) ⊂ BL ( AB ) . (3) When ρ AB = P dk =1 π k ρ Ak ⊗ ρ Bk is separable, especially, classically-classicallycorrelated [33–35], we have for any M A ,tr A [( M a | x ⊗ B ) ρ AB ] = d X k =1 π k P A ( a | x, k ) ρ Bk , ∀ x, a, where P A ( a | x, k ) = tr( M a | x ρ Ak ) . Thus, ρ AB is unsteerable from A to B with respectto any M A . Thus, ρ AB is unsteerable from A to B with respect to any M A . Thus, ρ AB is unsteerable from A to B . Similarly, ρ AB is also unsteerable from B to A . Astate which is steerable both from A to B and from B to A is said to be two-way teerable . A state which is steerable either from A to B , or from B to A is said to be one-way steerable .With the discussion above, we have the following relationships. CC ( AB ) ) Sep ( AB ) ) US ( A → B ) ∩ US ( B → A ) ) US ( A → B ) ∪ US ( B → A ) ) BL ( AB ) , where CC ( AB ) and Sep ( AB ) are sets of all classically-classically (CC) correlated andseparable states of AB , respectively. Hence, QC ( AB ) ) Ent ( AB ) ) S ( A → B ) ∪ S ( B → A ) ) S ( A → B ) ∩ S ( B → A ) ⊃ BN L ( AB ) , where QC ( AB ) = D ( AB ) \ CC ( AB ) , the set of all quantum correlated states of AB ; Ent ( AB ) = D ( AB ) \ Sep ( AB ) , the set of all entangled states of AB ; BN L ( AB ) = D ( AB ) \ BL ( AB ), the set of all Bell nonlocal states of AB .Consequently,Bell locality ⇐ Unsteerability ⇐ Separability ⇐ Classical correlation,equivalently,Bell nonlocality ⇒ Steerability ⇒ Entanglement ⇒ Quantum correlation Definition 4.1. Two bases e = {| e i i} ni =1 and f = {| f i i} ni =1 for an n -dimensionalHilbert space H are said to be disjoint and denoted by e V f = 0 if | e i ih e i | 6 = | f j ih f j | for all i, j , equivalently, ( C | e i i ) ∩ ( C | f j i ) = { } , ∀ i, j. (4 . e = {| e i i} ni =1 for H , if U = [ u ij ] is an n × n unitarymatrix such that | u ij | < i, j , then the bases U e := { P nj =1 u ij | e j i} ni =1 and e are disjoint. Especially, if F n is the n -order quantum Fourier transform, i.e. F n = 1 √ n [ ω ( k − j − n ] ( ω n = e πn i ) , whose ( k, j )-entry is u kj = 1 √ n ω ( k − j − n ( k, j = 1 , , . . . , n ) , then e and F n e are disjoint. Lemma 4.1. If | x i is a pure state in a Hilbert state H (dim( H ) ≥ and T is abounded linear operator on H with ≤ T ≤ | x ih x | , then T = r | x ih x | for real number ≤ r ≤ . Proof. Put M = C | x i , then H = M ⊕ M ⊥ . In this decomposition, we have | x ih x | = (cid:18) (cid:19) , T = (cid:18) r 00 0 (cid:19) since ker( | x ih x | ) ⊂ ker( T ). Since 0 ≤ T ≤ | x ih x | , we have 0 ≤ r ≤ 1. From theserepresentations, we see that T = r | x ih x | . The proof is completed.13 heorem 4.1. Let M A be a set of POVMs on A and ρ AB ∈ D AB . Suppose thatthere exist two disjoint bases e = {| e i i} d B i =1 and f = {| f i i} d B i =1 for H B and there aretwo POVMs P = { P i : i = 1 , , . . . , d B } and Q = { Q i : i = 1 , , . . . , d B } such that tr A (( P i ⊗ B ) ρ AB ) = c i | e i ih e i | ( i = 1 , , . . . , d B ) , (4 . A (( Q i ⊗ B ) ρ AB ) = d i | f i ih f i | ( i = 1 , , . . . , d B ) , (4 . with c i d i > i = 1 , , . . . , d B ) . Then ρ AB is steerable from A to B with respect toany M A containing POVMs P and Q . Proof. In our setting, m A = o A = d B . Suppose that the state ρ AB is unsteerablefrom A to B with respect to some M A containing P and Q . Then by definition,there exists a PD { π k } dk =1 and a group of states { σ k } dk =1 ⊂ D B such that for every M = { E i } mi =1 in M A , it holds thattr A (( E i ⊗ B ) ρ AB ) = d X k =1 π k P A ( i | M, k ) σ k , ∀ i = 1 , , . . . , m, (4 . P A ( i | M, k ) ≥ P mi =1 P A ( i | M, k ) = 1( k = 1 , , . . . , d ) . In this case, d X k =1 π k σ k = ρ B . (4 . P = { P i } d B i =1 and Q = { Q i } d B i =1 , respectively, and combiningEqs. (4.2) and (4.3), we obtain that d X k =1 π k P A ( i | P, k ) σ k = c i | e i ih e i | ( i = 1 , , . . . , d B ) , (4 . d X k =1 π k P A ( i | Q, k ) σ k = d i | f i ih f i | ( i = 1 , , . . . , d B ) . (4 . ≤ c − i π k P A ( i | P, k ) σ k ≤ | e i ih e i | ( i = 1 , , . . . , d B )for each k = 1 , , . . . , d . Therefore, Lemma 4.1, we know that for each k = 1 , , . . . , d and each i = 1 , , . . . , d B , there exists a ik ∈ [0 , 1] such that c − i π k P A ( i | P, k ) σ k = a ki | e i ih e i | . Because that P d B i =1 P A ( i | P, k ) = 1 for all k = 1 , , . . . , d, we conclude that for each k ,there exists an i k such that P A ( i k | P, k ) = 0 and so π k σ k = c i k a ki P A ( i k | P, k ) | e i k ih e i k | . This shows that { π σ , π σ , . . . , π d σ d } ⊂ d B [ i =1 ( R | e i ih e i | ) := S P . Similarly, { π σ , π σ , . . . , π d σ d } ⊂ d B [ i =1 ( R | f i ih f i | ) := S Q . { π σ , π σ , . . . , π d σ d } ⊂ S P T S Q . Since e and f are disjoint, S P T S Q = { } and so π k σ k = 0 for all k = 1 , , . . . , d . This contradicts Eq. (4.5). The proof iscompleted. Corollary 4.1. Let {| ε i i} ni =1 be a real orthonormal basis for H A = H B = C n and | ψ i = √ n P ni =1 | ε i i| ε i i . Then ρ AB = | ψ ih ψ | is steerable from A to B with re-spect to any M A containing POVMs P and Q , in which P = {| e i ih e i |} ni =1 and Q = {| f j ih f j |} nj =1 where e = {| e i i} ni =1 is any basis for H A = C n and f = F n e = {| f i i} ni =1 . Proof. First we compute thattr A [( | x ∗ ih x ∗ | ⊗ I B ) ρ AB ] = 1 n | x ih x | , ∀| x i ∈ C n , where | x ∗ i denotes the conjugation of | x i . Since f and e are disjoint bases andtr A (( | e ∗ i ih e ∗ i | ⊗ I B ) ρ AB ) = 1 n | e i ih e i | ( i = 1 , , . . . , n ) , tr A (( | f ∗ i ih f ∗ i | ⊗ I B ) ρ AB ) = 1 n | f i ih f i | ( i = 1 , , . . . , n ) , we see from Theorem 4.1 that ρ AB = | ψ ih ψ | is steerable from A to B with respect toany M A containing POVMs P and Q . The proof is completed. Example 4.1. The bipartite maximally entangled state | ψ i AB = 1 √ | i + | i ) , i.e. ρ AB = | ψ ih ψ | , is steerable from A to B with respect to any M A containingPOVMs {| ih | , | ih |} and {| f ih f | , | f ih f |} where | f i = 1 √ , T , | f i = 1 √ , − T . Proof. Use Corollary 4.1 for | ε i = | e i = | i , | ε i = | e i = | i . The proof iscompleted.The following result shows that steerability is invariant under a local unitarytransformation. Theorem 4.2. Let ρ ∈ D AB and let U : H A → K A and V : H B → K B be unitaryoperators, ρ ′ = ( U ⊗ V ) ρ ( U † ⊗ V † ) , and let M = { M x : x = 1 , , . . . , m A } be a set ofPOVMs (resp. projection measurements) of system H A . Then(1) ρ ′ ∈ D ( K A ⊗ K B ) .(2) U M U † := { U M x U † : x = 1 , , . . . , m A } is a set of POVMs (resp. resp. pro-jection measurements) of system K A where U M x U † = { U M a | x U † : a = 1 , , . . . , o A } if M x = { M a | x : a = 1 , , . . . , o A } .(3) ρ is unsteerable from A to B with M if and only if ρ ′ is unsteerable from A to B with U M U † .(4) ρ is unsteerable from A to B if and only if ρ ′ is unsteerable from A to B .(5) ρ is unsteerable if and only if ρ ′ is unsteerable.(6) ρ is steerable if and only if ρ ′ is steerable. Proof. (1) Denote ρ ′ = ( U ⊗ V ) ρ ( U † ⊗ V † ). For any | x i ∈ K A ⊗ K B , by writing | y i = ( U † ⊗ V † ) | x i = | ( U † ⊗ V † ) x i we have h x | ρ ′ | x i = h ( U † ⊗ V † ) x | ρ | ( U † ⊗ V † ) x i = h y | ρ | y i ≥ , and so ρ ′ ≥ 0. For any orthonormal basis {| x i i} for K A ⊗K B , we have { ( U † ⊗ V † ) | x i i} is an orthonormal basis for H A ⊗ H B and sotr ρ ′ = X i h x i | ( U ⊗ V ) ρ ( U † ⊗ V † ) | x i i = X i h ( U † ⊗ V † ) x i | ρ | ( U † ⊗ V † ) x i i = 1 . ρ ′ ∈ D ( K A ⊗ K B ).(2) Clearly.(3) Suppose that ρ is unsteerable from A to B with respect to M , then thereexists a PD { π k } and states σ k ∈ D B such thattr A (( M a | x ⊗ I B ) ρ ) = X k π k P A ( M a | x , k ) σ k , ∀ a, x (4 . P A ( M a | x , k ) ≥ P a P A ( M a | x , k ) = 1 . When ρ = C ⊗ D , we computethat ρ ′ = U CU † ⊗ V DV † and so for every operator T on H A , tr A [( U T U † ⊗ I B ) ρ ′ ] = tr A ( U T CU † ⊗ V DV † )= tr( T C ) · V DV † = V · tr A [( T ⊗ I B )( C ⊗ D )] · V † = V · tr A [( T ⊗ I B ) ρ ] · V † . Generally, by writing ρ = P j C j ⊗ D j we get thattr A [( U T U † ⊗ I B ) ρ ′ ] = V · tr A [( T ⊗ I B ) ρ ] · V † . By using this identity for T = M a | x and Eq. (4.8), we see thattr A (( U M a | x U † ⊗ I B ) ρ ′ ) = X k π k P A ( M a | x , k ) V σ k V † , ∀ a, x. Since { V σ k V † } ⊂ D ( K B ), we conclude that ρ ′ is unsteerable from A to B with respectto U M U † . By using this conclusion, we see that if ρ ′ is unsteerable from A to B with U M x U † , then ρ is unsteerable from A to B with respect to U † ( M x U † ) U = M .(4)-(6): Use (1)-(3). The proof is completed. Corollary 4.2. Suppose that {| θ i i} ni =1 and {| η i i} ni =1 are bases for H A = H B = C n , | ϕ i = √ n P ni =1 | θ i i| η i i . For any basis e = {| e i i} ni =1 for C n , let P = {| e i ih e i |} ni =1 , Q = {| f j ih f j |} nj =1 where f = F n e = {| f i i} ni =1 . Then ρ AB = | ϕ ih ϕ | is steerable from A to B with respect to any M A containing U † P U and U † QU where U is the unitary operatoron C n satisfying U | θ i i = | ε i i ( ∀ i ) and {| ε i i} ni =1 is a real ONB for C n . Proof. Let V be the unitary operator on C n such that V | η i i = | ε i i for all i = 1 , , . . . , n . Since ( U ⊗ V ) | ϕ i = 1 √ n n X i =1 | ε i i| ε i i := | ψ i ,ρ ′ := ( U ⊗ V ) ρ ( U † ⊗ V † ) = | ψ ih ψ | , which is steerable from A to B with respect toany M A containing P and Q (Corollary 4.1). Therefore, Theorem 4.2 yields that ρ is steerable from A to B with respect to any M A containing U † P U and U † QU . Theproof is completed. Example 4.2. The bipartite maximally entangled state | ψ i AB = 1 √ | i + | i ) , i.e. ρ AB = | ψ ih ψ | is steerable from A to B with respect to any M A containing {| ih | , | ih |} and {| f ih f | , | f ih f |} where | f i = 1 √ , T , | f i = 1 √ , − T . roof. Use Corollary 4.2 for | θ i = | e i = | ε i = | i , | θ i = | e i = | ε i = | i , | η i = | i , | η i = | i , U = I. The proof is completed. Lemma 4.2. Let ε = {| i i} ni =1 be the canonical − basis for C n , | ψ i = P ri =1 µ i | i i| i i with < r ≤ n be an entangled pure state of C n ⊗ C n . Put f = F n ε = {| f i i} ni =1 , P = {| i ih i |} ni =1 and Q = {| f j ih f j |} nj =1 . Then ρ AB = | ψ ih ψ | is steerablefrom A to B with respect to any M A containing POVMs P and Q . Proof. We compute thattr A [( | x ih x | ⊗ I B ) ρ AB ] = | x ⋆ ih x ⋆ | , ∀| x i = n X k =1 a k | k i ∈ C n , (4 . | x ⋆ i = P rk =1 µ k a k ∗ | k i . Especially,tr A [( | i ih i | ⊗ I B ) ρ AB ] = | i ⋆ ih i ⋆ | , tr A [( | f j ih f j | ⊗ I B ) ρ AB ] = | f ⋆j ih f ⋆j | , (4 . i, j = 1 , , . . . , n .Suppose that the state ρ AB is unsteerable from A to B with respect to some M A containing P and Q . Then by definition, there exists a PD { π k } dk =1 with positiveprobabilities and a group of states { σ k } dk =1 ⊂ D B such that for every M = { E i } mi =1 in M A , it holds thattr A [( E i ⊗ B ) ρ AB ] = d X k =1 π k P A ( i | M, k ) σ k , ∀ i = 1 , , . . . , m, (4 . P A ( i | M, k ) ≥ P mi =1 P A ( i | M, k ) = 1( k = 1 , , . . . , d ) . In this case, d X k =1 π k σ k = ρ B . (4 . P and Q , respectively, and combining Eq. (4.10), we obtainthat d X k =1 π k P A ( i | P, k ) σ k = | i ⋆ ih i ⋆ | ( i = 1 , , . . . , n ) , (4 . d X k =1 π k P A ( j | Q, k ) σ k = | f ⋆j ih f ⋆j | ( j = 1 , , . . . , n ) . (4 . | i ⋆ i = µ i | i i (1 ≤ i ≤ r ) , | i ⋆ i = 0( r < i ≤ n ); | f ⋆j i = r X k =1 µ k b ( j ) k | k i (1 ≤ j ≤ n ) , where b ( j ) k = h j |F n | k i satisfy | f j i = P nk =1 b ( j ) k | k i . From the structure of The Fouriertransformation F n , we know that each b ( j ) k is not zero.From Eq. (4.14) and Lemma 4.1, we know that for each k = 1 , , . . . , d and each i = 1 , , . . . , r , there exists a ik = 0 such that π k P A ( i | P, k ) σ k = a ik | i ⋆ ih i ⋆ | . Because that P ri =1 P A ( i | P, k ) = 1 for all k = 1 , , . . . , d, we conclude that for each k ,there exists an 1 ≤ i k ≤ r such that P A ( i k | P, k ) > π k σ k = a ki P A ( i k | P, k ) | ε ⋆i k ih ε ⋆i k | . k , there exists an 1 ≤ j k ≤ n such that P A ( j k | Q, k ) > π k σ k = b ki P A ( j k | Q, k ) | f ⋆j k ih f ⋆j k | , where b ki = 0. Thus, | ε ⋆i k i = c k | f ⋆j k i ( k = 1 , , . . . , d ) for some nonzero constants c k ,that is, µ i k | ε i k i = c k r X m =1 µ m b ( j k ) m | ε m i . This shows that b ( j k ) m = 0 for all m = i k . Since r > 1, such an m does exist. Thiscontradicts the fact that b ( j ) k = 0 for all k, j . Therefore, ρ AB = | ψ ih ψ | is steerablefrom A to B with respect to any M A containing POVMs P and Q . The proof iscompleted. Theorem 4.3. Let | ψ i be an entangled pure state of C n ⊗ C n . Then there existtwo POVMs P and Q such that ρ AB = | ψ ih ψ | is steerable from A to B with respectto any M A containing POVMs P and Q . Proof. Since | ψ i is an entangled pure state of C n ⊗ C n , it has Schmidt decompo-sition | ψ i = P ri =1 µ i | ε i i| η i i where {| ε i i} ni =1 and {| η i i} ni =1 are orthonormal bases for C n and µ i > i = 1 , , . . . , r with 1 < r ≤ n . Choose unitary operators U and V on C n such that | ε i i = U | i i , | η i i = V | i i ( i = 1 , , . . . , n )where {| i i} ni =1 is the canonical 0 − C n . Since ( U † ⊗ V † ) | ψ i = P ri =1 µ i | i i| i i ,it follows from Lemma 4.2 that ( U † ⊗ V † ) ρ AB ( U ⊗ V ) is steerable from A to B withany M A containing POVMs {| i ih i |} ni =1 and {F n | j ih j |F † n } nj =1 . By using Theorem 4.2,we know that the state ρ AB is steerable from A to B with any M A containing POVMs P = { U | i ih i | U † } ni =1 and Q = { U F n | j ih j |F † n U † } nj =1 . The proof is completed. In this note, we have obtained some characterizations of Bell locality and EPR steer-ability of bipartite states and proved that the set of all Bell local states and the setof all unsteerable states are both convex and compact. The compactness of thesesets are useful for quantifying Bell locality and EPR steerability. From the convexityof US ( A → B, M A ), we see that when a mixed state with spectral decomposition ρ = P i λ i | ψ i ih ψ i | is steerable from A to B with respect to M A , there exists an i such that | ψ i ih ψ i | is steerable from A to B with respect to M A . From the convex-ity of US ( A → B ), we see that when a mixed state with spectral decomposition ρ = P i λ i | ψ i ih ψ i | is steerable from A to B , there exists an M A and an i such that | ψ i ih ψ i | is steerable from A to B with respect to M A . Since S ( A → B ) is open, weconclude that when a state ρ AB is steerable from A to B , all states close to ρ AB aresteerable from A to B .We have also proved that any locally unitary operation do not change steerability.By using this fact and proving a EPR-steering criteria, we prove that any maximallyentangled pure state of C n ⊗ C n is steerable from A to B with respect to two projectionmeasurements.Moreover, convexity and compactness of BL ( AB ) implies that for every Bell non-local state σ AB , there exists a Hermitian operator L on H A ⊗ H B such thattr( L ρ AB ) ≥ ∀ ρ AB ∈ BL ( AB )) and tr( L σ AB ) < . Such an L is said to be a Bell nonlocality witness. The steerability witness can bedefined similarly. 18 eferences [1] Bell, J.S., On the Einstein Podolsky Rosen paradox, Physics 1964, 1, 195.[2] Brunner N, Cavalcanti D, Pironio S, Scarani V and Wehner S, Bell nonlocality,Rev. Mod. Phys. Rev. Mod. Phys. 2014, , 419-478[3] J.F. Clauser, A. 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