Mapping Criteria Between Nonlocality and Steerability in Qudit-Qubit Systems and Between Steerability and Entanglement in Qubit-Qudit Systems

PHYSICAL REVIEW A 98, 052114 (2018)

Mapping criteria between nonlocality and steerability in qudit-qubit systems and between steerability and entanglement in qubit-qudit systems

Changbo Chen,1,*,† Changliang Ren,2,3,*,‡ Xiang-Jun Ye,3 and Jing-Ling Chen4,5,§ 1Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, People’s Republic of China 2Center for Nanofabrication and System Integration, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, People’s Republic of China 3CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of China 4Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China 5Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543

(Received 16 March 2018; revised manuscript received 15 July 2018; published 12 November 2018)

Entanglement, quantum steering, and nonlocality are distinct quantum correlations which are the resources behind various quantum information and quantum computation applications. However, a central question of determining the precise quantitative relation among them is still unresolved. Here we present a mapping criterion between Bell nonlocality and quantum steering in the bipartite qudit-qubit system, as well as a mapping criterion between quantum steering and quantum entanglement in the bipartite qubit-qudit system, starting from the fundamental concepts of quantum correlations. Precise quantitative mapping criteria are derived analytically. Such mapping criteria are independent of the form of the state. In particular, they cover several previous well-known research results which are only special cases in our simple mapping criteria.

DOI: 10.1103/PhysRevA.98.052114

I. INTRODUCTION hierarchy of nonlocality, the set of EPR steerable states is a strict subset of entangled states and a strict superset of The distinctive nonclassical features of quantum physics Bell nonlocal states [25]. In simplicity, the strongest concept were first discussed in the seminal paper [1]byEinstein is Bell nonlocality, which implies nonclassical correlations et al., which indicated that there were some conflicts be- that cannot be described by local hidden variable (LHV) tween quantum mechanics and local realism. Immediately, the Einstein-Podolsky-Rosen (EPR) paper provoked an in- theory; quantum steering describes correlations beyond ones teresting response from Schrödinger [2,3], who introduced constrained by local hidden state (LHS) theory; the strictly the notion of entanglement and steering. Three fundamental weaker concept is that of nonseparability or entanglement, definitions, quantum entanglement [4], EPR steering [5], and where a nonseparable state is one whose joint probability Bell nonlocality [6], were intuitively elaborated, which have cannot be simulated by any separable model (SM). How- since opened an epoch of unrelenting exploration of quantum ever, the above is basically the complete knowledge of the correlations. Entanglement and Bell nonlocality have attained relations among these three different quantum correlations. flourishing developments, while EPR steering had even lacked In particular, there are very few quantitative results on the a rigorous formulation until the work of Wiseman et al. [7]. relation of such quantum correlations. Quantitatively deter- Over 80 years of investigation, physicists have scrupulously mining their difference and relation is an important task; it distinguished the notions and clarified the concepts out of helps toward a deep understanding of the nonclassical physics chaos. These concepts have nowadays become the center described by quantum mechanics and provides a verification of quantum foundations and have found many practical ap- of them in terms of their usefulness for various quantum plications in modern quantum information theory including information applications. In this paper, a mapping criterion quantum key distribution [8–12], communication complexity of entanglement-steerability in the qudit-qubit system and a [13,14], cloning of correlations [15,16], quantum metrology mapping criterion of steerability-nonlocality in the qubit-qudit [17], quantum state merging [18,19], remote state preparation system are derived from their fundamental definitions. As a [20], and random number generation [21]. result, we are able to prove that a difficultly verified quantum Through decades of investigation, a great number of fruit- correlation can be translated into an easily verified problem. ful results on characterizing the properties of these quantum This result connects the previous research of detecting Bell’s correlations have been obtained [3,22–25]. According to the nonlocality by quantum steering inequality in [26,27]tothe more recent research direction of steering. It is shown that part of these previously known result in [26,27] is only a special *These authors contributed equally to this work. case for the two-qubit system in our simple mapping criterion. †[email protected] Moreover, the perspective in this research supplies a simple ‡[email protected] way of exploring the relation of such quantum correlations §[email protected] quantitatively.

II. PRELIMINARY NOTIONS Consider a bipartite scenario composed of Alice and Bob sharing an arbitrary quantum state τAB . Suppose Alice performs a measurement A with outcome a and Bob performs a measurement B with outcome b; then these outcomes are in general governed by a joint probability distribution P (a,b | A, B, τAB ), where this joint probability distribution predicted by quantum theory is defined by | = A ⊗ B P (a,b A, B, τAB ) Tr a b τAB , (1) A B where a and b are the projective operators for Alice and Bob, respectively. Definition 1. If the joint probability satisfies FIG. 1. Venn diagram of a mapping relation between quantum nonlocality and quantum steering in Theorem 1. The states in the

P (a,b | A, B, τAB ) = P (a | A, ξ)P (b | B,ξ)Pξ dξ (2) green region are nonlocal and those in the blue region are steerable but not nonlocal. All of the states which can be described by LHS for any measurements A and B, τ has a LHV model. model are in the red region. The state ρAB is mixed by an arbitrary AB Definition 2. If the joint probability and the marginal unsteerable state τAB and the other arbitrary state τAB . Theorem 1 probability satisfy gives a mapping criterion between ρAB and τAB . The purple arrows show that if ρAB is EPR steerable, then τAB is Bell nonlocal; equivalently, if τAB is not nonlocal, then ρAB is unsteerable. P (a,b | A, B, τAB ) = P (a | A, ξ)PQ(b | B,ξ)Pξ dξ, (3) | = B B PQ(b B,ξ) Tr b ρξ (4) Note that Eqs. (8) and (9) contain the same Pξ . If there 2 + 2 + 2 exists a range of μ such that rx ry rz 1 holds for any for any measurements A and B, τAB has a LHS model, where probability distributions 0 P (a | A, ξ) 1 and 0 P (b | P ρB dξ = Tr [τ ]. ξ ξ A AB B,ξ) 1, where A is an arbitrary projective measurement, Definition 3. If the joint probability and the marginal B ∈{x,y,z}, and probability satisfy = 2η(x) − | = | | rx 1, P (a,b A, B, τAB ) PQ(a A, ξ)PQ(b B,ξ)Pξ dξ, (5) ℘(a | A, ξ) A A 2η(y) PQ(a | A, ξ) = Tr ρ , (6) r = − , a ξ y | 1 (10) ℘(a A, ξ) P (b | B,ξ) = Tr B ρB (7) 2η(z) Q b ξ r = − 1, z ℘(a | A, ξ) for any measurements A and B, τAB has a SM. η(B)=μP (a|A, ξ)P (0|B,ξ) + (1−μ)P (a|A, ξ)P (0|B,ξ), III. MAPPING CRITERION BETWEEN BELL with B ∈{x,y,z}, and ℘(a | A, ξ) = μP (a | A, ξ) + (1 − NONLOCALITY AND QUANTUM STEERING μ)P (a | A, ξ), then when μ falls into this range, one can = M In what follows we present a mapping criterion between construct a LHS model for ρAB (τAB ). Bell nonlocality and quantum steering. A curious quantum Proof. Let the measurement settings at Bob’s side be phenomenon directly connecting these two different types chosen as x,y,z. Since the state τAB has a LHV model = of quantum correlations was proposed. We find that Bell description, based on Eq. (8), we explicitly have (with B nonlocal states can be constructed from some EPR steerable x,y,z) states, which indicates that Bell’s nonlocality can be detected | = | | indirectly through EPR steering (see Fig. 1) and offers a P (a,0 A, B, τAB ) P (a A, ξ)P (0 B,ξ)Pξ dξ, distinctive way to study Bell’s nonlocality. The result can be expressed as the following theorem. P (a,1|A, B, τ ) = P (a|A, ξ)P (1|B,ξ)P dξ. (11) Theorem 1. In a bipartite qudit-qubit system, we define AB ξ M → + − amap : τAB μτAB (1 μ)τAB ,0 μ 1, where We now turn to study the EPR steerability of ρAB .After τAB is an arbitrary bipartite qudit-qubit state shared by Alice Alice performs the projective measurement on her qubit, the and Bob and τAB is a bipartite qudit-qubit state constructed in state ρAB collapses to Bob’s conditional states (unnormalized) such a way that whenever τAB has a LHV model as | = | | P (a,b A, B, τAB ) P (a A, ξ)P (b B,ξ)Pξ dξ, (8) A = A ⊗ = − ρã TrA a 1 ρAB ,a 0,...,d 1. (12) τ also has a LHV model AB To prove that there exists a LHS model for ρAB , it suffices to A prove that, for any projective measurement a and outcome P (a,b | A, B, τ ) = P (a | A, ξ)P (b | B,ξ)Pξ dξ. (9) AB a, one can always find a hidden state ensemble {℘ξ ρξ } and the

052114-2 MAPPING CRITERIA BETWEEN NONLOCALITY AND … PHYSICAL REVIEW A 98, 052114 (2018) conditional probabilities ℘(a | A, ξ) such that the relation Similarly, because ν + ν A y A 11 22 ρ˜ = ℘(a | A, ξ)ρ ℘ dξ (13) Tr ρ˜ = − Im[ν12], a ξ ξ 0 a 2

with Im[ν12] being the imaginary part of ν12,wehave is always satisfied. Here ξ is a local hidden variable, ρξ is a | hidden state, ℘ξ is a probability density function, and ℘(a 1 = | −Im[ν12] = η(y) − ℘(a | A, ξ) Pξ dξ. A, ξ) are probabilities satisfying ℘ξ dξ 1 and a ℘(a 2 A, ξ) = 1. Indeed, if Eq. (13) is satisfied, then Eq. (3) holds B A On the other hand, we have by calculating Tr[ b ρã ]. × Each ρξ is a 2 2 density matrix which can be written ν − ν 1 + 11 22 1 σ rξ × = η(z) − ℘(a | A, ξ) Pξ dξ. in the form of 2 , where 1 is the 2 2 identity matrix, 2 2 σ = (σx ,σy ,σz ) is the vector of the Pauli matrices, and rξ = 2 + 2 + 2 Note that the following decomposition holds: (rx,ry ,rz ) is the Bloch vector satisfying rx ry rz 1. A solution of Eq. (13) can be given as ν + ν A = 11 22 + ρã 1 Re[ν12]σx ℘(a | A, ξ) = μP (a | A, ξ) + (1 − μ)P (a | A, ξ), 2 ν − ν 1 +σ ·r − Im[ν ] σ + 11 22 σ . ℘ = P ,ρ= ξ , (14) 12 y 2 z ξ ξ ξ 2 By combining the above equations, we finally deduce that where the hidden state ρξ has been parametrized in the Bloch- Eq. (15) holds. Thus, if there is a LHV model description = vector form, with rξ (rx,ry ,rz ) defined in Eq. (10). The for τ , then there is a LHS model description for ρ .This | | AB AB assumption that rξ 1 ensures that ρξ is a density matrix. completes the proof. By substituting Eq. (14) into Eq. (13), we obtain Remark 1. Provided the conditions in Theorem 1 are met, +· Theorem 1 actually provides a way to prove the following A 1 σ rξ ρ˜ = ℘(a | A, ξ) Pξ dξ. (15) important property: If ρ is EPR steerable from A to B, then a 2 AB τAB is Bell nonlocal. Otherwise, if τAB is not Bell nonlocal, To prove the theorem is to verify that the relation (15)is there will be a LHS model for ρAB . satisfied. As a direct application of Theorem 1, we have the follow- Let us calculate the left-hand side of Eq. (15). One has ing corollary. A A Corollary 1. For any bipartite qudit-qubit state τAB shared ρ˜ = TrA ⊗ 1 ρAB a a by Alice and Bob, define another state = A ⊗ + − A ⊗ μ TrA a 1 τAB (1 μ)TrA a 1 τAB . = + − ρAB μτAB (1 μ)τAB , (16) × A + For convenience, let us define the 2 2matrix˜ρa as = ⊗ 1 cσ3 = with τAB τA , τA TrB [τAB ], and the conditions √ 2 √ ν ν 3 1−2μ2−μ A = 11 12 0 μ and 0 c .IfρAB is EPR steerable ρã 3 1−μ ν21 ν22 from A to B, then τAB is Bell nonlocal. and calculate its each element. Obviously, we get Proof. Assume that τAB is not Bell nonlocal, that is, it has a LHV model. Then we have = z A ν11 Tr ρã 0 | = | | = | + − | P (a,b A, B, τAB ) P (a A, ξ)P (b B,ξ)Pξ dξ. μP(a,0 A, z, τAB ) (1 μ)P (a,0 A, z, τAB ) Note that we have and similarly = | + − | 1 b 1 ν22 μP (a,1 A, z, τAB ) (1 μ)P (a,1 A, z, τAB ). P (a,b | A, B, τ )= P (a | A, ξ) (−1) cB + P dξ. AB 2 z 2 ξ + = A = | Note that we have ν11 ν22 Tr[ρ ã ] μP (a A, τAB ) + − | If we let (1 μ)P (a A, τAB ). With the help of Eq. (11) and using P (a|A, ξ)Pξ dξ = P (a|A, τAB ) and P (a|A, ξ)Pξ dξ = P (a | A, ξ) = P (a | A, ξ), | P (a A, τAB ), we have | = 1 − b + 1 P (b B,ξ) 2 ( 1) cBz 2

(ν11 + ν22 ) = ℘(a | A, ξ)Pξ dξ. and substitute them into Eq. (10) in Theorem 1, we have rx = 2μP (0 | x,ξ) − μ, ry = 2μP (0 | y,ξ) − μ, and Because rz = 2μP (0 | z, ξ ) + c − μc − μ. To satisfy the assumption + |rξ | 1 in Theorem 1, it is equivalent to solving the following x A ν11 ν12 Tr ρ˜ = + Re[ν12], real quantiﬁer elimination [28] problem: 0 a 2 0c 1∧0μ1∧[∀P (0 | x,ξ),P(0 | y,ξ),P(0 | z, ξ ), with Re[ν12] being the real part of ν12,wehave

0P (0 | x,ξ)1∧0P (0 | y,ξ)1∧0P (0 | z, ξ ) = − 1 | Re[ν12] η(x) ℘(a A, ξ) Pξ dξ. ⇒ 2 + 2 + 2 2 1 rx ry rz 1].

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shown that steerable states can be constructed from some entangled states, which indicates that EPR steering can be detected indirectly through entanglement (see Fig. 2) and offers a distinctive way to study EPR steering. The result can be expressed as the following theorem. Theorem 2. In a bipartite qubit-qudit system, we deﬁne N → + − amap : ρAB μρAB (1 μ)ρAB ,0 μ 1, where ρAB is an arbitrary bipartite qubit-qudit state shared by Alice and Bob and ρAB is a bipartite qubit-qudit state constructed in such a way that whenever ρAB has a LHS model

P (a,b | A, B, ρAB ) = P (a | A, ξ)PQ(b | B,ξ)Pξ dξ, FIG. 2. Venn diagram of a mapping relation between quantum steering and quantum entanglement in Theorem 2. The states in ρ also has a LHS model the blue region are steerable and those in the yellow region are AB unsteerable but entangled. All of the states which can be described by the SM are in the red region. The state σAB is mixed by an arbitrary P (a,b | A, B, ρ ) = P (a | A, ξ)PQ(b | B,ξ)Pξ dξ. AB separable state ρAB and the other arbitrary state ρAB . Theorem 2 gives a mapping criterion between ρAB and σAB . The purple arrows mean | σ ρ Note that the above two equations have the same PQ(b B,ξ) that if AB is entangled, then AB is EPR steerable; equivalently, if 2 2 2 and Pξ . If there exists a range of μ such that r + r + r 1 ρAB is unsteerable, then σAB is separable. x y z holds for any probability distributions 0 P (0 | A, ξ) 1, √ where A ∈{x,y,z} and 3 It is not hard√ to show that the solution is 0 μ 3 and = | + − | − 1−2μ2−μ rx 2[μP (0 x,ξ) (1 μ)P (0 x,ξ)] 1, 0 c 1−μ . Since μ falls into the range required, the r = 2[μP (0 | y,ξ) + (1 − μ)P (0 | y,ξ)] − 1, (18) conditions of Theorem 1 are met. Thus ρAB has a LHS model, y which is a contradiction. rz = 2[μP (0 | z, ξ ) + (1 − μ)P (0 | z, ξ )] − 1, Remark 2. This inspiring result clearly explore a curious quantum phenomenon: Bell nonlocal states can be constructed then when μ falls into this range, one can construct a SM for from steerable states. Such a ﬁnding not only offers a distinc- σAB = N (ρAB ). tive way to study Bell’s nonlocality without Bell’s inequality Proof. To prove that σAB has a SM description is equivalent but with steering inequality, but also may avoid the locality to proving that the following equation has a solution: loophole in Bell’s tests and make Bell’s nonlocality easier for demonstration. Interestingly, we can easily extract a simple | = | | corollary, namely, Corollary 2, from Corollary 1, which is a P (a,b A, B, σAB ) ℘Q(a A, ξ)℘Q(b B,ξ)℘ξ dξ. well-known result derived in [26,27]. (19) Corollary 2. For any any bipartite qudit-qubit state τAB shared by Alice and Bob, deﬁne another state A solution is given by ρAB = μτAB + (1 − μ)τ , (17) AB A A ℘Q(a | A, ξ) = Tr ρ , with τ = τ ⊗ 1/2, τ = Tr [τ ] = Tr [ρ ] being the a ξ AB A A B AB B AB | = | = reduced density matrix at Alice’s side, and μ = √1 .Ifρ is ℘Q(b B,ξ) PQ(b B,ξ),℘ξ Pξ , 3 AB EPR steerable from A to B, then τ is Bell nonlocal. It is a AB 1+σ ·rA = A = ξ A = special case of Corollary 1. When d 2, it will give us the where ρξ 2 and rξ (rx ,ry ,rz ), with rx ,ry ,rz given derived results [26,27] for the two-qubit system. in Eq. (18). The assumption that |rξ | 1 ensures that ρξ is a Proof. It is proved by setting μ = √1 and c = 0in density matrix. 3 Corollary 1. Next we prove that the above solution makes Eq. (19) hold. It is easy (by hand or a computer algebra system) to check that IV. MAPPING CRITERION BETWEEN QUANTUM 1 + (−1)aA ·rA STEERING AND QUANTUM ENTANGLEMENT ℘ (a | A, ξ) = ξ . Q 2 Similarly, a mapping criterion between quantum steering and quantum entanglement can be precisely derived. It is Thus we have

1 (−1)a ℘ (a | A, ξ)℘ (b | B,ξ)℘ dξ = P (b | B,ξ)P dξ + (A r + A r + A r )P (b | B,ξ)P dξ. (20) Q Q ξ 2 Q ξ 2 x x y y z z Q ξ

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On the other hand, we have | = | + − | − | Ax rxPQ(b B,ξ)Pξ dξ 2μAx P (0,b x,B,ρAB ) 2(1 μ)Ax P (0,b x,B,ρAB ) Ax PQ(b B,ξ)Pξ dξ,

(−1)a 1 (−1)aA 1 + (−1)aA σ A P (0,b | x,B,ρ ) = Tr x x x ⊗ B ρ , 2 x AB 2 2 b AB × B = | Tr 1 b ρAB PQ(b B,ξ)Pξ dξ,

2 + 2 + 2 combining with Eqs. (20) and (21), we have Eq. (19). This To satisfy the assumption rx ry rz 1 in Theorem 2, proves the theorem. it is equivalent to solving exactly the same real quantifier Remark 3. Provided the conditions in Theorem 2 are met, elimination problem as the one in Corollary 1, whose solution √ √ Theorem 2 provides a way to prove the following important 3 1−2μ2−μ is 0 μ and 0 c − . Since μ falls into the property: If σAB is entangled, ρAB is EPR steerable in the 3 1 μ range required, the conditions of Theorem 2 are met. Thus sense that Alice can steer Bob. Otherwise, if ρAB is not EPR σAB has a SM, which is a contradiction. steerable from A to B, there will be a SM description for σAB . As a direct application of Theorem 2, we have the follow- Corollary 4. For an arbitrary bipartite qubit-qudit state ρAB shared by Alice and Bob, one can map it into a new state ing result. 1 defined by σAB = μρAB + (1 − μ)ρ , with ρ = ⊗ ρB , Corollary 3. For an arbitrary bipartite qubit-qudit state ρAB √AB AB 2 shared by Alice and Bob, define where ρB = TrA[ρAB ] and μ = 1/ 3; if σAB is entangled = + − state, then ρAB is the steerable state in the sense that Alice σAB μρAB (1 μ)ρAB , (22) can steer Bob. When d = 2, it will reduce to the two-qubit + = 1 cσ3 ⊗ = system. with ρAB ρB and ρB TrA[ρAB ], with the condi- 2√ √ 1−2μ2−μ Proof. It can be deduced directly from Corollary 3 by 3 1 tions 0 μ and 0 c − .IfσAB is entan- setting μ = √ and c = 0. 3 1 μ 3 gled state, then ρAB is the steerable state in the sense that Alice can steer Bob. Proof. Assume that ρAB is not steerable, that is, it has a V. CONCLUSION LHS model We presented not only a mapping criterion between Bell nonlocality and quantum steering, but also a mapping criterion P (a,b | A, B, ρAB ) = P (a | A, ξ)PQ(b | B,ξ)Pξ dξ. between quantum steering and quantum entanglement, start- Since the marginal probability satisfies ing from these fundamental concepts of quantum correlations. Many quantitative results on the relation of such quantum P (b | B,ρB ) = PQ(b | B,ξ)Pξ dξ, correlations were derived. It was shown that part of these previously known results in [26,27] is only a special case in we have our simple mapping criterion. Our result not only pinpoints a deep connection among quantum entanglement, quantum P (a,b | A, B, ρ ) AB steering, and Bell nonlocality, but also provides a feasible 1 + (−1)acA approach to experimentally test a difficultly verified quantum = z P (b | B,ξ)P dξ. 2 Q ξ correlation by translating it into an easily verified problem. + − a The method we use in the present paper provides a different | = 1 ( 1) cAz If we set P (a A, ξ) 2 and substitute it into perspective to understand various quantum correlations and Eq. (18), we have shines light on the intricate relations among them. As we showed with concrete examples, this connection allows us rx = 2μP (0 | x,ξ) − μ, ry = 2μP (0 | y,ξ) − μ, to translate results from one concept to another. There is = | + − − rz 2μP (0 z, ξ ) c μc μ. no doubt that this method is easily extendable, so for future

052114-5 CHEN, REN, YE, AND CHEN PHYSICAL REVIEW A 98, 052114 (2018) work it would be very interesting to use such a method to China (Grants No. 11771421, No. 11471307, No. 61572024, explore many different mapping criteria especially in higher and No. 11671377), the Key Research Program of Fron- dimensions. Another open question is that such a mapping tier Sciences of CAS (Grant No. QYZDB-SSW-SYS026), criterion between Bell nonlocality and quantum entanglement and EU H2020-FETOPEN-2016-2017-CSA project SC2 is still unknown. If such a mapping criterion exists, which in- (No. 712689). C.R. was supported by National key re- dicates that Bell nonlocality can be detected indirectly through search and development program (No. 2017YFA0305200), quantum entanglement, deﬁnitely, it will supply a distinctive the Youth Innovation Promotion Association (CAS) (Grant way to avoid the locality loophole in Bell tests and make No. 2015317), the National Natural Science Foundation of Bell nonlocality easier for demonstration. Hence, this open China (Grant No. 11605205), the Natural Science Founda- question is also important enough to deeply explore. tion of Chongqing (Grants No. cstc2015jcyjA00021 and No. cstc2018jcyjA2509), the Entrepreneurship and Innovation Support Program for Chongqing Overseas Returnees (Grants ACKNOWLEDGMENTS No. cx2017134 and No. cx2018040), and the fund of CAS The authors would like to thank anonymous referees Key Laboratory of Quantum Information. J.C. was supported for valuable comments and suggestions. C.C. was sup- by National Natural Science Foundations of China (Grants ported by the National Natural Science Foundation of No. 11475089 and No. 11875167).

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