Operational nonclassicality of local multipartite correlations in the limited-dimensional simulation scenario
C. Jebaratnam, Debarshi Das, Suchetana Goswami, R. Srikanth, A. S. Majumdar
OOperational nonclassicality of local multipartitecorrelations in the limited-dimensional simulationscenario
C. Jebaratnam
E-mail: [email protected]
S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700 098, India
Debarshi Das
E-mail: [email protected]
Centre for Astroparticle Physics and Space Science (CAPSS), Bose Institute,Block EN, Sector V, Salt Lake, Kolkata 700 091, India
Suchetana Goswami
S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700 098, India
R. Srikanth
E-mail: [email protected]
Poornaprajna Institute of Scientific Research Bangalore- 560 080, Karnataka,India
A. S. Majumdar
E-mail: [email protected]
S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700 098, IndiaFebruary 2014
Abstract.
For a bipartite local quantum correlation, superlocality refers tothe requirement for a larger dimension of the random variable in the classicalsimulation protocol than that of the quantum states that generate the correlations.In this work, we consider the classical simulation of local tripartite quantumcorrelations P among three parties A, B and C . If at least one of the bipartitions( A | BC ), ( B | AC ) and ( C | AB ) is superlocal, then P is said to be absolutelysuperlocal, whereas if all three bipartitions are superlocal, then P is said tobe genuinely superlocal. We present specific examples of genuine superlocalityfor tripartite correlations derived from three-qubit states. It is argued thatgenuine quantumness as captured by the notion of genuine discord is necessary fordemonstrating genuine superlocality. Finally, the notions of absolute and genuinesuperlocality are also defined for multipartite correlations. a r X i v : . [ qu a n t - ph ] J u l perational nonclassicality of local multipartite correlations
1. Introduction
Quantum composite systems exhibit nonclassical features such as entanglement [1]and Bell nonlocality [2]. In the Bell scenario, a correlation arising from local quantummeasurements on a composite system is nonlocal if it cannot be simulated by sharingclassical randomness, i.e., if it does not admit a local hidden variable (LHV) model [3].It is well-known that quantum mechanics is not maximally nonlocal as there arenonsignaling (NS) correlations, e.g., Popescu-Rohrlich (PR) box [4], which are morenonlocal than that allowed by quantum theory. In the framework of generalizednonsignaling theory (GNST), there have been various attempts to find out the minimalconstraints of quantum theory [5]. GNST has also been used in other contexts suchas quantum cryptography [6] and quantifying nonlocality [7].Recently, it has been shown that even some separable states demonstrateadvantage in certain quantum information tasks if they have quantum discord whichis a generalized measure of quantum correlations [8]. This initiated the study ofnonclassicality going beyond nonlocality [9]. In Ref. [10], it has been shown thatcertain separable states which have quantumness may improve certain informationtheoretic protocols if there is a constraint on the possible local hidden variable models,i.e., shared randomness between the parties is limited to be finite. This provides away for obtaining an operational meaning of the measures of quantumness such asquantum discord .While entanglement is necessary for nonlocality, not all entangled states canbe used to demonstrate nonlocality. In the context of classical simulation of localentangled states, Bowles et al [11] have defined a measure which is the minimaldimension of shared classical randomness that is needed to reproduce the statistics of agiven local entangled state. Unlike the previous works, which used unbounded sharedrandomness to simulate a given local entangled state, Bowles et al have shown thatall local entangled states can be simulated by using only finite shared randomness.In Ref. [12], Donohue and Wolfe (DW) have provided upper bounds on theminimal dimension of the shared classical randomness required to simulate any localcorrelation in a given Bell scenario. DW have demonstrated an interesting feature ofcertain local boxes called superlocality: there exist local boxes which can be simulatedby certain quantum systems of local dimension lower than that of the shared classicalrandomness needed to simulate them. That is, superlocality refers to the dimensionaladvantage in simulating certain local boxes by using quantum systems. In particular,DW [12] and also Goh et al [13] have shown that entanglement enables superlocality,however, superlocality occurs even for separable states.In Ref. [14], Jebaratnam et al have pointed out that superlocality cannot occur forarbitrary separable states. In particular, Jebaratnam et al have argued that separablestates which are a classical-quantum state [15] or its permutation can never lead tosuperlocality. Note that bipartite quantum states which are not a classical-quantumstate must have quantumness as quantified by quantum discord [8]. The observationthat the limited dimensional quantum simulation of certain local correlations requiresquantumness in the states motivated the study of nonclassicality going beyond Bellnonlocality [16]. In Ref. [14], Jebaratnam et al have considered a measure ofnonclassicality in the context of GNST called Bell strength for a family of localcorrelations and identified nonzero Bell strength as a quantification of superlocalityas well.The extension of the Bell-type scenario to more than two parties was first perational nonclassicality of local multipartite correlations genuine multipartite nonlocalty for amultipartite system occurs precisely when every bi-partition is nonlocal. By contrast, absolute nonlocality occurs when at least one bi-partition is nonlocal. Absolutemultipartite nonlocality is indicated by the violation of a Mermin inequality [19].In the present work, we are interested in providing an operational characterizationof genuine and absolute nonclassicality via superlocality in the scenario of localtripartite systems and its generalization to higher-particle systems. It is arguedthat genuine quantum discord is necessary for demonstrating genuine superlocality.This implies that genuine superlocality provides an operational characterization ofgenuine quantum discord. The concept of superlocality is generalized for multipartitecorrelations also. As specific examples, we consider tripartite correlations that havenon-vanishing Svetlichny strength or Mermin strength [26].The organization of the paper is as follows. In Sec. 2, we present the mathematicaltool that we use for the purpose of studying superlocality of local tripartite boxes,namely a polytope of tripartite nonsignaling boxes with two-binary-inputs-two-binary-outputs. In Sec. 3, we define absolute and genuine superlocality for tripartite boxesand we present examples of two families of tripartite local correlations that areabsolutely superlocal or genuinely superlocal, having their nonclassicality quantifiedin terms of nonzero
Svetlichny strength and nonzero
Mermin strength , respectively.In Sec. 4, we demonstrate the connection between genuine superlocality and genuinequantum discord. In Sec. 5, we generalize the notion of superlocality to n -partitecorrelations. Sec. 6 is reserved for certain concluding remarks.
2. Preliminaries
We are interested in quantum correlations arising from the following tripartite Bellscenario. Three spatially separated parties, say Alice, Bob, and Charlie, perform threedichotomic measurements on a shared tripartite quantum state ρ ∈ H A ⊗ H B ⊗ H C ,where H K denotes Hilbert space of k th party. In this scenario, a correlationbetween the outcomes is described by the set of conditional probability distributions P ( abc | xyz ), where x , y , and z denote the inputs (measurement choices) and a , b and c denote the outputs (measurement outcomes) of Alice, Bob and Charlie respectively(with x, y, z, a, b, c ∈ { , } ). Suppose M aA x , M bB y and M cC z denote the measurement perational nonclassicality of local multipartite correlations P ( abc | xyz ) = Tr (cid:16) ρM aA x ⊗ M bB y ⊗ M cC z (cid:17) . (1)In particular, we focus on the scenario where the parties share a three-qubit stateand perform spin projective measurements A x = ˆ a x .(cid:126)σ , B y = ˆ b y .(cid:126)σ , and C z = ˆ c z .(cid:126)σ .Here ˆ a x , ˆ b y , and ˆ c z are unit Bloch vectors denoting the measurement directions and (cid:126)σ = { σ , σ , σ } , with { σ i } i =1 , , being the Pauli matrices.The set of nonsignaling (NS) boxes with two binary inputs and two binary outputsforms a convex polytope N in a 26 dimensional space [31] and includes the set ofquantum correlations Q as a proper subset. Any box belonging to this polytope canbe fully specified by 6 single-party, 12 two-party and 8 three-party expectations, P ( abc | xyz ) = 18 [1 + ( − a (cid:104) A x (cid:105) + ( − b (cid:104) B y (cid:105) + ( − c (cid:104) C z (cid:105) + ( − a ⊕ b (cid:104) A x B y (cid:105) + ( − a ⊕ c (cid:104) A x C z (cid:105) + ( − b ⊕ c (cid:104) B y C z (cid:105) + ( − a ⊕ b ⊕ c (cid:104) A x B y C z (cid:105) ] , (2)where (cid:104) A x (cid:105) = (cid:80) a ( − a P ( a | x ), (cid:104) A x B y (cid:105) = (cid:80) a,b ( − a ⊕ b P ( ab | xy ) and (cid:104) A x B y C z (cid:105) = (cid:80) a,b,c ( − a ⊕ b ⊕ c P ( abc | xyz ), ⊕ denotes modulo sum 2. The set of boxes that can besimulated by a fully LHV model are of the form, P ( abc | xyz ) = d λ − (cid:88) λ =0 p λ P λ ( a | x ) P λ ( b | y ) P λ ( c | z ) , (3)with (cid:80) d λ − λ =0 p λ = 1 and it forms the fully local (or, 3-local) polytope [32, 33] denotedby L . Here λ denotes shared classical randomness which occurs with probability p λ .For a given fully local box, the form (3) determines a classical simulation protocolwith dimension d λ [12]. The extremal boxes of L are 64 local vertices which are fullydeterministic boxes, P αβγ(cid:15)ζηD ( abc | xyz ) = , a = αx ⊕ βb = γy ⊕ (cid:15)c = ζz ⊕ η , otherwise . (4)Here, α, β, γ, (cid:15), ζ, η ∈ { , } . The above boxes can be written as theproduct of deterministic distributions corresponding to Alice and Bob-Charlie, i.e., P αβγ(cid:15)ζηD ( abc | xyz ) = P αβD ( a | x ) P γ(cid:15)ζηD ( bc | yz ), here P αβD ( a | x ) = (cid:26) , a = αx ⊕ β , otherwise (5)and P γ(cid:15)ζηD ( bc | yz ) = , b = γy ⊕ (cid:15)c = ζz ⊕ η , otherwise , (6) perational nonclassicality of local multipartite correlations P γ(cid:15)ζηD ( bc | yz ) = P γ(cid:15)D ( b | y ) P ζηD ( c | z ), where P γ(cid:15)D ( b | y ) = (cid:26) , b = γy ⊕ (cid:15) , otherwise (7)and P ζηD ( c | z ) = (cid:26) , c = ζz ⊕ η , otherwise . (8)Note that the set of 3-local boxes and quantum boxes satisfy L ⊂ Q ⊂ N . Boxes lyingoutside L are called absolutely nonlocal boxes and they cannot be written as a convexmixture of the local deterministic boxes alone.A tripartite correlation is said to be “2-local across the bipartite cut ( AB | C )” ifit has the following form: P ( abc | xyz ) = (cid:88) λ p λ P λ ( ab | xy ) P λ ( c | z ) , (9)where P λ ( ab | xy ) can have arbitrary nonlocality consistent with the NS principle. “2-locality across other bipartite cuts” for tripartite correlations can be defined similarly.Note that any 3-local box can always be written in 2-local form across any possiblebipartition. The general form [34] is, therefore: P ( abc | xyz ) = s (cid:88) λ p λ P AB | Cλ + s (cid:88) λ q λ P AC | Bλ + s (cid:88) λ r λ P A | BCλ , (10)where P AB | Cλ = P λ ( ab | xy ) P λ ( c | z ), and, where P AC | Bλ and P A | BCλ are similarly defined.Here s + s + s = 1; (cid:80) λ p λ = 1; (cid:80) λ q λ = 1; (cid:80) λ r λ = 1. Each bipartite distribution inthe decomposition (10) can have arbitrary nonlocality consistent with the NS principle.A tripartite nonlocal box is genuinely tripartite nonlocal if it cannot be written in the2-local form given by Eq. (10). Hence, a genuinely tripartite nonlocal box is nonlocalwith respect to every bipartition ( A | BC ), ( B | CA ), ( C | AB ).The set of boxes that admit a decomposition as in Eq. (10) again forms a convexpolytope which is called , denoted by L . The extremal boxes of thispolytope are the 64 local vertices and 48 2-local vertices. There are 16 2-local verticesin which a PR-box [4] is shared between A and B , P αβγ(cid:15) ( abc | xyz )= (cid:26) , a ⊕ b = x · y ⊕ αx ⊕ βy ⊕ γ & c = γz ⊕ (cid:15) , otherwise , (11)and the other 32 two-way local vertices, P αβγ(cid:15) and P αβγ(cid:15) , in which a PR-box is sharedby AC and BC , are similarly defined. The extremal boxes in Eq. (11) can be writtenin the factorized form, P αβγ(cid:15) ( abc | xyz ) = P αβγP R ( ab | xy ) P γ(cid:15)D ( c | z ), where P αβγP R ( ab | xy )= (cid:26) , a ⊕ b = x.y ⊕ αx ⊕ βy ⊕ γ , otherwise , (12) perational nonclassicality of local multipartite correlations P γ(cid:15)D ( c | z ) is the aforementioned deterministic box. The set of 2-local boxes satisfy, L ⊂ L ⊂ N . Absolute tripartite nonlocal boxes can be either 2-local or genuinelytripartite nonlocal. A genuinely tripartite nonlocal box cannot be written as a convexmixture of the extremal boxes of L and violates a facet inequality of L given inRef. [34].The Svetlichny inequalities [18] which are given by S αβγ(cid:15) = (cid:88) xyz ( − x · y ⊕ x · z ⊕ y · z ⊕ αx ⊕ βy ⊕ γz ⊕ (cid:15) (cid:104) A x B y C z (cid:105) ≤ , (13)are one of the classes of facet inequalities of the 2-local polytope. The violation of aSvetlichny inequality implies one of the forms of genuine tripartite nonlocality [34].The following extremal tripartite nonlocal boxes: P αβγ(cid:15) Sv ( abc | xyz )= (cid:26) , a ⊕ b ⊕ c = x · y ⊕ x · z ⊕ y · z ⊕ αx ⊕ βy ⊕ γz ⊕ (cid:15) , otherwise , (14)which violate a Svetlichny inequality to its algebraic maximum are called Svetlichnyboxes.Mermin inequalities [19] are one of the classes of facet inequalities of the fullylocal (or, 3-local) polytope [35, 36]. One of the Mermin inequalities is given by, (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) ≤ , (15)and the other 15 Mermin inequalities can be obtained from the above inequality bylocal reversible operations which are analogous to local unitary operations in quantumtheory and include local relabeling of the inputs and outputs. All the 16 Mermininequalities are given by M αβγ(cid:15) = ( α ⊕ β ⊕ γ ⊕ M + αβγ(cid:15) + ( α ⊕ β ⊕ γ ) M − αβγ(cid:15) ≤ , (16)where M + αβγ(cid:15) := ( − γ ⊕ (cid:15) (cid:104) A B C (cid:105) + ( − β ⊕ (cid:15) (cid:104) A B C (cid:105) + ( − α ⊕ (cid:15) (cid:104) A B C (cid:105) +( − α ⊕ β ⊕ γ ⊕ (cid:15) ⊕ (cid:104) A B C (cid:105) and M − αβγ(cid:15) := ( − α ⊕ β ⊕ (cid:15) ⊕ (cid:104) A B C (cid:105) +( − α ⊕ γ ⊕ (cid:15) ⊕ (cid:104) A B C (cid:105) +( − β ⊕ γ ⊕ (cid:15) ⊕ (cid:104) A B C (cid:105) +( − (cid:15) (cid:104) A B C (cid:105) . Mermin inequalities detect absolute nonlocality, i.e., it guaranteesthe box to lie outside L . Quantum correlations that violate a Mermin inequality toits algebraic maximum demonstrate Greenberger–Horne–Zeilinger (GHZ) paradox [17]and are called Mermin boxes.In the tripartite case, while the dimension of the NS polytope N is 26, the numberof extreme boxes are 53,856 [37]. Thus, a given P , expressed as a convex combinationof the extreme boxes, does not have a unique decomposition. The correlations P of interest to us, belong to a subpolytope of N , called Svetlichny-box polytope inRef. [26], denoted R , having 128 extreme boxes (64 3-local boxes, 48 2-local boxesthat are not 3-local, and finally, 16 Svetlichny boxes). The Svetlichny-box polytopecan be seen as a generalization of the bipartite PR-box polytope.Even in the case of the polytope R , since it has a more extreme boxes thanits dimension of 26, therefore it has no unique decomposition. However, a unique, canonical , decomposition for any P ∈ R can be given, as shown in Ref. [26], bymaking use of the non-trivial symmetry properties among the extreme boxes of R . perational nonclassicality of local multipartite correlations P ∈ R violating a Svetlichny inequality can be brought tothe form [26]: P = p Sv P αβγ(cid:15)Sv + (1 − p Sv ) P SvL , (17)where a single Svetlichny-box, P αβγ(cid:15)Sv , is dominant and p Sv has been maximized and P SvL is a Svetlichny-local box.Eq. (17) is called the canonical decomposition for any box P belonging tothe Svetlichny-box polytope. Following Ref. [26], we refer to p Sv in the canonicaldecomposition as Svetlichny strength. In a given situation, suppose P ↑ Sv denotes thedominant Svetlichny-box. The canonical form (17) becomes P = µ P ↑ Sv + (1 − µ ) P G =0 SvL . (18)The fact that µ represents the maximal fraction of P ↑ Sv over all decompositions hasthe following consequence. The quantity G in Eq. (18) is given by G := min {G , ..., G } , (19)where G := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |S − S | − |S − S | (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) |S − S | − |S − S | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and theother eight G i are similarly defined and can be obtained by interchanging S αβγ ’s in G .The motivation for this quantity is the “monoandrous” nature of Svetlichny boxes,whereby they are maximal for precisely one of the Svetlichny inequalities. On theother hand, the deterministic boxes have the same absolute values. As a result, G canbe used to witness the canonicality of the decomposition, by checking that the localpart has G = 0 [26].The Svetlichny strength µ of the box given by Eq. (18) satisfies the relationship G ( P ) = 8 µ as shown in Ref. [26]. The box P G =0 SvL in Eq.(18) is a Svetlichny-local boxhaving G ( P ) = 0. Thus, the canonical decomposition (18) is irreducible in the sensethat the full weight of genuine tripartite nonlocality has been transferred to the P ↑ Sv part and no further reduction of this weight is possible in the local part.Analogously, it can be shown (see Section 3.2) that the canonical decompositionof P SvL in Eq. (18) wherein p Sv = 0 is one which is a convex combination of adominant Mermin box (an equal mixture of two Svetlichny boxes) and a Mermin-local box (one that doesn’t violate the Mermin inequality), such that the weight ofthe dominant Mermin box has been maximized. Following Ref. [26], we define thismaximized weight as Mermin strength of the correlation.Accordingly, the Svetlichny-local part in Eq. (18), which is multiplydecomposable, can itself be canonically decomposed such that the weight of theMermin part has been maximized. Therefore, the boxes that we consider in thiswork belong to the family of boxes which have the following canonical form: P = µ P ↑ Sv + ν P ↑ M + (1 − µ − ν ) P G = Q =0 SvL , (20)where P ↑ M is the dominant Mermin-box. The quantity Q in Eq. (20), which witnessesthe canonicality of the Svetlichny-local decomposition into a Mermin-nonlocal andMermin-local part, is given by Q := min { Q , ..., Q } , (21)where Q = |||M −M |−|M −M ||−||M −M |−|M −M ||| , andother Q i ’s are obtained by permutations (Here, M αβγ are given by Eq. (16)). The perational nonclassicality of local multipartite correlations ν of the box given by Eq. (20) satisfies the relationship Q ( P ) = 4 ν as shown in Ref. [26]. The box P G = Q =0 SvL in Eq.(20) is a box having G ( P ) = 0 and Q ( P ) = 0. Unlike the nonlocal cost [7], Svetlichny strength and/or Mermin strengthcan also be nonzero for certain fully local as well as 2-local correlations.In the following, we study the non-classicality as captured by the notion ofsuperlocality of two families of fully local and 2-local tripartite correlations whichhave Svetlichny strength and Mermin strength, respectively.
3. Superlocality of tripartite fully local and -local boxes For a given local bipartite or n -partite box, let d λ denote the minimal dimension ofthe shared classical randomness. In Ref. [12], DW have derived the upper on d λ withthe assumption that n − λ , whereas the other party can use nondeterministic strategy on each λ in the classicalsimulation model. Before we define superlocality for fully local tripartite boxes, let usdefine superlocality for local bipartite boxes. Definition.
Suppose we have a quantum state in C d A ⊗ C d B and measurements whichproduce a local bipartite box P ( ab | xy ). The correlation P is sublocal precisely if P ( ab | xy ) = d λ − (cid:88) λ =0 p λ P λ ( a | x ) P λ ( b | y ) , (22)with d λ ≤ d , with d = min { d A , d B } . Then, superlocality holds iff there is no sublocaldecomposition of the given box. Here, (cid:80) d λ − λ =0 p λ = 1. (cid:4) Note that, suppose the given local box is produced by a quantum state in C d A ⊗ C d B .Then it can also be produced by a quantum state in C d ⊗ C d with d = min { d A , d B } [12].Note that a bipartite correlation that is nonlocal is obviously non-sublocal becausethere is no dimensionally bounded preshared randomness that can reproduce it in aclassical simulation protocol. The interesting cases where superlocality can witnessquantumness will then pertain only to local correlations. Hence in this work, we shallbe mainly concerned with superlocality as applied to local correlations.We can now extend superlocality to a tripartite system. Definition.
Suppose we have a quantum state in C d A ⊗ C d B ⊗ C d C and measurementswhich produce a fully local tripartite box P ( abc | xyz ). The correlation P is (or, fully sublocal ) precisely if P ( abc | xyz ) = d λ − (cid:88) λ =0 p λ P λ ( a | x ) P λ ( b | y ) P λ ( c | z ) , (23)with d λ ≤ d , with d = min { d A , d B , d C } . Then, absolute tripartite superlocalityholds iff there is no 3-sublocal decomposition of the given fully local box. Here, (cid:80) d λ − λ =0 p λ = 1. (cid:4) In other words, if a fully local correlation P ( abc | xyz ) is such that at least one ofthe bipartitions ( A | BC ), ( B | AC ) and ( C | AB ) is superlocal, then P ( abc | xyz ) is saidto have absolute tripartite superlocality. The set of all fully sublocal tripartite boxesforms a nonconvex subset of the fully local polytope L and is denoted by L ∗ ≡ L ∗ .We now define the concept of across a given bipartite cut. perational nonclassicality of local multipartite correlations A B C λ BC λ A -‐ BC X Y Z Figure 1. A 2-sublocal box:
Directional acyclic graph illustrating simulationof a local tripartite box in the following classical protocol: Alice shares hiddenvariable λ A − BC with Bob-Charlie. For each λ A − BC , Bob and Charlie can useshared randomness λ BC of arbitrary dimension. Definition.
Suppose we have a quantum state in C d A ⊗ C d B ⊗ C d C and measurementsproducing a tripartite box P ( abc | xyz ), which is 2-local across the bipartite cut ( A | BC ).Then, the correlation P is ( A | BC ) precisely if P ( abc | xyz ) = d λ − (cid:88) λ =0 p λ P λ ( a | x ) P λ ( bc | yz ) , (24)where (cid:80) d λ − λ =0 p λ = 1; d λ ≤ min { d A , d B d C } . Here, P λ ( bc | yz ) is an arbitrary NS box (If P λ ( bc | yz ) is local box, then P ( abc | xyz ) is fully local). Tripartite superlocality acrossthe bipartite cut ( A | BC ) holds iff there is no 2-sublocal decomposition across thebipartite cut ( A | BC ). (cid:4) Tripartite superlocality across other possible bipartite cuts can be defined in a similarway. The nonconvex set of all 2-sublocal boxes across different possible bipartite cutsdenoted L ∗ satisfies L ∗ ⊂ L ∗ .Now we define genuine tripartite superlocality. Definition.
A tripartite local (fully local or 2-local) correlation box is called genuinely superlocal iff it is superlocal across all possible bipartitions. (cid:4)
A fully local correlation P ( abc | xyz ) that isn’t 3-sublocal has absolute superlocality. In such a correlation, there is at least one partition for which 2-sublocality doesn’t hold. Obviously, a fully local correlation that is genuinelysuperlocal must be absolutely superlocal also. Here we may remark that genuinesuperlocality can occur for absolutely nonlocal (i.e., which are not fully local), but2-local correlations.Two different classical simulation protocols of tripartite 2-local boxes (they maybe fully local as well) are depicted in Fig. 1 and Fig. 2. Fig. 1 is applicable when the2-local tripartite box is fully local as well; on the other hand, Fig. 2 is applicable whenthe 2-local tripartite box may or may not be fully local. We also use the notation λ to denote λ A − BC for our convenience in calculations. perational nonclassicality of local multipartite correlations A B C λ
A-‐BC
X Y Z
No signaling box
Figure 2.
Directional acyclic graph illustrating simulation of a 2-local tripartitebox in the following classical protocol: Alice shares hidden variable λ A − BC withBob-Charlie. For each λ A − BC , the correlations of Bob and Charlie are NS boxes. The generalized GHZ (GGHZ) state in C ⊗ C ⊗ C , | ψ GGHZ (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) , (25)gives rise to the Svetlichny family of quantum correlations, defined by: P µSvF = 2 + ( − a ⊕ b ⊕ c ⊕ xy ⊕ xz ⊕ yz √ µ
16 ; 0 < µ ≤ µ > √ , and for µ ≤ √ it is fully local (i.e., 3-local)as in this range the correlation does not violate any Bell inequality.The canonical decomposition of the Svetlichny family (26) is the “noisySvetlichny-box” P µSvF = p Sv P Sv + (1 − p Sv ) P N ; 0 < p Sv ≤ , (27)with p Sv = µ/ √
2, which is a special case of Eq. (18). Here, P N is the maximally mixedbox, i.e., P N ( abc | xyz ) = 1 / x, y, z, a, b, c . That Eq. (27) is indeed canonicaldecomposition is proved in detail in [26]. Briefly, the proof makes use of non-trivialsymmetry properties of the extreme boxes of the polytope R , such as for examplethat every Svetlichny box (say, P Sv ) has a “complement” (here: P Sv ) such thattheir uniform mixture yields the maximally mixed box, i.e., P N = ( P Sv + P Sv ).Note that the Svetlichny family (26) has G = 4 √ µ and the Svetlichny-local box P N in the decomposition (27) of this family has G = 0, indicating that this decompositionis indeed canonical. This implies that the Svetlichny strength of the Svetlichny familycan be calculated from G and is given by G ( P µSvF ) / µ √ as stated above.Therefore, the Svetlichny-box fraction in Eq. (27) can be read off as the Svetlichnystrength of the Svetlichny family. Since the Svetlichny family has nonzero Svetlichnystrength for any µ >
0, the quantum simulation of these correlations by using a three-qubit system necessarily requires genuine quantumness in the state [26]. In this light,the Svetlichny strength p Sv satisfies the relation p Sv = √ τ √ , where τ = sin θ is the perational nonclassicality of local multipartite correlations | ψ GGHZ (cid:105) , when the Svetlichny family is simulated by | ψ GGHZ (cid:105) forthe noncommuting projective measurements corresponding to the operators: A = σ x , A = σ y , B = ( σ x − σ y ) / √ B = ( σ x + σ y ) / √ C = σ x and C = σ y .As noted above, for 0 < µ ≤ √ the fully local Svetlichny family can bedecomposed as a convex mixture of the 3-local deterministic boxes. In this range,the fully local Svetlichny family can be decomposed in the following 2-local formacross the bipartition ( A | BC ): P µSvF = (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) , (28)where P Svλ ( a | x ) are different deterministic distributions and P Svλ ( bc | yz ) are local boxes(see Appendix A for the derivation of the above decomposition). For the fully localSvetlichny family (0 < µ ≤ √ ), the decomposition (28) defines a classical simulationprotocol where Alice shares hidden variable λ A − BC of dimension 4 with Bob-Charlieas in Fig. 1. Theorem 1.
The fully local Svetlichny family P µSvF ( < µ ≤ √ ) is genuinelytripartite superlocal.Proof. Let us try to reproduce the fully local Svetlichny family P µSvF (0 < µ ≤ √ )in the scenario as in Fig. 1 where Alice preshares the hidden variable λ A − BC ofdimension 2 with Bob-Charlie. Before proceeding, we want to mention that in case ofSvetlichny family, all the marginal probability distributions of Alice, Bob and Charlieare maximally mixed: P SvF ( a | x ) = P SvF ( b | y ) = P SvF ( c | z ) = 12 ∀ a, b, c, x, y, z. (29)Let us now try to check whether the fully local Svetlichny family P µSvF (0 < µ ≤ √ )can be decomposed in the following form: P µSvF = (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) , (30)where p = x , p = x (0 < x <
1, 0 < x < x + x = 1) and P Svλ ( bc | yz ) are localboxes. Let us assume that Alice’s strategy to be deterministic one, i.e., each of the twoprobability distributions P Sv ( a | x ) and P Sv ( a | x ) in the above decomposition belongsto any one among P D , P D , P D and P D . In order to satisfy the marginal probabilitiesfor Alice P SvF ( a | x ), the only two possible choices of P Sv ( a | x ) and P Sv ( a | x ) are:(i) P D and P D with x = x = (ii) P D and P D with x = x = .Now, it can be easily checked that none of these two possible choices will satisfyall the tripartite joint probability distributions P µSvF simultaneously (for detailedcalculations, see Appendix B). It is, therefore, impossible to reproduce the fully localbox P µSvF (0 < µ ≤ √ ) in the scenario as in Fig. 1 where Alice preshares the hiddenvariable λ A − BC of dimension 2 with Bob-Charlie and uses deterministic strategy foreach λ A − BC . perational nonclassicality of local multipartite correlations P µSvF (0 < µ ≤ √ ) cannot be reproduced by a classicalsimulation model as in Eq. (30) with hidden variable λ A − BC of dimension 2 even ifAlice uses nondeterministic strategy for each λ A − BC . To see this, we note that fromany decomposition of the fully local box in terms of fully deterministic boxes (4), onemay derive a classical simulation protocol as in Fig. 1 with different deterministicdistributions at Alice’s side. Any such classical simulation protocol does not requireAlice to preshare the hidden variable λ A − BC of dimension more than 4 since thereare only 4 possible different deterministic distributions given by Eq. (5) at Alice’sside. Hence, a classical simulation model with hidden variable λ A − BC of dimension 2of the fully local box P µSvF (0 < µ ≤ √ ) can be achieved by constructing a classicalsimulation model of the fully local box P µSvF with hidden variable λ A − BC of dimension3 or 4 with different deterministic distributions at Alice’s side followed by takingequal joint probability distributions at Bob-Charlie’s side as common and making thecorresponding distributions at Alice’s side nondeterministic.Let us now try check whether the fully local noisy Svetlichny-box P µSvF (0 < µ ≤ √ ) can be simulated by a classical simulation model in the scenario as in Fig. 1where Alice shares the hidden variable of dimension d λ A − BC = 3 and uses differentdeterministic strategy at each λ A − BC . In this case, we assume that the box can bedecomposed in the following way: P µSvF = (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) . (31)Here, p = x , p = x , p = x (0 < x <
1, 0 < x <
1, 0 < x < x + x + x = 1)and P Svλ ( a | x ) are deterministic distributions and P Svλ ( bc | yz ) are local boxes. SinceAlice’s distributions are deterministic, the three probability distributions P Sv ( a | x ), P Sv ( a | x ) and P Sv ( a | x ) must be equal to any three among P D , P D , P D and P D .But any such combination will not satisfy the marginal probabilities P SvF ( a | x ) forAlice. This implies that the fully local box P µSvF (0 < µ ≤ √ ) cannot be reproducedin any classical simulation protocol with different deterministic distributions P Svλ ( a | x )at Alice’s side, where Alice preshares the hidden variable λ A − BC of dimension 3 withBob-Charlie.Therefore, in the classical simulation model for the fully local Svetlichny family inthe scenario as in Fig. 1 where Alice uses deterministic strategies, Alice has to sharethe hidden variable of dimension d λ A − BC = 4.Suppose the fully local Svetlichny family P µSvF (0 < µ ≤ √ ) can be reproducedby the following classical simulation model: P µSvF = (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) , (32)where P Svλ ( a | x ) are different deterministic distributions and either any three of the fourjoint probability distributions P Svλ ( bc | yz ) are equal to each other, or there exists twosets each containing two equal joint probability distributions P Svλ ( bc | yz ); 0 < p λ < λ = 0 , , , (cid:80) λ =0 p λ = 1. Then taking equal joint probability distributions P Svλ ( bc | yz ) at Bob-Charlie’s side as common and making corresponding distributionat Alice’s side non-deterministic will reduce the dimension of the hidden variable perational nonclassicality of local multipartite correlations λ A − BC from 4 to 2. For example, let us consider P Sv ( bc | yz ) = P Sv ( bc | yz ) = P Sv ( bc | yz ) . (33)Now in order to satisfy Alice’s marginal given by Eq. (29), one must take p = p = p = p = . Hence, the decomposition (32) can be written as, P µSvF = q P Sv ( a | x ) P Sv ( bc | yz ) + p P Sv ( a | x ) P Sv ( bc | yz ) , (34)where P Sv ( a | x ) = P Sv ( a | x )+ P Sv ( a | x )+ P Sv ( a | x )3 , which is a non-deterministic distributionat Alice’s side, and q = . The decomposition (34) represents a classical simulationprotocol of the fully local Svetlichny family P µSvF (0 < µ ≤ √ ) with differentdeterministic/non-deterministic distributions at Alice’s side, where Alice shares hiddenvariable λ A − BC of dimension 2 with Bob-Charlie. Now in this protocol, consideringarbitrary joint probability distributions P Svλ ( bc | yz ) at Bob-Charlie’s side (withoutconsidering any constraint as in the case presented in Appendix B), it can be checkedthat all the tripartite distributions of P µSvF are not reproduced simultaneously.There are the following other cases in which the dimension of the hidden variable λ A − BC can be reduced from 4 to 2 in the classical simulation model as in Eq. (32): P Sv ( bc | yz ) = P Sv ( bc | yz ) = P Sv ( bc | yz ); P Sv ( bc | yz ) = P Sv ( bc | yz ) = P Sv ( bc | yz ); P Sv ( bc | yz ) = P Sv ( bc | yz ) = P Sv ( bc | yz ); P Sv ( bc | yz ) = P Sv ( bc | yz ) as well as P Sv ( bc | yz ) = P Sv ( bc | yz ); P Sv ( bc | yz ) = P Sv ( bc | yz ) as well as P Sv ( bc | yz ) = P Sv ( bc | yz ); P Sv ( bc | yz ) = P Sv ( bc | yz ) as well as P Sv ( bc | yz ) = P Sv ( bc | yz ).Now in any of these possible cases, considering arbitrary joint probability distributions P Svλ ( bc | yz ) at Bob-Charlie’s side (without considering any constraint), it can bechecked that all the tripartite distribution P µSvF are not reproduced simultaneously.Hence, this also holds when the boxes P Svλ ( bc | yz ) satisfy NS principle as well as localitycondition.Hence, one can conclude that it is impossible to reduce the dimension from 4to 2 in the classical simulation protocol of the fully local Svetlichny family P µSvF (0 < µ ≤ √ ) in the scenario as in Fig. 1 .It is, therefore, impossible to reproduce the fully local box P µSvF (0 < µ ≤ √ ) with deterministic/non-deterministic distributions at Alice’s side, where Alicepreshares the hidden variable λ A − BC of dimension 2 with Bob-Charlie in the scenarioas in Fig. 1.It can be checked that the fully local Svetlichny box P µSvF (0 < µ ≤ √ )is non-product. It is, therefore, impossible to reproduce the fully local box P µSvF (0 < µ ≤ √ ) in the scenario where Alice preshares the hidden variable λ A − BC ofdimension 1 with Bob-Charlie.Hence, the dimension of the hidden variable λ A − BC , which Alice preshares withBob-Charlie to reproduce the fully local box P µSvF (0 < µ ≤ √ ), must be greater than
2. Therefore, the fully local Svetlichny box P µSvF (0 < µ ≤ √ ) has 2-local form acrossthe bipartite cut ( A | BC ) and is not 2-sublocal across that bipartite cut. Hence, thefully local Svetlichny box P µSvF (0 < µ ≤ √ ) is superlocal across that bipartite cut. perational nonclassicality of local multipartite correlations C | AB ) and ( B | CA ), i.e., this box is superlocal across all three possible bipartitecuts and, hence, must be genuinely tripartite superlocal.Since by definition, fully local correlations, which are genuinely superlocal, mustbe absolutely superlocal as well, it follows from Theorem 1 that the fully localSvetlichny box P µSvF (0 < µ ≤ √ ) is absolutely superlocal also. We are now interested in quantum correlations that belong to the Mermin familydefined as P νMF = 1 + ( − a ⊕ b ⊕ c ⊕ xy ⊕ xz ⊕ yz δ x ⊕ y ⊕ ,z ν < ν ≤ . (35)For ν ≤ , the above box is fully local as in this range the correlation does not violateany Bell inequality. The above box is absolutely nonlocal, but 2-local for ν > asit violates the Mermin inequality (given in Eq. (15)) for ν > , but not any of theSvetlichny inequalities. Thus, it isn’t obvious that this correlation would be genuinelysuperlocal, yet this is what will be established below.The Mermin family (35) has the canonical decomposition as the noisy Mermin-box P νMF = p M P M + (1 − p M ) P N ; 0 < p M ≤ , (36)with p M = ν , which is a special case of Eq. (20). Here, the Mermin-box P M = (cid:0) P Sv + P Sv (cid:1) . That Eq. (36) is indeed the canonical decomposition forEq. (35) can be shown, as with the case of the noisy Svetlichny box, by makinguse of the non-trivial symmetry properties of the extremal boxes of the polytope R , such as the fact that any given Mermin box (say, P M ) has a complement suchthat their uniform mixture yields the white noise, i.e., P N = P M + P (cid:48) M , where P (cid:48) M = (cid:0) P Sv + P Sv (cid:1) [26]. Therefore, the Mermin-box fraction in Eq. (36) indeedgives Mermin strength [26] of the Mermin family. Since the Mermin family has adecomposition as in Eq. (20), its Mermin strength can be calculated from Q and isgiven by Q ( P νMF ) / ν as stated above.For any ν >
0, the quantum simulation of the Mermin family by using a three-qubit system necessarily requires genuine quantumness in the state, even if it is fullylocal or 2-local. This is due to the fact that the Mermin family has nonzero Merminstrength for any ν > | ψ GGHZ (cid:105) (given by Eq. (25)),gives rise to the Mermin family with Mermin strength ν = √ τ for the noncommutingprojective measurements corresponding to the operators: A = σ x , A = σ y , B = σ x , B = σ y , C = σ x and C = σ y that demonstrates the GHZ paradox.As noted above, for 0 < ν ≤ the fully local noisy Mermin box can bedecomposed in a convex mixture of the 3-local deterministic boxes. In this range,the fully local Mermin family can be decomposed in the following 2-local form acrossthe bipartition ( A | BC ): P νMF = (cid:88) λ =0 r λ P Mλ ( a | x ) P Mλ ( bc | yz ) , (37) perational nonclassicality of local multipartite correlations P Mλ ( a | x ) are different deterministic distributions and P Mλ ( bc | yz ) are local boxes(see Appendix C for the derivation of the above decomposition). For the fully localnoisy Mermin-box (0 < ν ≤ ), the decomposition (37) defines a classical simulationprotocol where Alice shares hidden variable λ A − BC of dimension 4 as in Fig. 1. Theorem 2.
The fully local noisy Mermin box P νMF ( < ν ≤ ) is genuinelytripartite superlocal.Proof. In case of noisy Mermin-box also, all the marginal probability distributions ofAlice, Bob and Charlie are maximally mixed: P MF ( a | x ) = P MF ( b | y ) = P MF ( c | z ) = 12 ∀ a, b, c, x, y, z. (38)Let us try to construct a classical simulation protocol for the fully local Merminfamily P νMF (0 < ν ≤ ) with different deterministic distributions P Mλ ( a | x ) at Alice’sside, where Alice shares hidden variable λ A − BC of dimension 2 with Bob-Charlie.In this case, the fully local Mermin family (0 < ν ≤ ) can be decomposed in thefollowing way: P νMF = (cid:88) λ =0 r λ P Mλ ( a | x ) P Mλ ( bc | yz ) . (39)Here, r = a , r = a (0 < a <
1, 0 < a < a + a = 1). SinceAlice’s distributions are deterministic, the two probability distributions P M ( a | x ) and P M ( a | x ) must be equal to any two among P D , P D , P D and P D . In order tosatisfy the marginal probabilities for Alice P MF ( a | x ), the only two possible choices of P M ( a | x ) and P M ( a | x ) are:1) P D and P D with a = a = P D and P D with a = a = .Now, in a similar process as adopted in case of Svetlichny family, it can be easilychecked that none of these two possible choices will satisfy all the tripartite jointprobability distributions P νMF simultaneously.Since Alice’s marginal distributions are maximally mixed, it can be shown in asimilar way as presented in case of noisy Svetlichny box that there does not exist aclassical simulation model for the fully local Mermin family in the scenario as in Fig.1 where Alice shares the hidden variable of dimension d λ A − BC = 3 and uses differentdeterministic strategies at each λ A − BC .Thus, in the classical simulation model for the fully local Mermin family in thescenario as in Fig. 1 where Alice uses different deterministic strategies, Alice has toshare the hidden variable of dimension d λ A − BC = 4. Let us now try to check whetherthere exists such a classical simulation model for the fully local noisy Mermin boxwhere d λ A − BC can be reduced from 4 to 2 by allowing non-deterministic strategies onAlice’s side. That is we try to construct the following classical simulation protocol forthe fully local noisy Mermin-box (0 < ν ≤ ) with different deterministic distributions P Mλ ( a | x ) at Alice’s side, where Alice shares hidden variable λ A − BC of dimension 4 withBob-Charlie in the scenario as in Fig. 1: P νMF = (cid:88) λ =0 p λ P Mλ ( a | x ) P Mλ ( bc | yz ) , (40) perational nonclassicality of local multipartite correlations P SMλ ( bc | yz ) are equalto each other, or there exists two sets each containing two equal joint probabilitydistributions P Mλ ( bc | yz ); 0 < p λ < λ = 0 , , , (cid:80) λ =0 p λ = 1. Then, asdescribed earlier in the case of noisy Svetlichny-box, taking equal joint probabilitydistributions P Mλ ( bc | yz ) at Bob-Charlie’s side as common and making correspondingdistribution at Alice’s side non-deterministic will reduce the dimension of the hiddenvariable λ A − BC from 4 to 2. Now following the similar procedure adopted in caseof noisy Svetlichny box, one can show that it is impossible to reduce the dimensionfrom 4 to 2 in the classical simulation protocol of the fully local noisy Mermin-box(0 < ν ≤ ) in the scenario as in Fig. 1.It is, therefore, impossible to reproduce the fully local Mermin family P νMF (0 < ν ≤ ) in any classical simulation protocol with deterministic/non-deterministicdistributions P Mλ ( a | x ) at Alice’s side, where Alice preshare the hidden variable λ A − BC of dimension 2 with Bob-Charlie in the scenario as in Fig. 1 .It can be checked that the fully local Mermin family P νMF (0 < ν ≤ ) is non-product. It is, therefore, impossible to reproduce the fully local Mermin family P νMF (0 < ν ≤ ) in the scenario where Alice preshares the hidden variable λ A − BC ofdimension 1 with Bob-Charlie.Hence, the dimension of the hidden variable λ A − BC , which Alice preshares withBob-Charlie to reproduce the fully local Mermin family P νMF (0 < ν ≤ ), must be greater than
2. Therefore, the fully local Mermin family P νMF (0 < ν ≤ ) has 2-local form across the bipartite cut ( A | BC ) and is not 2-sublocal across that bipartitecut. Hence, the fully local Mermin family P νMF (0 < ν ≤ ) is superlocal across thebipartite cut ( A | BC ).It can be checked that a similar argument holds across the remaining two bipartitecuts ( C | AB ) and ( B | CA ), i.e., this box is superlocal across all three possible bipartitecuts and, hence, must be genuinely tripartite superlocal.Since the fully local noisy Mermin box P νMF (0 < ν ≤ ) is genuinely superlocal,it is absolutely superlocal as well.Now, as noted before, for 0 < ν ≤ A | BC ): P <ν< MF = (cid:88) λ =0 r λ P Mλ ( a | x ) P Mλ ( bc | yz ) , (41)where P Mλ ( a | x ) are different deterministic distributions and P Mλ ( bc | yz ) are NS boxes(see Appendix D for the derivation of the above decomposition). For the 2-localnoisy Mermin-box (0 < ν ≤ λ A − BC of dimension 4.Since the fully local noisy Mermin box is a special case of the 2-local noisy Merminbox and the proof of Theorem 2 is independent of the locality condition of the bipartitedistributions P Mλ ( bc | yz ) at Bob-Charlie’s side, the proof is also valid when the bipartitedistributions P Mλ ( bc | yz ) at Bob-Charlie’s side are NS (local or nonlocal) boxes. Hence,it is not difficult to see that the proof of the Theorem 2 can be straightforwardlyadopted to obtain the following result: Theorem 3.
The 2-local noisy Mermin box P νMF ( < ν ≤ ) is genuinely tripartitesuperlocal.perational nonclassicality of local multipartite correlations
4. Connection between genuine super-locality and genuine nonclassicality
In Ref. [22], Giorgi et al defined genuine tripartite quantum discord to quantify thequantum part of genuine tripartite correlations in a tripartite quantum state. As thename suggests, this kind of quantifier also captures genuine quantumness of separablestates. Genuine tripartite quantum discord defined in Ref. [24] goes to zero iff thereexists a bipartite cut of the tripartite system such that no quantum correlation existsbetween the two parts. It is known that a bipartite quantum state has no quantumcorrelation as quantified by Alice to Bob quantum discord iff it can be written inthe classical-quantum state form [39]. We define tripartite classical-quantum states asfollows. Definition.
A fully separable tripartite state has a classical-quantum state formacross the bipartite cut ( A | BC ) if it can be decomposed as ρ A | BCCQ = (cid:88) i p i | i (cid:105) A (cid:104) i | ⊗ ρ Bi ⊗ ρ Ci , (42)where {| i (cid:105) A } is some orthonormal basis of Alice’s Hilbert space H A .Note that the classical-quantum states as defined above do not have nonzero genuinequantum discord since Alice’s subsystem is always classically correlated with Bob andCharlie’s subsystem. We characterize a (the fully separable) state as having genuinequantumness, if it cannot be written in the classical-quantum state in any bipartitecut as in Eq. (42).The Svetlichny family and Mermin family violate a three-qubit biseparabilityinequality for µ > / ν > / µ ≤ / ν ≤ /
2, the Svetlichny and Mermin familiescan also be reproduced by separable three-qubit states since they do not violate anybiseparability inequality in this range.However, for µ, ν ∈ (0 , ], the simulation of the Svetlichny family and Merminfamily by using three-qubit separable states serves to witness genuine quantumness inthe form of genuine quantum discord as they have nonzero Svetlichny strength andnonzero Mermin strength, respectively [26]. This observation prompts us to make thefollowing observation. Observation 1.
Genuine quantumness (i.e., nonzero discord across any bipartite cut)of any correlation P is necessary for genuine superlocality.Proof. Consider tripartite boxes arising from three-qubit classical-quantum stateswhich have the form as given in Eq. (42) with i = 0 ,
1. In the Bell scenario thatwe have considered, for Alice measuring in basis {| i (cid:105)} , it is clear that the resultingbox can be simulated by a probabilistic strategy using dimension d λ A = 2 on Alice’sside. This observation holds even when Alice measures in any other basis (except thather random number generator will be possibly be more randomized). This impliesthat for any three-qubit state which do not have genuine quantumness, there existsa bipartite cut in which it is not superlocal. Therefore, genuine quantum discord isnecessary for implying genuine superlocality. perational nonclassicality of local multipartite correlations
5. Genuine multipartite nonclassicality
Generalizing the definitions presented in Section 2, an n -partite correlation is said tobe fully local or n -local if the box has a decomposition of the form: P ( a , a , · · · , a n | x , x , · · · , x n ) = (cid:88) λ p λ P λ ( a | x ) P λ ( a | x ) · · · P λ ( a n | x n ) , (43)where (cid:80) λ p λ = 1, and x , x , · · · , x n denote the inputs (measurement choices) and a , a , · · · , a n denote the outputs (measurement outcomes) of the parties q , q , ..., q n respectively. An n -partite box that is not n -local is said to have “absolute nonlocality”.A n -partite correlation is said to be k -local if it can be decomposed as a convexcombination of k -partitions such that these k parts (defined by each k -partition) arelocally correlated with each other. For example, a 4-partite correlation is 3-local if itcan be decomposed as a convex combination of tripartitions such as ( q | q q | q ) suchthat these parts are locally correlated each other. However, q q may be nonlocal initself. Similarly, a 4-partite correlation is 2-local if it can be decomposed as a convexcombination of probability distributions over bipartitions such that in each bipartition,the two parts are locally correlated, though within a part, even nonlocality may hold.Any k -local correlation is also k (cid:48) -local where k (cid:48) < k . Thus, a 4-local correlation is also3-local and a 3-local correlation is also 2-local. But the converse is not true.Therefore, the weakest form of locality is 2-locality, and the strongest form ofnonlocality for an n -partite system is that which is not 2-local. This is called genuine n -partite nonlocality, for which all bipartitions are nonlocal.An n -partite system is n -sublocal (or, fully sublocal) if each of the n particlesare locally correlated, with the shared classical randomness dimension being less thanor equal to the smallest local Hilbert space dimension among all the n particles, inanalogy with Eq. (23): Definition.
Suppose we have an n -partite quantum state in C d s ⊗ C d s ⊗ · · · ⊗ C d sn and measurements which produce a fully local n -partite box P ( a a · · · a n | x x · · · x n ).The correlation P is n -sublocal (or, fully sublocal ) precisely if there exists adecomposition such that P ( a a · · · a n | x x · · · x n ) = d λ − (cid:88) λ =0 p λ P λ ( a | x ) P λ ( a | x ) · · · P λ ( a n | x n ) , (44)with d λ ≤ d , and d = min { d s , d s , · · · , d s n } , where d s j is the local Hilbert spacedimension of the j th particle. The fully local correlation P is absolutely superlocal ifit is not n -sublocal. In other words, absolute n -partite superlocality holds iff there isno n -sublocal decomposition (44) of the given fully local box. Here (cid:80) d λ − λ =0 p λ = 1. (cid:4) A n -partite correlation P is said to be “ k -sublocal across a particular k -partition”if these k parts are sublocally correlated with each other. For example, a 4-partitecorrelation is 3-sublocal across the tripartite cut ( q | q q | q ) if these three parts aresublocally correlated with each other. Note that the part ( q q ) may be sublocal orsuperlocal or even nonlocal. perational nonclassicality of local multipartite correlations Q ( ||| ) , is also 3-sublocal across any tripartition Q ( || ) obtained by merging any two partitions of Q ( ||| ) ,and also 2-sublocal across any bipartition Q ( | ) obtained by merging any two partitionsof Q ( || ) .In general, any correlation, which is k -sublocal across some k -partition, isalso k (cid:48) -sublocal across some k (cid:48) -partition where k (cid:48) < k , where the k (cid:48) -partition hasbeen obtained by merging parts of the k -partition. Therefore, the weakest formof sublocality is 2-sublocality. An n -partite correlation P that isn’t sublocal – orequivalently, is superlocal– across every bipartition, is genuinely n -partite superlocal. Note that, because of the non-convexity associated with sublocal sets, a convexcombination of 2-sublocal correlations needn’t itself be 2-sublocal.In line with our definition for superlocality for a multipartite system, we maydefine n -concord as the absence of quantum discord across all cuts splitting all n particles, i.e., ( q | q | · · · | q n ). For example, the system q q q q has 4-concord ifthere is no quantum discord across each of the cuts q | q | q | q . Absolute n -partitediscord [42] holds when the state in question is not n -concordant. Genuine n -partitediscord holds when the state in question lacks 2-concord form across all possiblebipartitions [22–25]. The relation between different such measures of discord andsuperlocality for multipartite systems, such as that noted for the tripartite system inSection 4, is an interesting issue meriting further studies.
6. Conclusions
In Ref. [26], two quantities called, Svetlichny strength and Mermin strength, have beenintroduced to study genuine nonclassicality of tripartite correlations. By using thesetwo quantities, it has been demonstrated that genuine tripartite quantum discord isnecessary to simulate certain fully local or 2-local tripartite correlations if the measuredtripartite systems are restricted to be three-qubit states.Genuine multipartite quantum nonlocality occurs if the multipartite correlationcannot be expressed as a convex combination of all possible bipartitions where the 2-parts defined by each of these bipartitions are locally correlated with each other. Ourmotivation has been to perform the characterization of genuine nonclassicality of local(fully or partially local) tripartite correlations arising from the concept of superlocality,and relating this to genuine quantum discord. Thus, genuine superlocality, i.e., theoccurrence of superlocality across all bipartitions, provides an operational definitionof genuine nonclassicality. We have studied how genuine superlocality occurs for twofamilies of local tripartite correlations having their nonclassicality quantified in termsof nonzero Svetlichny strength and nonzero Mermin strength, respectively.In Ref. [10], it was demonstrated that certain bipartite separable states havingquantum discord may improve the so-called random access codes (RAC), whichis a class of communication problem, if the shared randomness between the twoparties is limited to be finite. Recently, in Ref. [43], a family of RAC protocolshave been considered in tripartite quantum networks and is associated with genuinetripartite nonlocality. Our work on characterizing genuine quantumness of certainlocal tripartite or multipartite correlations in the limited dimensional simulationscenario and its link with genuine quantum discord motivates the following study.It would be interesting to investigate quantum advantage for the above RAC protocolassociated with tripartite or multipartite quantum networks in the presence of limitedshared randomness by using tripartite or multipartite separable states with genuine perational nonclassicality of local multipartite correlations
Acknowledgements
CJ thanks Manik Banik and Arup Roy for discussions. DD acknowledges the financialsupport from University Grants Commission (UGC), Government of India. CJ, SGand ASM acknowledge support through Project SR/S2/LOP-08/2013 of the DST,Govt. of India. perational nonclassicality of local multipartite correlations [1] Ryszard Horodecki, Pawe(cid:32)l Horodecki, Micha(cid:32)l Horodecki, and Karol Horodecki. Quantumentanglement. Rev. Mod. Phys. , 81:865–942, Jun 2009.[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner.Bell nonlocality.
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Phys. Rev. A , 97:062335, Jun 2018. perational nonclassicality of local multipartite correlations Appendix A. -local form across the bipartition ( A | BC ) for the fully localSvetlichny family P µSvF in the range < µ ≤ √ For 0 < µ ≤ √ the fully local Svetlichny family can be decomposed as a convexmixture of the 3-local deterministic boxes. In this range, we consider the followingdecomposition for the Svetlichny family in terms of the 3-local deterministic boxes: P µSvF = 14 P D (cid:40) √ µ (cid:0) P D + P D + P D + P D (cid:1) + 1 − √ µ (cid:0) P D + P D + P D + P D (cid:1)(cid:41) + 14 P D (cid:40) √ µ (cid:0) P D + P D + P D + P D (cid:1) + 1 − √ µ (cid:0) P D + P D + P D + P D (cid:1)(cid:41) + 14 P D (cid:40) √ µ (cid:0) P D + P D + P D + P D (cid:1) + 1 − √ µ (cid:0) P D + P D + P D + P D (cid:1)(cid:41) + 14 P D (cid:40) √ µ (cid:0) P D + P D + P D + P D (cid:1) + 1 − √ µ (cid:0) P D + P D + P D + P D (cid:1)(cid:41) := (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) , (A.1)where p = p = p = p = ; P Sv ( a | x ) = P D , P Sv ( a | x ) = P D ,P Sv ( a | x ) = P D , P Sv ( a | x ) = P D ; perational nonclassicality of local multipartite correlations P Sv ( bc | yz ) = (cid:40) √ µ (cid:0) P D + P D + P D + P D (cid:1) + 1 − √ µ (cid:0) P D + P D + P D + P D (cid:1)(cid:41) := bcyz
00 01 10 1100 √ µ −√ µ −√ µ √ µ
14 14 14 14
14 14 14 14 −√ µ √ µ √ µ −√ µ , where each row and column corresponds to a fixed measurement ( yz ) and a fixedoutcome ( bc ) respectively ‡ , P Sv ( bc | yz ) = −√ µ √ µ √ µ −√ µ
414 14 14 1414 14 14 141+ √ µ −√ µ −√ µ √ µ , P Sv ( bc | yz ) =
14 14 14 141+ √ µ −√ µ −√ µ √ µ √ µ −√ µ −√ µ √ µ
414 14 14 14 , P Sv ( bc | yz ) =
14 14 14 141 −√ µ √ µ √ µ −√ µ −√ µ √ µ √ µ −√ µ
414 14 14 14 .Note that, in the range 0 < µ ≤ √ , each of the P Svλ ( bc | yz ) given above belongsto BB84 family [13] defined as P BB ( bc | yz ) = 1 + ( − b ⊕ c ⊕ y · z δ y,z V , (A.2)with 0 < V = √ µ ≤
1, upto local reversible operations. The above family islocal, as it satisfies the complete set of Bell-CHSH (Bell-Clauser-Horne-Shimony-Holt)inequalities [44, 45]. In Ref. [46], it has been demonstrated that the BB84 familycannot be reproduced by shared classical randomness of dimension d λ ≤
3. On theother hand, the BB84 family can be reproduced by performing appropriate quantummeasurements on 2 ⊗ P Svλ ( bc | yz ) is superlocalin the range 0 < µ ≤ √ .For the fully local Svetlichny family (0 < µ ≤ √ ), the decomposition(A.1) defines a classical simulation protocol with different deterministic distributions P Svλ ( a | x ) at Alice’s side, where Alice shares hidden variable λ A − BC of dimension 4with Bob-Charlie as in Fig. 1. Decomposition (A.1) represents 2-local form across thebipartite cut ( A | BC ) of the fully local Svetlichny family (0 < µ ≤ √ ). ‡ Throughout the paper we will follow the same convention perational nonclassicality of local multipartite correlations Appendix B. Demonstrating impossibility to reproduce fully local noisySvetlichny family in the scenario where Alice preshares the hiddenvariable λ A − BC of dimension with Bob-Charlie and uses differentdeterministic strategies for each λ A − BC . Let us try to reproduce the fully local noisy Svetlichny family P µSvF (0 < µ ≤ √ )in the scenario as in Fig. 1 where Alice preshares the hidden variable λ A − BC ofdimension 2 with Bob-Charlie and she uses different deterministic strategies for each λ A − BC . In this case, we assume that the fully local noisy Svetlichny family P µSvF (0 < µ ≤ √ ) can be decomposed in the following way: P µSvF = (cid:88) λ =0 p λ P Svλ ( a | x ) P Svλ ( bc | yz ) . (B.1)Here, p = x , p = x (0 < x <
1, 0 < x < x + x = 1). Since Alice’s strategyis deterministic one, each of the two probability distributions P Sv ( a | x ) and P Sv ( a | x )must be equal to any one among P D , P D , P D and P D . In order to satisfy themarginal probabilities for Alice P SvF ( a | x ) = ∀ a, x , the only two possible choices of P Svλ ( a | x ) are:1) P D and P D with x = x = P D and P D with x = x = .In case of the first choice, let us assume that P Sv ( a | x ) = P D , P Sv ( a | x ) = P D ; P Sv ( bc | yz ) and P Sv ( bc | yz ) are given by, P Sv ( bc | yz ) := u u u u u u u u u u u u u u u u ,where 0 ≤ u ij ≤ ∀ i, j , and (cid:80) j u ij = 1 ∀ i , and P Sv ( bc | yz ) := w w w w w w w w w w w w w w w w ,where 0 ≤ w ij ≤ ∀ i, j , and (cid:80) j w ij = 1 ∀ i .Now, with this choice, the box P µSvF given by the model (B.1) has perational nonclassicality of local multipartite correlations P µSvF = abcxyz
000 001 010 011 100 101 110 111000 u u u u w w w w u u u u w w w w u u u u w w w w u u u u w w w w u u u u w w w w u u u u w w w w u u u u w w w w u u u u w w w w , (B.2)where each row and column corresponds to a fixed measurement ( xyz ) and a fixedoutcome ( abc ) respectively.From Eq. (B.2), it can be seen that P µSvF ( abc | P µSvF ( abc | , which is not true for the fully local Svetlichny family as given in Eq. (26) with0 < µ ≤ √ . Because, in case of fully local P µSvF (0 < µ ≤ √ ) given in Eq. (26), P µSvF ( abc | − a ⊕ b ⊕ c √ µ P µSvF ( abc | − a ⊕ b ⊕ c ⊕ √ µ . Again, from Eq. (B.2), it can be seen that P µSvF ( abc | P µSvF ( abc | , which is not true for the fully local Svetlichny family as given in Eq. (26) with0 < µ ≤ √ . Because, in case of fully local P µSvF (0 < µ ≤ √ ) given in Eq. (26), P µSvF ( abc | − a ⊕ b ⊕ c √ µ P µSvF ( abc | − a ⊕ b ⊕ c ⊕ √ µ . Hence, in this case, though the marginal probabilities for Alice P SvF ( a | x ) aresatisfied, all the tripartite joint probability distributions P µSvF are not satisfied simul-taneously.Similarly, in case of the first choice, if we assume that P Sv ( a | x ) = P D , P Sv ( a | x ) = P D , then the marginal probabilities for Alice P SvF ( a | x ) are satisfied,but all the tripartite joint probability distributions P µSvF are not satisfied simultane-ously. perational nonclassicality of local multipartite correlations P Sv ( a | x ) = P D , P Sv ( a | x ) = P D ; P Sv ( bc | yz ) and P Sv ( bc | yz ) are given by, P Sv ( bc | yz ) = u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) ,where 0 ≤ u (cid:48) ij ≤ ∀ i, j , and (cid:80) j u (cid:48) ij = 1 ∀ i , and P Sv ( bc | yz ) = w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) ,where 0 ≤ w (cid:48) ij ≤ ∀ i, j , and (cid:80) j w (cid:48) ij = 1 ∀ i .Now, with this choice, the box P µSvF (0 < µ ≤ √ ) given by the model (B.1) has, P µSvF = abcxyz
000 001 010 011 100 101 110 111000 u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) w (cid:48) w (cid:48) w (cid:48) w (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) . (B.3)From Eq. (B.3), it can be seen that P µSvF (0 bc | P µSvF (1 bc | , which is not true for the fully local Svetlichny family as given in Eq. (26) with0 < µ ≤ √ . Because, in case of fully local P µSvF (0 < µ ≤ √ ) given in Eq. (26), P µSvF (0 bc | − b ⊕ c √ µ P µSvF (1 bc | − b ⊕ c ⊕ √ µ . From Eq. (B.3), it can be seen that P µSvF (1 bc | P µSvF (0 bc | , perational nonclassicality of local multipartite correlations < µ ≤ √ . Because, in case of fully local P µSvF (0 < µ ≤ √ ) as given in Eq.(26), P µSvF (1 bc | − b ⊕ c ⊕ √ µ P µSvF (0 bc | − b ⊕ c √ µ . Again, from Eq. (B.3), it can be seen that P µSvF (0 bc | P µSvF (1 bc | , which is not true for fully local P µSvF as given in Eq. (26) for 0 < µ ≤ √ . Because,in case of fully local P µSvF (0 < µ ≤ √ ) as given in Eq. (26), P µSvF (0 bc | − b ⊕ c ⊕ √ µ P µSvF (1 bc | − b ⊕ c √ µ P µSvF (1 bc | P µSvF (0 bc | , (B.4)which is not true for fully local P µSvF as given in Eq. (26) for 0 < µ ≤ √ . Because,in case of fully local P µSvF (0 < µ ≤ √ ) as given in Eq. (26), P µSvF (1 bc | − b ⊕ c √ µ P µSvF (0 bc | − b ⊕ c ⊕ √ µ . Hence, in this case, though the marginal probabilities for Alice P SvF ( a | x ) aresatisfied, all the tripartite joint probability distributions P µSvF are not satisfied simul-taneously.Similarly, in case of the second choice, if we assume that P Sv ( a | x ) = P D , P Sv ( a | x ) = P D , then the marginal probabilities for Alice P SvF ( a | x ) are satisfied,but all the tripartite joint probability distributions P µSvF are not satisfied simultane-ously.Note that this proof is valid for any two bipartite correlations P Svλ ( bc | yz )( λ = 0 ,
1) shared between Bob and Charlie at each λ A − BC without any constrainton the correlations. Hence, it is obvious that this proof will also be valid when thesetwo bipartite correlations P Svλ ( bc | yz ) ( λ = 0 ,
1) at Bob-Charlie’s side are NS boxesand satisfy the marginal probabilities P SvF ( b | y ), P SvF ( c | z ) at Bob and Charlie’s side, perational nonclassicality of local multipartite correlations P µSvF (0 < µ ≤ √ )in the scenario as in Fig. 1 where Alice preshares the hidden variable λ A − BC ofdimension 2 with Bob-Charlie. Appendix C. -local form across the bipartition ( A | BC ) for the fully localMermin family P νMF in the range < ν ≤ For 0 < ν ≤ the fully local noisy Mermin box can be decomposed in a convexmixture of the 3-local deterministic boxes. In this range, we consider the followingdecomposition for the Mermin family in terms of the 3-local deterministic boxes: P νMF = 14 P D (cid:40) ν (cid:16) P D + P D + P D + P D + P D + P D + P D + P D (cid:17) + 1 − ν (cid:16) P D + P D + P D + P D (cid:17)(cid:41) + 14 P D (cid:40) ν (cid:16) P D + P D + P D + P D + P D + P D + P D + P D (cid:17) + 1 − ν (cid:16) P D + P D + P D + P D (cid:17)(cid:41) + 14 P D (cid:40) ν (cid:16) P D + P D + P D + P D + P D + P D + P D + P D (cid:17) + 1 − ν (cid:16) P D + P D + P D + P D (cid:17)(cid:41) + 14 P D (cid:40) ν (cid:16) P D + P D + P D + P D + P D + P D + P D + P D (cid:17) + 1 − ν (cid:16) P D + P D + P D + P D (cid:17)(cid:41) := (cid:88) λ =0 r λ P Mλ ( a | x ) P Mλ ( bc | yz ) , (C.1) perational nonclassicality of local multipartite correlations r = r = r = r = ; P M ( a | x ) = P D , P M ( a | x ) = P D ,P M ( a | x ) = P D , P M ( a | x ) = P D ;and P M ( bc | yz ) = (cid:40) ν (cid:16) P D + P D + P D + P D + P D + P D + P D + P D (cid:17) + 1 − ν (cid:16) P D + P D + P D + P D (cid:17)(cid:41) = ν − ν − ν ν ν − ν − ν ν ν − ν − ν ν − ν ν ν − ν ,P M ( bc | yz ) = − ν ν ν − ν − ν ν ν − ν − ν ν ν − ν ν − ν − ν ν ,P M ( bc | yz ) = − ν ν ν − ν ν − ν − ν ν ν − ν − ν ν ν − ν − ν ν ,P M ( bc | yz ) = ν − ν − ν ν − ν ν ν − ν − ν ν ν − ν − ν ν ν − ν . Note that, each of the probability distributions P Mλ ( bc | yz ) is local for 0 < ν ≤ ,as they satisfy the complete set of Bell-CHSH inequalities [44, 45] in this range. Infact, each of the P Mλ ( bc | yz ) is superlocal in the range 0 < ν ≤ . Because each of the P Mλ ( bc | yz ) belongs to CHSH family [13], P CHSH ( bc | yz ) = 2 + ( − b ⊕ c ⊕ yz √ V , (C.2)with 0 < V = √ ν ≤
1, upto local reversible operations. In Ref. [14], it has beenshown that the above local box is superlocal for any
V > < ν ≤ ), the decomposition (C.1) definesa classical simulation protocol with different deterministic distributions P Mλ ( bc | yz ) atAlice’s side, where Alice shares hidden variable λ A − BC of dimension 4 as in Fig. 1.Decomposition (C.1) represents 2-local form across the bipartite cut ( A | BC ) of thefully local noisy Mermin-box (0 < ν ≤ ). perational nonclassicality of local multipartite correlations Appendix D. -local form across the bipartition ( A | BC ) for the Merminfamily P νMF in the range < ν ≤ < ν ≤ P <ν< MF = 14 P D ( a | x ) νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) + 14 P D ( a | x ) νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) + 14 P D ( a | x ) νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) + 14 P D ( a | x ) νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) := (cid:88) λ =0 r λ P Mλ ( a | x ) P Mλ ( bc | yz ) , (D.1)where r = r = r = r = ; P M ( a | x ) = P D , P M ( a | x ) = P D ,P M ( a | x ) = P D , P M ( a | x ) = P D ;and P M ( bc | yz ) = νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) ,P M ( bc | yz ) = νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) ,P M ( bc | yz ) = νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) ,P M ( bc | yz ) = νP P R ( bc | yz ) + 1 − ν (cid:88) γ,(cid:15),ζ,η P γ(cid:15)ζηD ( bc | yz ) . Note that, each of the probability distributions P Mλ ( bc | yz ) satisfy NS principle for0 < ν ≤
1, Hence, the noisy Mermin box has the 2-local form across the bipartite cut( A | BC ) in the range 0 < ν ≤