On the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site

Abstract

We consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature. This allows us: (1) to specify in probabilistic terms the difference between nonsignaling, the Einstein-Podolsky-Rosen (EPR) locality and Bell's locality; (2) to introduce the notion of an LHV model for an S 1 × ... × S Nsetting N -partite correlation experiment with outcomes of any spectral type, discrete or continuous, and to prove both general and specifically "quantum" statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for a multipartite quantum state, in particular, to show that any N -partite quantum state, pure or mixed, admits an arbitrary S 1 × 1 × ... × 1-setting LHV description; (4) to evaluate a threshold visibility for an arbitrary bipartite noisy quantum state to admit an S 1 × S 2 -setting LHV description under any generalized quantum measurements of two parties.

Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations

“…If an LHV model (12) for joint probabilities is deterministic [38,37], then the values of functions f ai , f b k , i, k = 1, 2, constitute outcomes under Alice and Bob corresponding measurements with settings a i and b k , respectively. However, in a stochastic LHV model [38,37], functions f ai , f b k may take any values in [−1, 1] even in a dichotomic case.…”

Quantifying Tolerance of a Nonlocal Multi-Qudit State to Any Local Noise

“…Let us shortly recall the notion of a general multipartite Bell inequality [25] with arbitrary numbers of settings and outcomes per site. For the general framework on the probabilistic description of an arbitrary multipartite correlation scenario with any number of settings and any spectral type of outcomes at each site, see [26]. Consider a correlation scenario, where each n-th of N parties performs S n ≥ 1 measurements with outcomes λ n ∈ [−1, 1] and every measurement at n-th site is specified by a positive integer s n = 1, ..., S n .…”

“…. · P N,s N (dλ N |ω)ν E S (dω) 4 For the general statements on the LHV modelling, see section 4 in [26].…”

No-signaling, perfect bipartite dichotomic correlations and local randomness

“…In this space the no-signaling polytope is the convex hull of the 16 local extreme points (12) and of those given by (20). Its facet inequalities are just the 8 trivial inequalities in (13) and therefore it is in fact just the four-dimensional unit cube [26].…”

“…These are thus the trivial facets that follow from the positivity conditions as well as the non-trivial ones that follow from the no-signaling requirements (6). The importance of the non-trivial facets of the no-signaling polytope is that if a point, representing some experimental data, lies within the polytope, then a model that 2 Loubenets [20] claims that this latter claim is not true in general, and that one should distinguish the, where the first is called 'no-signaling' and the second 'EPR locality'. However, if one quantifies over all b and b ′ -as one should-these conditions in fact become identical.…”