We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S 1 × · · · × S Nsetting Bell-type inequalities -either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S 1 × · · · × S N -setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either on correlation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1) N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes. C 2012 American Institute of Physics.
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.
We introduce a single general representation incorporating in a unique manner all Bell-type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site. Specifying this general representation for correlation functions, we prove that the form of any correlation Bell-type inequality does not depend on a spectral type of outcomes, in particular, on their numbers at different sites, and is determined only by extremal values of outcomes at each site. We also specify the general form of bounds in Bell-type inequalities on joint probabilities. Our approach to the derivation of Bell-type inequalities is universal, concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type, discrete or continuous. We, in particular, prove that, for an N-partite quantum state, possibly, infinite dimensional, admitting the 2 × ... × 2 N -setting LHV description, the Mermin-Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.
In contrast to the wide-spread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where a separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements.In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system in a separable state. We derive quantum analogues of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended CHSH inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.
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