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arXiv:0804.4046 · Journal of Physics A Mathematical and Theoretical

Multipartite Bell-type inequalities for arbitrary numbers of settings and outcomes per site

Elena R. Loubenets

Published Oct 7, 2008• 22 citations
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Abstract

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 spectral types 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 -setting LHV description, the Mermin–Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.

DOI
10.1088/1751-8113/41/44/445304

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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 spectral types 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 -setting LHV description, the Mermin–Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.
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 a local hidden variable (LHV) model for an S 1 × ⋅ ⋅ ⋅ × S N -setting 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 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 S1 × ⋯ × SN-setting 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 S1 × ⋯ × SN-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.
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.
Derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and N possible local settings is not as easy as it seemed. The proof of v1 is wrong. Additionaly one can find a counterexample, which will be presented soon. Thus our thesis is dead. Still the series of Bell inequalities discussed in the manuscript (v1) form a necessary condition for local realism, and are tight. They are tight and complete (sufficient) only for N=3 settings per observer (as shown in quant-ph/0611086, fortunately using an entirely different approach).
Facet inequalities play an important role in detecting the nonlocality of a quantum state. The number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, measurement outcomes, or/and the number of measurement settings, there are more nontrivial facet inequalities. For several Bell scenarios, by involving two dichotomic measurement settings for two parties and one dichotomic measurement by other parties, we show that the local polytope has only one nontrivial facet. For three parties, we have three variants of this inequality, depending upon which party is doing one dichotomic measurement. This measurement scenario for a multipartite state may be considered as the minimal scenario involving multipartite correlations that can detect nonlocality. We show that this inequality is violated by all generalized GHZ states. Being the only facet Bell inequality, this inequality is also violated by any entangled three-qubit pure state. We also show that for noisy [Formula: see text] states, our inequality is more effective than the well-known Mermin inequality.