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arXiv:quant-ph/0603094 · Journal of Mathematical Physics

Bell-type inequalities for nonlocal resources

Nicolas Brunner, Valerio Scarani, Nicolas Gisin

Published Nov 1, 2006• 9 citations
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Abstract

We present bipartite Bell-type inequalities which allow the two partners to use some nonlocal resource. Such inequalities can only be violated if the parties use a resource which is more nonlocal than the one permitted by the inequality. We introduce a family of N-input nonlocal machines, which are generalizations of the well-known PR (Popescu-Rohrlich) box. Then we construct Bell-type inequalities that cannot be violated by strategies that use one of these new machines. Finally we discuss implications for the simulation of quantum states.

DOI
10.1063/1.2352857

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We present bipartite Bell-type inequalities which allow the two partners to use some nonlocal resource. Such inequalities can only be violated if the parties use a resource which is more nonlocal than the one permitted by the inequality. We introduce a family of N-input nonlocal machines, which are generalizations of the well-known PR (Popescu-Rohrlich) box. Then we construct Bell-type inequalities that cannot be violated by strategies that use one of these new machines. Finally we discuss implications for the simulation of quantum states.
We present bipartite Bell-type inequalities which allow the two partners to use some non-local resource. Such inequality can only be violated if the parties use a resource which is more non-local than the one permitted by the inequality. We introduce a family of N-inputs non-local machines, which are generalizations of the well-known PR-box. Then we construct Bell-type inequalities that cannot be violated by strategies that use one these new machines. Finally we discuss implications for the simulation of quantum states.
The propensity for certain quantum system to consistently violate Bell inequalities—dubbed quantum nonlocality—has recently been found to be an information theoretic resource in quantum communication that is distinct from entanglement. Accordingly, a useful measurement of “how much” of this resource is present in a given system is needed. One candidate, the PR‐box (sometimes called a nonlocal machine), has been suggested as a possible benchmark, but one of the difficulties in using it has been to determine exactly what kinds of quantum systems are simulable by it. The present work is a first step towards a graph‐theoretic understanding of PR‐boxes that may prove useful in this endeavor.
We consider general settings of Bell inequality experiments with many parties, where each party chooses from a finite number of measurement settings each with a finite number of outcomes. We investigate the constraints that Bell inequalities place upon the correlations possible in local hidden variable theories using a geometrical picture of correlations. We show that local hidden variable theories can be characterized in terms of limited computational expressiveness, which allows us to characterize families of Bell inequalities. The limited computational expressiveness for many settings (each with many outcomes) generalizes previous results about the many-party situation each with a choice of two possible measurements (each with two outcomes). Using this computational picture we present generalizations of the Popescu-Rohrlich nonlocal box for many parties and nonbinary inputs and outputs at each site. Finally, we comment on the effect of preprocessing on measurement data in our generalized setting and show that it becomes problematic outside of the binary setting, in that it allows local hidden variable theories to simulate maximally nonlocal correlations such as those of these generalized Popescu-Rohrlich nonlocal boxes.
Imagine a task in which a group of separated players aim to simulate a statistic that violates a Bell inequality. Given measurement choices the players shall announce an output based solely on the results of local operations---which they can discuss before the separation---on shared random data and shared copies of a so-called unit correlation. In the first part of this paper we show that in such a setting the simulation of any bipartite correlation, not containing the possibility of signaling, can be made arbitrarily accurate by increasing the number of shared Popescu-Rohrlich (PR) boxes. This establishes the PR box as a simple asymptotic unit of bipartite nonlocality. In the second part we study whether this property extends to the multipartite case. More generally, we ask if it is possible for separated players to asymptotically reproduce any nonsignaling statistic by local operations on bipartite unit correlations. We find that nonadaptive strategies are limited by a constant accuracy and that arbitrary strategies on nnn resource correlations make a mistake with a probability greater or equal to c/nc/nc/n, for some constant ccc.
In this paper, we will study the behavior of nonlocal boxes and non-signaling boxes by calculating a Bell Inequalities (BI) for all possible logical combinations of inputs and outputs. We have considered the case of equal number n of inputs and outputs, first for n = 2 using the CHSH BI and then generalized to the case n = 3 using the Svetlichny BI. n = 2 corresponds to 256 cases and increases exponentially for n = 3 (256x 256 = 65536 cases). We thus developped a computer program for processing the different cases by calculating the inequalities and also testing non-signaling in a systematic way. Maximal Bell parameter of 4 is obtained for the no-signaling family of 16 nonlocal boxes the PR Box being part of this family. Another family of 32 nonlocal boxes (green cases) is signaling and giving a BI violation 10/3 > 2sqrt(2)> 2, exceeding the quantum limit. The three-party correlation logical boxes give 65536 combinations of logical functions. Calculation (computer) of the Svetlichny leads to nonlocality for six different values.