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Logical Families of Nonlocal Boxes

Zeno Toffano, Cassandre Chatard, Kaicheng Ding, Alice Portalier, Gregoire Vilde

Published Nov 18, 2015• 0 citations
OpenAlex

Abstract

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.

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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.
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.
We quantify the amount of non-locality contained in n noisy versions of the so-called Popescu–Rohrlich boxes ( PRBs ), i.e. bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a local behaviour, with maximal possible weight, and a non-signalling system. We show that the local part of n systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε ⌈ n /2⌉ ) in the isotropic, and equal to (3ε) n in the maximally biased case.
The intriguing and powerful capability of nonlocality in communication field ignites the research of the nonlocality distillation. The first protocol presented in Ref[Phys. Rev. Lett. 102, 120401] shows that the nonlocality of bipartite binary-input and binary-output nonsignaling correlated boxes could be amplified by 'wiring' two copies of weaker-nonlocality boxes. Several optimized distillation protocols were presented later for bipartite binary-input and binary-output nonsignaling correlated boxes. In this paper, we focus on the bipartite binary-input and multi-nary-output nonsignaling correlated boxes---high-dimensional boxes, and design comparators-based protocols to achieve the distillation of high-dimensional nonlocality. The results show that the high-dimensional nonlocality can be distilled in different ways, and we find that the efficiencies of the protocols are influenced not only by the wirings but also by the classes the initial nonlocality boxes belongs to. Here, the initial nonlcalities may have the same violation of the high-dimensional Bell-type inequality, but they can fall into different classes, which shows that the value of the violation of the high-dimensional Bell-type inequality is not the only representation of nonlocality, and the combination manner(classes) in the expression of the correlated boxes is another important nonlocal representation too. The current protocols are compatible with the previous two-dimensional nonlocality distillation protocols, but our protocols are more powerful and universal than previous ones in the sense that the current protocols can be applied to the system with any dimension rather than the only two-dimension system in the previous protocols.
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
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.