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arXiv:0811.1649 · Journal of Physics A Mathematical and Theoretical

The non-locality ofnnoisy Popescu–Rohrlich boxes

Matthias Fitzi, Esther Hänggi, Valerio Scarani, Stefan Wolf

Published Oct 29, 2010• 17 citations
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Abstract

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.

DOI
10.1088/1751-8113/43/46/465305

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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.
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 a no-signaling world, the outputs of a nonlocal box cannot be completely predetermined, a feature that is exploited in many quantum information protocols exploiting nonlocality, such as device-independent randomness generation and quantum key distribution. This relation between nonlocality and randomness can be formally quantified through the min-entropy, a measure of the unpredictability of the outputs that holds conditioned on the knowledge of any adversary that is limited only by the no-signaling principle. This quantity can easily be computed for the noisy Popescu-Rohrlich (PR) box, the paradigmatic example of nonlocality. In this paper, we consider the min-entropy associated to several copies of noisy PR boxes. In the case where nnn noisy PR boxes are implemented using nnn noncommunicating pairs of devices, it is known that each PR box behaves as an independent biased coin: the min-entropy per PR box is constant with the number of copies. We show that this does not hold in more general scenarios where several noisy PR boxes are implemented from a single pair of devices, either used sequentially nnn times or producing nnn outcome bits in a single run. In this case, the min-entropy per PR box is smaller than the min-entropy of a single PR box, and it decreases as the number of copies increases.
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.
Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.
One of the most fascinating consequences of quantum theory is non‐locality, i.e., the fact that the measurement outcomes of (spatially separated) parts of a system can have a correlation unexplainable by shared classical information. Note that at the same time, these correlations are non‐signaling and do not allow for message transmission. Popescu and Rohrlich have defined a non‐local box as a “basic building block of non‐locality” and initiated a systematic study of non‐local correlations and their applications. However, they left open the question of whether any non‐signaling correlation can be simulated by such non‐local boxes. We show that the answer is yes with respect to arbitrarily accurate approximations.
Supraquantum nonlocality refers to correlations that are more nonlocal than allowed by quantum theory but still physically conceivable in postquantum theories, in the sense of respecting the basic no-faster-than-light communication principle. While supraquantum correlations are relatively well understood for finite-dimensional systems, little is known in the infinite-dimensional case. Here, we study supraquantum nonlocality for bipartite systems with two measurement settings and infinitely many outcomes per subsystem. We develop a formalism for generic no-signaling black-box measurement devices with continuous outputs in terms of probability measures, instead of probability distributions, which involves a few technical subtleties. We show the existence of a class of supraquantum Gaussian correlations, which violate the Tsirelson bound of an adequate continuous-variable Bell inequality. We then introduce the continuous-variable version of the celebrated Popescu--Rohrlich (PR) boxes, as a limiting case of the above-mentioned Gaussian ones. Finally, we characterize the geometry of the set of continuous-variable no-signaling correlations. Namely, we show that that the convex hull of the continuous-variable PR boxes is dense in the no-signaling set. We also show that these boxes are extreme in the set of no-signaling behaviors and provide evidence suggesting that they are indeed the only extreme points of the no-signaling set. Our results lay the grounds for studying generalized-probability theories in continuous-variable systems.