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Journal of Mathematical Physics

Small violations of Bell inequalities for multipartite pure random states

Raphael C. Drumond, Cristhiano Duarte, Roberto I. Oliveira

Published May 1, 2018• 4 citations
Publisher

Abstract

For any finite number of parts, measurements, and outcomes in a Bell scenario, we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension, the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.

DOI
10.1063/1.5026690

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For any finite number of parts, measurements, and outcomes in a Bell scenario, we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension, the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.
For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random NNN-qudddit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the the local Hilbert space dimension goes to infinity.
We estimate the probability of random NNN-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with NNN, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with NNN is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing any violation is already small. For the qubits case we discuss, furthermore, the consequences of this result for the probability of seeing arbitrary violations (i.e., of any order of magnitude) when experimental imperfections are considered.
In this work we investigate the probability of violation of local realism under random measurements in parallel with the strength of these violations as described by resistance to white noise admixture. We address multisetting Bell scenarios involving up to seven qubits. As a result, in the first part of this manuscript, we report statistical distributions of a quantity reciprocal to the critical visibility for various multipartite quantum states subjected to random measurements. The statistical relevance of different classes of multipartite tight Bell inequalities violated with random measurements is investigated. We also introduce the concept of typicality of quantum correlations for pure states as the probability to generate a nonlocal behavior with both random state and measurement. Although this typicality is slightly above 5.3% for two qubits, for a modest increase in the number of involved particles it quickly surpasses 99.99%.
We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.
We study the explicit relation between violation of Bell inequalities and bipartite distillability of multiqubit states. It has been shown that even though for $N\ensuremath{\ge}8$ there exist NNN-qubit bound entangled states which violates a Bell inequality [W. D\"ur, Rev. Lett. 87, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [A. Ac\'{\i}n, Rev. Lett. 88, 027901 (2002)]. We here prove that for all NNN-qubit states violating the inequality the number of distillable bipartite splits increases exponentially with NNN, and hence the probability that a randomly chosen bipartite split is distillable approaches 1 exponentially with NNN, as NNN tends to infinity. We also show that there exists at least one NNN-qubit bound entangled state violating the inequality if and only if $N\ensuremath{\ge}6$.