Bell nonlocality using tensor networks and sparse recovery

I S Eliëns,S G A Brito,R Chaves

doi:10.1103/physrevresearch.2.023198

I S Eliëns, S G A Brito + Show 1 more

Open Access

https://doi.org/10.1103/physrevresearch.2.023198

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Abstract

Bell's theorem, stating that quantum predictions are incompatible with a local hidden variable description, is a cornerstone of quantum theory and at the center of many quantum information processing protocols. Over the years, different perspectives on non-locality have been put forward as well as different ways to to detect non-locality and quantify it. Unfortunately and in spite of its relevance, as the complexity of the Bell scenario increases, deciding whether a given observed correlation is non-local becomes computationally intractable. Here, we propose to analyse a Bell scenario as a tensor network, a perspective permitting to test and quantify non-locality resorting to very efficient algorithms originating from compressed sensing and that offer a significant speedup in comparison with standard linear programming methods. Furthermore, it allows to prove that non-signalling correlations can be described by hidden variable models governed by a quasi-probability.

Highlights

Bell’s theorem [1] shows that quantum predictions are at odds with the physical intuition from classical physics
That the correlations obtained by local measurements on distant but entangled systems cannot be reproduced by any local hidden variable model, the phenomenon generally known as Bell nonlocality [2]
Bell nonlocality is a cornerstone of quantum theory and central resource in a variety of quantum information processing protocols

Summary

INTRODUCTION

Bell’s theorem [1] shows that quantum predictions are at odds with the physical intuition from classical physics. Notwithstanding, the LP approach suffers from the curse of dimensionality, being of no use as the number of parties, measurements, settings, or measurement outcomes increase in the Bell scenario Motivated by these issues [9,14,15,16,17,18], our aim in this paper is to offer an alternative view on Bell nonlocality, based on tensor networks [19] and sparse recovery [20,21,22,23]. We show that nonsignaling correlations (including nonlocal correlations) can be mapped to hidden variable models governed by quasiprobabilities, that is, quantities that sum up to one but are not necessarily positive [33,34] This points out a way to detect and quantify nonlocality with tools originating from the field of compressed sensing [35]. We show that, formulated this way, sparse recovery algorithms allow for a significant speedup in the detection of nonlocality in comparison with the standard LP approach

BELL SCENARIO AS A TENSOR NETWORK

THE DECOMPOSITION OF NONSIGNALING

RECOVERING THE HIDDEN VARIABLE AND QUANTIFYING NONLOCALITY

CONCLUSIONS