Inferring Nonlinear Many-Body Bell Inequalities From Average Two-Body Correlations: Systematic Approach for Arbitrary Spin-j Ensembles

TL;DR

This paper introduces a novel, scalable data-driven method for inferring nonlinear Bell inequalities from average two-body correlations in many-body quantum systems, applicable to arbitrary spin-j ensembles, enhancing the detection of entanglement.

Contribution

The authors develop a flexible, quadratic-in-data approach that improves over previous methods by handling multiple outcomes and settings, enabling systematic discovery of Bell inequalities for complex quantum states.

Findings

Successfully applied to quantum spin-1/2 ensembles.

Discovered new Bell inequalities for spin-squeezed states.

Robustly violated inequalities for many-body spin singlets.

Abstract

Violating Bell's inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions -- in a device-independent manner. Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavour. In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties. For two-outcomes measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs. Data-driven methods -- reconstructing a violated BI from the data themselves -- have therefore been considered. Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data. Our approach offers two main improvements over the existing literature: 1) it is directly designed for any number of outcomes and settings; 2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments. This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously-known Bell's inequalities robustly violated by ensembles of quantum spin-$1/2$; and to discover novel families of Bell's inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-$j$ ensembles.