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.
Highlights
Multipartite entanglement is a central feature of quantum many-body systems, fundamentally challenging our ability to efficiently simulate them on classical computers [1,2]
We have presented a new data-driven method to detect multipartite entanglement in quantum simulators and computers
In order to do so, we have expressed the two-body coefficients of the Bell inequality as a positive semidefinite matrix, whose optimization allows for a systematic exploration of all potentially violated Bell inequalities of this form
Summary
Multipartite entanglement is a central feature of quantum many-body systems, fundamentally challenging our ability to efficiently simulate them on classical computers [1,2]. Failing to violate all known Bell inequalities does not imply that device-independent entanglement certification is impossible based on the available data This motivates the development of data-driven methods [5,7,11], where the data serve as input into an algorithm, which builds, from the data themselves, a tailored Bell inequality. The Bell inequalities inferred by our method are nonlinear in the input data, and tightly wrap around the polytope of local-variable models (see Fig. 2 for an example) This feature offers a fundamental scaling improvement regarding experimental requirements, including for all previously known Bell inequalities invariant under permutations [7,8,9,10,12].
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