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arXiv:2504.20953

Optimal Local Simulations of a Quantum Singlet

David Llamas, Dmitry V. Chistikov, Adrian Kent, Mike Paterson, Olga Goulko

Published Apr 29, 2025• 0 citations
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Abstract

Bell's seminal work showed that no local hidden variable (LHV) model can fully reproduce the quantum correlations of a two-qubit singlet state. His argument and later developments by Clauser et al. effectively rely on gaps between the anticorrelations achievable by classical models and quantum theory for projective measurements along randomly chosen axes separated by a fixed angle. However, the size of these gaps has to date remained unknown. Here we numerically determine the LHV models maximizing anticorrelations for random axes separated by any fixed angle, by mapping the problem onto ground state configurations of fixed-range spin models. We identify angles where this gap is largest and thus best suited for Bell tests. These findings enrich the understanding of Bell non-locality as a physical resource in quantum information theory and quantum cryptography.

DOI
10.48550/arxiv.2504.20953

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The essence of Bell’s theorem is that, in general, quantum statistics cannot be reproduced by a local hidden variable (LHV) model. This impossibility is strongly manifested when statistics collected by measuring certain local observables on a singlet state, violates the Bell inequality. In this work, we search for local POVMs with binary outcomes for which an LHV model can be constructed for a singlet state. We provide various subsets of observables for which an LHV model can be provided for singlet statistics.
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An alternative method of detection-loophole-free Bell test is proposed using local hidden variable (LHV) models with optimal detection efficiencies. A framework for constructing such optimal LHV models is presented. Optimal LHV models for maximally and non-maximally entangled twopartite states are constructed to reproduce the quantum correlations within the critical detection efficiencies. The LHV models are shown to be completely equivalent with the existing twopartite Bell inequalities in their optimized setups, and to have even lower critical efficiencies in the LHV modes' own optimized setups. Applications in Bell tests and in device-independent quantum information processing are discussed.
Constructing local hidden variable (LHV) models for entangled quantum states is a fundamental problem, with implications for the foundations of quantum theory and for quantum information processing. It is, however, a challenging problem, as the model should reproduce quantum predictions for all possible local measurements. Here we present a simple method for building LHV models, applicable to any entangled state and considering continuous sets of measurements. This leads to a sequence of tests which, in the limit, fully captures the set of quantum states admitting a LHV model. Similar methods are developed for local hidden state models. We illustrate the practical relevance of these methods with several examples.
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Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.