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arXiv:1412.1416 · Physical Review Letters

Local Hidden Variable Models for Entangled Quantum States Using Finite Shared Randomness

Joseph Bowles, Flavien Hirsch, Marco Túlio Quintino, Nicolas Brunner

Published Mar 24, 2015• 33 citations
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Abstract

The statistics of local measurements performed on certain entangled states can be reproduced using a local hidden variable (LHV) model. While all known models make use of an infinite amount of shared randomness, we show that essentially all entangled states admitting a LHV model can be simulated with finite shared randomness. Our most economical model simulates noisy two-qubit Werner states using only log_{2}(12)≃3.58 bits of shared randomness. We also discuss the case of positive operator valued measures, and the simulation of nonlocal states with finite shared randomness and finite communication. Our work represents a first step towards quantifying the cost of LHV models for entangled quantum states.

DOI
10.1103/physrevlett.114.120401

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The statistics of local measurements performed on certain entangled states can be reproduced using a local hidden variable (LHV) model. While all known models make use of an infinite amount of shared randomness, we show that essentially all entangled states admitting a LHV model can be simulated with finite shared randomness. Our most economical model simulates noisy two-qubit Werner states using only log_{2}(12)≃3.58 bits of shared randomness. We also discuss the case of positive operator valued measures, and the simulation of nonlocal states with finite shared randomness and finite communication. Our work represents a first step towards quantifying the cost of LHV models for entangled quantum states.
Measurement correlations in quantum systems can exhibit nonlocal behavior, a fundamental aspect of quantum mechanics with applications such as device-independent quantum information processing. However, it is in general not known which states are local and which ones are not. In particular, it remains an outstanding challenge to explicitly construct local hidden-variable (LHV) models for arbitrary multipartite entangled states. To address this, we use gradient-descent algorithms from machine learning to find LHV models that reproduce the statistics of arbitrary measurements for quantum many-body states. In contrast to previous approaches, our method employs a general ansatz, enabling it to discover LHV models for all local states. Therefore, it for example provides actual estimates for the critical noise levels at which two-qubit Werner states and three-qubit Greenberger-Horne-Zeilinger and W states become local. Furthermore, we find evidence suggesting that two-spin subsystems in the ground states of translationally invariant Hamiltonians are genuinely local, while bigger subsystems are in general not. Our method now offers a quantitative tool for determining the regimes of nonlocality in any given physical context, such as nonequilibrium, decoherence, or disorder.
The study of local models using finite shared randomness originates from the consideration about the cost of classically simulating entanglement in composite quantum systems. We construct explicitly two families of local-hidden-state (LHS) models for T-states, by mapping the problem to the Werner state. The continuous decreasing of shared randomness along with entanglement, as the anisotropy increases, can be observed in the one from the most economical model for the Werner state. The construction of the one for separable states shows that the separable boundary of T-states can be generated from the one of the Werner state, and the cost is 2 classical bits.
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.
We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state ρ=v|ψ−⟩⟨ψ−|+(1−v)14 via a local hidden variable (LHV) model, where |ψ−⟩ denotes the singlet state. We show analytically that these correlations are local for v=999×689×10−6 cos2⁡(π/50)≃0.6829 . In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant KG(3)≤1/v≃1.4644 . We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v≃0.4553 . The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.
We present a general criterion for deciding the existence of local-hidden-state (LHS) models and consequently local-hidden-variable (LHV) models for arbitrary states that can be implemented by semidefinite programming. This method can be applied to any given state and general (POVM) measurements and can also be adapted to generate random states with LHS models. As applications we provide an estimate of the fraction of two-qubit entangled states that have LHS models, and present new explicit examples of such states, including Bell-diagonal states, and noisy GHZ and W states. We also generate a list of 750000 new entangled two-qubit states with LHS model for projective measurements, 1500 for POVM measurements, and 150 genuine multipartite entangled three-qubit local states.
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.