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arXiv:1609.06114 · Quantum

Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant KG(3)

Flavien Hirsch, Marco Túlio Quintino, Tamás Vértesi, Miguel Navascués, Nicolas Brunner

Published Apr 25, 2017• 86 citations
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Abstract

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.

DOI
10.22331/q-2017-04-25-3

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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 relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ${\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4$, we show that there is a local model for projective measurements if and only if $p\ensuremath{\leqslant}1∕{K}_{G}(3)$, where KG(3){K}_{G}(3)KG​(3) is Grothendieck's constant of order 3. Known bounds on KG(3){K}_{G}(3)KG​(3) prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary 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 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.
Contemporary versions of Bell’s argument [Physics (Long Island City, N.Y.) 1, 195 (1964)] against local hidden variable (LHV) theories are based on the Clauser-Horne-Shimony-Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)] inequality and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper, we derive two-particle Bell inequalities for traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constants both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations and applying a recent result of Alon et al. [Invent. Math. 163, 499 (2006)], we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to infinity as the dimension and number of measurements grow. Technically, these result are possible because perfect correlations (or anticorrelations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%.
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.