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arXiv:1205.1475 · Journal of Physics A Mathematical and Theoretical

Local simulation of singlet statistics for a restricted set of measurements

Ashutosh Rai, Md. Rajjak Gazi, Manik Banik, Subhadipa Das, Samir Kunkri

Published Nov 8, 2012• 7 citations
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Abstract

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.

DOI
10.1088/1751-8113/45/47/475302

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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.
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.
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.
The experimental violation of the Bell inequality establishes necessary but not sufficient conditions that any theory must obey. Namely, a theory compatible with the experimental observations can satisfy at most two of the three hypotheses at the basis of Bell's theorem: free will, no signaling, and outcome independence. Quantum mechanics satisfies the first two hypotheses but not the latter. Experiments not only violate the Bell inequality, but show an excellent agreement with quantum mechanics. This fact restricts further the class of admissible theories. In this work, the author determines the form of the hidden-variable models that reproduce the quantum mechanical predictions for a spin singlet while satisfying both the hypotheses of free will and no signaling. Two classes of hidden-variable models are given as an example, and a general recipe to build infinitely many possible models is provided.
The observation of quantum nonlocality, i.e. quantum correlations violating a Bell inequality, implies the use of incompatible local quantum measurements. Here we consider the converse question. That is, can any set of incompatible measurements be used in order to demonstrate Bell inequality violation? Our main result is to construct a local hidden variable model for an incompatible set of qubit measurements. Specifically, we show that if Alice uses this set of measurements, then for any possible shared entangled state, and any possible dichotomic measurements performed by Bob, the resulting statistics are local. This represents significant progress towards proving that measurement incompatibility does not imply Bell nonlocality in general.
Bell’s theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell’s inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depend on measurement direction of both parties [Phys. Rev. Lett.105 250404 (2010)]. He also proved that for any model of singlet correlation, 86 % measurement independence is optimal. In this paper, we show that 59 % measurement independence is optimal for simulating singlet correlation when the underlying distribution of hidden variables depend only on measurements of one party. We also show that a distribution corresponding to this optimal lack of free will, already exists in literature which first appeared in the context of detection efficiency loophole. I.