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Simulating Bell violations without quantum computers

P. D. Drummond, Bogdan Opanchuk, L. Rosales-Zárate, M. D. Reid

Published Jan 1, 2024• 9 citations
Publisher

Abstract

We demonstrate that it is possible to simulate Bell violations using probabilistic methods. A quantum state corresponding to optical experiments that violate a Bell inequality is randomly sampled, demonstrating Bell's quantum paradox. This provides an explicit counter-example to Feynman's claim that such classical simulations could not be carried out.

DOI
10.25916/sut.26219219.v1

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•Dec 15, 2011•0 citations
We demonstrate that it is possible to simulate Bell violations using probabilistic methods. A quantum state corresponding to optical experiments that violate a Bell inequality is randomly sampled, demonstrating Bell's quantum paradox. This provides an explicit counter-example to Feynman's claim that such classical simulations could not be carried out.
Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive-PPP representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.
From the early days of quantum mechanics, there has been a discussion on the concept of reality, exemplified by the EPR paradox. To many, the idea of the paradox and the possibility of local hidden variables was dismissed by the Bell inequality. Yet, there remains considerable evidence that this inequality can be violated even by classical systems, so that experiments showing quantum behavior and the violation of the inequality must be questioned. Here, we demonstrate that classical optical polarization experiments can be shown to violate the Bell inequality. Hence, such experiments cannot be used to distinguish between classical and quantum theories.
We show that violation of genuine multipartite Bell inequalities can be obtained with sampled, probabilistic phase-space methods. These genuine Bell violations cannot be replicated if any part of the system is described by a local hidden variable theory. The Bell violations are simulated probabilistically using quantum phase-space representations. We treat mesoscopically large Greenberger-Horne-Zeilinger (GHZ) states having up to 60 qubits, using both a multipartite SU(2)-Q\text{SU(2)-}QSU(2)-Q representation and the positive-PPP representation. Surprisingly, we find that sampling with phase-space distributions can be exponentially faster than experiment. This is due to the classical parallelism inherent in the simulation of quantum measurements using phase-space methods. Our probabilistic sampling method predicts a contradiction with local realism of ``Schr\"odinger-cat'' states that can be realized as a GHZ spin state, either in ion traps or with photonic qubits. We also present a quantum simulation of the observed superdecoherence of the ion-trap ``cat'' state, using a phenomenological noise model.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such distribution experimentally, as set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to quantum probe, however, produces minimal back-action, and so enables weak measurement of the projector of one observable, followed by projective measurement of non-commuting observable. By this technique it is possible to empirically measure probabilities for all of the values of the observables relevant to Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0, 1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within a [local] hidden variable view of nature ... everything works fine if we permit negative probabilities. However, there are infinitely many such theories. Our method, involving weak-valued singles out unique set of probabilities, and moreover does so empirically.