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arXiv:quant-ph/9910058

Strengthening the Bell Theorem: conditions to falsify local realism in an experiment

Marek Żukowski, Dagomir Kaszlikowski, Adam Baturo, Jan-Åke Larsson

Published Oct 14, 1999• 12 citations
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Abstract

The two-particle correlation obtained from the quantum state used in the Bell inequality is sinusoidal, but the standard Bell inequality only uses two pairs of settings and not the whole sinusoidal curve. The highest to-date visibility of an explicit model reproducing sinusoidal fringes is 2/pi. We conjecture from a numerical approach presented in this paper that the highest possible visibility for a local hidden variable model reproducing the sinusoidal character of the quantum prediction for the two-particle Bell-type interference phenomena is 1/sqrt2. In addition, the approach can be applied directly to experimental data.

DOI
10.48550/arxiv.quant-ph/9910058

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The two-particle correlation obtained from the quantum state used in the Bell inequality is sinusoidal, but the standard Bell inequality only uses two pairs of settings and not the whole sinusoidal curve. The highest to-date visibility of an explicit model reproducing sinusoidal fringes is 2/pi. We conjecture from a numerical approach presented in this paper that the highest possible visibility for a local hidden variable model reproducing the sinusoidal character of the quantum prediction for the two-particle Bell-type interference phenomena is 1/sqrt2. In addition, the approach can be applied directly to experimental data.
A sequence of Bell inequalities for NNN-particle systems, which involve three settings of each of the local measuring apparatuses, is derived. For Greenberger-Horne-Zellinger states, quantum mechanics violates these inequalities by factors exponentially growing with NNN. The threshold visibilities of the multiparticle sinusoidal interference fringes, for which local realistic theories are ruled out, decrease as (2/3)N{(2/3)}^{N}(2/3)N.
We formulate the expectation value of the Bell-Zukowski operator acting on qubit states of a two-particle Bell experiment. By using the equivalence between a set of N copies of a two-qubit experiment and a standard two-setting Bell experiment in an entangled 2N-particle state, we obtain an inequality, which we may call the Bell-Zukowski inequality. It determines whether the measured correlation functions of two-particle states can be modeled locally and realistically. In this Bell experiment of two particles, the conflict between local realism and quantum mechanics is discussed in conjunction with the violation of the Bell-Zukowski inequality. The main point of the result is that the Bell-Zukowski operator can be represented by the Bell-Mermin operator. The threshold visibility of two-particle interference analyzed in this scheme shows good agreement with the value to cause a violation of the Bell-Zukowski inequality.
Bell's inequalities are important to our understanding of quantum foundations and critical to several quantum technologies. A recent work [E. Wolfe and S. F. Yelin, Phys. Rev. A 86, 012123 (2012)] derived three parametrized families of two-particle, two-setting Bell inequalities. These inequalities are important as they theoretically explore a larger volume of allowed quantum correlations over local hidden-variable models than previous results [A. Cabello, Phys. Rev. A 72, 012113 (2005)] by exploiting marginal, or single particle measurements. In this work we subject those predictions to experimental test using nonmaximally entangled photon pairs to optimize the expected violation. We find excellent agreement with the upper bounds predicted by quantum mechanics with violations of the limits imposed by local hidden-variable models as large as almost 30$\ensuremath{\sigma}$ for the optimal parameters and a significant violation over a wide range of parameters.
Bell’s theorem presents a fundamental limit on the correlations achievable by any local hidden variable theory, seemingly precluding such models from reproducing quantum mechanical predictions. In this paper, we introduce a local hidden variable model based on phenomenological velocity and curvature operator techniques. By constructing a mathematical framework that maintains locality and realism, we demonstrate that it is possible to reproduce the quantum mechanical correlation functions that violate Bell’s inequalities. This challenges the conventional understanding imposed by Bell’s theorem and suggests the need for a re-examination of the foundational assumptions in quantum mechanics.