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arXiv:quant-ph/0302090 · Modern Physics Letters A

THE CONFLICT BETWEEN BELL–ŻUKOWSKI INEQUALITY AND BELL–MERMIN INEQUALITY

Koji Nagata, Jaewook Ahn

Published Nov 19, 2008• 4 citations
arXivHTMLPDF

Abstract

We consider a two-particle/two-setting Bell experiment to visualize the conflict between Bell–Żukowski inequality and Bell–Mermin inequality. The experiment is reproducible by local realistic theories which are not rotationally invariant. We find that the average value of Bell–Żukowski operator is evaluated by the two-particle/two-setting Bell experiment in question. Bell–Żukowski inequality reveals that the constructed local realistic theories for the experiment are not rotationally invariant. The two-particle Bell experiment in question reveals the conflict between Bell–Żukowski inequality and Bell–Mermin inequality. Our analysis finds the threshold visibility for the two-particle interference to reveal the conflict noted above. The threshold visibility agrees with the value to obtain a violation of Bell–Żukowski inequality.

DOI
10.1142/s0217732308028727

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We consider a two-particle/two-setting Bell experiment to visualize the conflict between Bell–Żukowski inequality and Bell–Mermin inequality. The experiment is reproducible by local realistic theories which are not rotationally invariant. We find that the average value of Bell–Żukowski operator is evaluated by the two-particle/two-setting Bell experiment in question. Bell–Żukowski inequality reveals that the constructed local realistic theories for the experiment are not rotationally invariant. The two-particle Bell experiment in question reveals the conflict between Bell–Żukowski inequality and Bell–Mermin inequality. Our analysis finds the threshold visibility for the two-particle interference to reveal the conflict noted above. The threshold visibility agrees with the value to obtain a violation of Bell–Żukowski inequality.
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We formulate the expectation value of the Bell-Żukowski 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-Żukowski 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-Żukowski inequality. The main point of the result is that the Bell-Żukowski operator can be represented by the Bell-Mermin operator. PACS numbers: 03.65.Ud, 03.67.Mn
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Chapter 7 discusses the Bell theorem and its demonstration using Mermin’s machine. The Bell theorem proves that a theory based on local realism (one with hidden variables) must satisfy certain inequalities; quantum mechanics predicts these are violated. Mermin designed a gedanken experiment having a double-particle source and two detectors each having three settings, which can correspond to measurements along three spin orientations. A simple hidden-variable assumption is shown to violate quantum predictions. The CHSH inequality, a form of the general Bell inequality, is proved. This relation has been shown to be violated experimentally in quantum experiments, implying that quantum mechanics indeed has “spooky action at a distance.” Many experiments over the years had loopholes, ways used to explain away the data in order to preserve local reality, but recent experiments have been able to eliminate most of these.
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.