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Frontiers in Optics

Multipartite quantum nonlocality using functional Bell inequalities

Qiongyi He, Eric G. Cavalcanti, M. D. Reid, P. D. Drummond

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We show that arbitrary functions of continuous variables can be used to generate tests of local hidden variable theories. The effect of nonideal detectors and noise is included, revealing that optimized functional inequalities are robust.
DOI
10.1364/ls.2009.lstua4

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We show that arbitrary functions of continuous variables, e.g., position and momentum, can be used to generate tests that distinguish quantum theory from local hidden variable theories. By optimizing these functions, we obtain more robust violations of local causality than obtained previously. We analytically calculate the optimal function and include the effect of nonideal detectors and noise, revealing that optimized functional inequalities are resistant to standard forms of decoherence. These inequalities could allow a loophole-free Bell test with efficient homodyne detection.
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Quantum theory is inconsistent with any local hidden variable model as was first shown by Bell. To test Bell inequalities two separated observers extract correlations from a common ensemble of identical systems. Since quantum theory does not allow simultaneous measurements of noncommuting observables, on each system every party measures a single randomly chosen observable out of a given set. Here we suggest a different approach for testing Bell inequalities that is experimentally realizable by current methods. We show that Bell inequalities can be maximally violated even when all observables are measured on each member of the ensemble. This is possible by using weak measurements that produce small disturbance, at the expense of accuracy. However, our approach does not constitute an independent test of quantum nonlocality since the local hidden variables may correlate the noise of the measurement instruments. Nevertheless, by adding a randomly chosen precise measurement at the end of every cycle of weak measurements, the parties can verify that the hidden variables were not interfering with the noise, and thus validate the suggested test.
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
Standard Bell inequalities apply to correlations arising when two or more macroscopically separated systems are each subjected to a single ideal measurement. While these inequalities demonstrate that quantum mechanics and local hidden variable theories yield incompatible predictions for pure states, they fail for mixed states: For mixed states, correlations arising from a single ideal measurement on each system may obey standard Bell inequalities, yet when each system is subjected to a sequence of ideal measurements the correlations are nonlocal. For some situations even this last procedure fails, and we must consider more complex (``nonideal'') measurements.
In a recent experiment, Freedman and Clauser have found that Bell's inequality for local hidden-variable theories is violated by over six standard deviations. We derive an inequality similar to Bell's for nonlocal hidden-variable theories in which the hidden-variable density distribution of the particles measured is nonlocal, but where the detectors are contained within some finite volume. The measurement of Freedman and Clauser is used to put a lower limit on the nonlocality of the hidden-variable density distributions of the photons detected in their experiment.