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arXiv:quant-ph/0002001 · Physical Review A

Optimal states for Bell-inequality violations using quadrature-phase homodyne measurements

William J. Munro

Published Jun 1, 1999• 71 citations
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Abstract

We identify what ideal correlated photon number states are required to maximize the discrepancy between local realism and quantum mechanics when a quadrature homodyne phase measurement is used. Various Bell-inequality tests are considered.
DOI
10.1103/physreva.59.4197

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We present a scheme that produces a conditionally prepared state that can be used for a Bell test based on homodyne detection. The state is near optimal for Bell-inequality violations based on quadrature-phase homodyne measurements that use correlated photon-number states. The scheme utilizes a Gaussian entanglement distillation protocol and uses only beam splitters and photodetection to conditionally prepare a non-Gaussian state from a source of two-mode squeezed states with low squeezing parameter.
We analyze Bell inequalities violations in photonic experiments for which the measurement apparatuses are restricted to homodyne measurements. Through numerical optimization of the Clauser-Horne-Shimony-Holt inequality over homodyne measurements and binning choices, we demonstrate large violations for states with a bounded number of photons. When considering states defined within qubit local subspaces of two Fock states, such as NOON states, a violation is observed solely within the qubit Fock space spanned by zero and two photons. For more generic states, large violations are obtained. Significant violations are observed even for states containing three photons locally and under realistic values of noise and losses. We propose concrete implementations to achieve such violations, opening new avenues for Bell experiments with homodyne detectors.
A correlation inequality is derived from local realism and a supplementary assumption. This inequality is violated by a factor of 1.5 in the case of real experiments, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of 2\sqrt 22​. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, the present inequality can be used to test locality very simply because it requires the measurements of only two detection probabilities. In contrast, Clauser-Horne inequality requires the measurements of five detection probabilities.