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arXiv:quant-ph/0610180 · Physical Review A

Strong violations of Bell-type inequalities for path-entangled number states

Christoph F. Wildfeuer, Austin P. Lund, Jonathan P. Dowling

Published Nov 6, 2007• 63 citations
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Abstract

We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number $N$. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any $N$.

DOI
10.1103/physreva.76.052101

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We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number NNN. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any NNN.
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The Bell inequalities for azimuthal quantum number, i.e., orbital angular momentum, have been extensively reported to show quantum correlations between the spatial degree of freedom of entangled photon pairs. Here we investigate instead the Bell inequalities for radial quantum number. We demonstrate, both theoretically and experimentally, the violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the radial index ppp of the Laguerre-Gaussian mode, thus manifesting the entanglement in the radial structure of two-photon wave functions.
The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).
To reproduce in a local hidden variables theory correlations that violate Bell inequalities, communication must occur between the parties. We show that the amount of violation of a Bell inequality imposes a lower bound on the average communication needed to produce these correlations. Moreover, for every probability distribution there exists an optimal inequality for which the degree of violation gives the minimal average communication. As an example, to produce using classical resources the correlations that maximally violate the Clauser-Horne-Shimony-Holt inequality, $\sqrt{2}\ensuremath{-}1\ensuremath{\simeq}0.4142$ bits of communication are necessary and sufficient. For Bell tests performed on two entangled states of dimension d> 3d>~3d> 3 where each party has the choice between two measurements, our results suggest that more communication is needed to simulate outcomes obtained from certain nonmaximally entangled states than maximally entangled ones.