We demonstrate that the Wigner function of the Einstein-Podolsky-Rosen state, though positive definite, provides a direct evidence of the nonlocal character of this state. The proof is based on an observation that the Wigner function describes correlations in the joint measurement of the phase space displaced parity operator.

A collection of static atoms is fixed in a crystal at a low temperature and prepared by a pulse of incident radiation of wave vector . The atoms are well described by an entangled Dicke-like state, in which each atom carries a characteristic phase factor exp(ik0.r(j)), where is the atomic position in the crystal. It is shown that a single photon absorbed by the N atoms will be followed by spontaneous emission in the same direction. Furthermore, phase matched emission is found when one photon is absorbed by N atoms followed by two-photon down-conversion.

Phys. Rev. 165, 1579. 12 The quoted ranges include values of A T and A LS from Refs. 8, 10, and 11 and R. A. Arndt, R. H. Hackman, and L. D. Roper, Phys. Rev. C 9, 555 (1974). From the latter reference we include the phase shifts from the 1-500-MeV analysis and two sets of phase shifts obtained from the analysis of the 1-27.6-MeV data.13 Arndt, Hackman, and Roper, Ref. 12. The caption of Fig. 3 is incorrect. The floated curves are the upper one for Fig. 3(a) and the lower ones for Figs. 3(b) and 3(c). The numbers on the vertical scale in Figs. 2(b) and 3(b) should be negative and the values in Fig. 3(b) should be integer multiples of 0.2 [R. A. Arndt, private communication].14

We propose an experiment demonstrating the nonlocality of a quantum singlet-like state generated from a single photon incident on a beam splitter. Each of the two spatially separated apparatuses in the setup performs a strongly unbalanced homodyning, employing a single photon counting detector. We show that the correlation functions violating the Bell inequalities in the proposed experiment are given by the joint two-mode Q-function and the Wigner function of the optical singlet-like state. This establishes a direct relationship between two intriguing aspects of quantum mechanics: the nonlocality of entangled states and the noncommutativity of quantum observables, which underlies the nonclassical structure of phase space quasidistribution functions. PACS Number(s): 03.65.Bz, 42.50.Dv A fundamental step providing a bridge between classical and quantum physics has been given by Wigner in form of a quantum mechanical phase space distribution: the Wigner function [1]. From the pioneering work of Weyl, Wigner and Moyal, it follows that the noncommutativity of quantum observables leads to a real abundance of different in form quantum mechanical phase space quasidistributions. A description of quantum phenomena in terms of the Wigner or the positive-Q quasidistributions, provided a milestone step towards a c-number formulation of quantum effects in phase space [2]. Due to Einstein, Podolsky, and Rosen (EPR) [3], followed by the seminal contribution of Bell [4], the meaning of quantum reality and quantum nonlocality has become a central issue of the modern interpretation and understanding of quantum phenomena [5]. Such concepts like entanglement and quantum nonlocality have generated a real flood of theoretical work devoted to various connections of the quantum description with different views or representations of the quantum formalism.Despite all these theoretical works a direct link between various phase space distributions and the nonlocality of quantum mechanics has been missing. In some works [6] the quantum phase space has been treated as a model for a hidden variable theory, and the incompatibility of quantum mechanics with local theories has been attributed to the nonpositive character of the Wigner function. In this context it has been argued that the original EPR wave function cannot violate the positionmomentum Bell inequality, because the corresponding Wigner function is positive everywhere.It is the purpose of this Letter to propose an experimental demonstration of nonlocal effects in phase space exhibited by a quantum optical singlet-like state generated from a single photon. The entanglement will be represented by a correlated state of light, which refers to two spatially separated modes of the electromagnetic field. We show that the proposed experiment establishes a direct relationship between quantum nonlocality and the positive phase space Q-function, as well as the nonpositive Wigner function. We demonstrate that for a certain class of experiments these two quasiprobability distributions are nonlocal ...

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