Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation

Abstract

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.

Collapse and revival of quantum coherence for a harmonic oscillator interacting with a classical fluctuating environment

“…The notion of nonclassicality arising from the negativity of the Wigner function is not equivalent to the nonclassical depth and it has been linked to nonlocal properties [89,90]. More precisely, it has been shown that positivity of the Wigner function implies that the corresponding quantum state cannot violate any Bell inequality involving only position and momentum measurements.…”

Tomographic test of Bell’s inequality for a time-delocalized single photon

“…[10]: the strict connection between the displaced parity operator (a dichotomic observable) and the Wigner function [16], suggests that the Wigner function of any two-mode/two-particle system (whether positive or negative) may play the role of a nonlocal correlation function. Bell's inequality can thus be recast in the form:…”

“…Intensive studies and debates have been dedicated to the meaningfulness of extending both the concepts of entanglement and nonlocality to TWC-type single-photon two-mode systems (see, e.g., [8,9,10,11,12,13,14,15]). Quite recently, Babichev, et al [11] and Zavatta, et al [14] have experimentally characterized a source of two-mode spatially and temporally delocalized single photons, respectively, by employing time-domain homodyne tomography [16] and reconstructing both density matrix and Wigner function of the measured system.…”

“…In particular, we evaluate the degree of entanglement characterizing our single-photon system by employing Vidal and Werner's reformulation [25] of Peres separability criterion [26]. Then, we employ the experimentally reconstructed two-mode Wigner function to perform an innovative continuous variable Bell's inequality test, based on the theoretical proposal by Banaszek and Wodkiewicz [10], which we briefly introduce in the next section.…”

Multicomponent Bell inequality and its violation for continuous-variable systems

“…They then showed that the EPR state and the two-mode squeezed vacuum state do not have a local realistic description in the sense that they violate Bell inequalities such as the Clauser and Horne inequality [5] and the Clauser-Horne-Shimony-Holt (CHSH) [6] inequality. The starting point of the demonstration in [4] is that the two-mode Wigner function can be interpreted as a correlation function for the joint measurement of the parity operator. In the limit r → ∞, when the original EPR state is recovered, a significant violation of Bell inequality takes place, however, the violation is not very strong.…”