Regularised Tripartite Continuous Variable EPR-type States with Wigner Functions and CHSH Violations

TL;DR

This paper introduces regularised tripartite continuous variable EPR-type states, derives their Wigner functions, compares them with NOPA states, and demonstrates CHSH inequality violations indicating quantum nonlocality.

Contribution

The work provides a new regularised formulation of tripartite CV EPR states, analyzes their Wigner functions, and shows their ability to violate CHSH inequalities, extending understanding of multipartite quantum entanglement.

Findings

Regularised tripartite EPR states have singular Wigner functions in specific regimes.

CHSH inequalities are violated with B3 values exceeding classical bounds.

Comparison with NOPA states highlights differences in singular behaviour and entanglement properties.

Abstract

We consider tripartite entangled states for continuous variable systems of EPR type, which generalise the famous bipartite CV EPR states (eigenvectors of conjugate choices X1 - X2, P1+ P2, of the systems' relative position and total momentum variables). We give the regularised forms of such tripartite EPR states in second-quantised formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r --> infinity, or tanh r --> 1^- (approaching the EPR states in the bipartite case), our regularised tripartite EPR states show singular behaviour not only in the approach to the EPR-type region (s --> 1 in our notation), but also for an additional, auxiliary regime of the regulator (s --> \sqrt{2}). While the s -->1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the s --> \sqrt{2} limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalised CHSH inequalities. Violations of the tripartite CHSH bound (B3 \le 2) are established, with B3 ~ 2.09 in the canonical regime (s -->1^+), as well as B3 ~ 2.32 in the auxiliary regime (s --> \sqrt{2^+}).