Regularised Continuous Variable EPR States with Wigner Functions and CHSH Violations

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 ’ total and relative position and momentum variables, respectively). We give the regularised forms of such tripartite EPR states in second-quantised formulation, and derive their Wigner functions. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r → ∞, 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 → √ 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 → √ 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 ≤ 2) are established, with B3 ∼ = 2.09 in the canonical regime (s → 1 +), as well as B3 ∼ = 2.32 in the auxiliary regime (s → √ 2 +).