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

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