Separable-entangled frontier in a bipartite harmonic system

Abstract:

We consider a statistical mixture based on that of two identical harmonic oscillators which is characterized by four parameters, namely, the concentrations (x and y) of diagonal and nondiagonal bipartite states, and their associated thermal-like noises (T/α and T, respectively). The fully random mixture of two spins 1/2 as well as the Einstein-Podolsky-Rosen (EPR) state are recovered as particular instances. By using the conditional nonextensive entropy as introduced by Abe and Rajagopal, we calculate a bound for the separable-entangled frontier. Although this procedure is known to provide a necessary but in general not sufficient condition for separability, it does recover, in the particular case x = T = 0 ( ∀α), the 1/3 exact result known as Peres' criterion. The x = 0 frontier remarkably resembles to the critical line associated with standard diluted ferromagnetism where the entangled region corresponds to the ordered one and the separable region to the paramagnetic one. The entangled region generically shrinks for increasing T or increasing α.