Peres criterion for separability through nonextensive entropy - art. no. 042104

A bipartite spin-1/2 system having the probabilities (1+3x)/4 of being in t he Einstein-Podolsky-Rosen (EPR) entangled state \psi (-)] = (1/root2)(\up arrow](A)\down arrow](B)-\down arrow](A)\up arrow](B)) and 3 (1 - x)/4 of b eing orthogonal is known to admit a local realistic description if and only if x < 1/3 (Peres criterion). We consider here a more general case where t he probabilities of being in the entangled states \<Phi>(+/-)] = (1/root2)( \up arrow](A)\up arrow](B)+/-\down arrow](A)\down arrow](B)) and \psi (+/-) ] = (1/root2)(\up arrow](A)\down arrow](B)+/-\down arrow](A)\up arrow (B)) (Bell basis) are given, respectively, by (1- x)/4, (1 - y)/4, (1-)/4, and ( 1 + x + y + z)/4. Following Abe and Rajagopal, we use the nonextensive entr opic form S-q = (1 - Tr rho (q))/(q - 1) (q is an element of R;S-1 = - Tr r ho 1n rho) which has enabled a current generalization of Boltzmann-Gibbs st atistical mechanics, and determine the entire region in the (x,y,z) space w here the system is separable. For instance, in the vicinity of the EPR stat e, separability occurs if and only if x + y + z < 1, which recovers Peres' criterion when x = y = z. In the vicinity of the other three states bf the Bell basis, the situation is identical. These results illustrate the comput ational power of this nonextensive-quantum-information procedure. In additi on to this, a critical-phenomenon-like scenario emerges which enrichens the discussion.