Peres criterion for separability through nonextensive entropy

A bipartite spin-1/2 system having the probabilities (1+3x)/4 of being in the Einstein-Podolsky-Rosen (EPR) entangled state |Ψ^-〉≡(1/√2)(|↑〉_A|↓〉_B-|↓〉_A|↑〉_B) and 3(1-x)/4 of being 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 the probabilities of being in the entangled states |Φ^±〉≡(1/√2)(|↑〉_A|↑〉_B±|↓〉_A|↓〉_B) and |Ψ^±〉≡(1/√2)(|↑〉_A|↓〉_B±|↓〉_A|↑〉_B) (Bell basis) are given, respectively, by (1-x)/4, (1-y)/4, (1-z)/4, and (1+x+y+z)/4. Following Abe and Rajagopal, we use the nonextensive entropic form S_q≡(1-Trρ^q)/(q-1)(q∈R;S₁=-Trρlnρ) which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the (x,y,z) space where the system is separable. For instance, in the vicinity of the EPR state, 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 of the Bell basis, the situation is identical. These results illustrate the computational power of this nonextensive-quantum-information procedure. In addition to this, a critical-phenomenon-like scenario emerges which enrichens the discussion.