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{-}Tr{{ρ}}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.