Peres criterion for separability through nonextensive entropy

A bipartite spin-1/2 system having the probabilities $\mathrm{}(1+3x)/4\mathrm{}$ of being in the Einstein-Podolsky-Rosen (EPR) entangled state $|{\Psi }^{-}〉\equiv (1/\sqrt{2})\left(|\uparrow {〉}_A|\downarrow {〉}_B-|\downarrow {〉}_A|\uparrow {〉}_B\right)$ and $3\left(1-x\right)/4\mathrm{}$ of being orthogonal is known to admit a local realistic description if and only if $x<\mathrm{}1/3\mathrm{}$ (Peres criterion). We consider here a more general case where the probabilities of being in the entangled states $|{\Phi }^{\pm }〉\equiv (1/\sqrt{2})(|\uparrow {〉}_A|\uparrow {〉}_B\pm |\downarrow {〉}_A|\downarrow {〉}_B)$ and $|{\Psi }^{\pm }〉\equiv (1/\sqrt{2})(|\uparrow {〉}_A|\downarrow {〉}_B\pm |\downarrow {〉}_A|\uparrow {〉}_B)$ (Bell basis) are given, respectively, by $\left(1\mathrm{}-x\right)/4,$ $\left(1\mathrm{}-y\right)/4,$ $\left(1\mathrm{}-z\right)/4,$ and $\mathrm{}(1+x+y+z)/4.\mathrm{}$ Following Abe and Rajagopal, we use the nonextensive entropic form $S_q\equiv \left(1\mathrm{}-\mathrm{Tr}{\rho }^q\right)/\left(q-1\right)$ $(q\in \mathcal{R}{;S}_1=-\mathrm{Tr}\rho \ln \rho )$ which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the $\mathrm{}(x,y,z)\mathrm{}$ space where the system is separable. For instance, in the vicinity of the EPR state, separability occurs if and only if $x+y+z<\mathrm{}1,$ which recovers Peres’ criterion when $x=y=z.\mathrm{}$ 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.