Peres criterion for separability through nonextensive entropy
Constantino Tsallis, Seth Lloyd, Michel Baranger · Physical Review A · 2001
A bipartite spin-1/2 system having the probabilities $(1+3x)/4$ of being in the Einstein-Podolsky-Rosen (EPR) entangled state $|{\ensuremath{\Psi}}^{\ensuremath{-}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ensuremath{-}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ and $3(1\ensuremath{-}x)/4$ 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 $|{\ensuremath{\Phi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B})$ and $|{\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}〉\ensuremath{\equiv}(1/\sqrt{2})(|\ensuremath{\uparrow}{〉}_{A}|\ensuremath{\downarrow}{〉}_{B}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}{〉}_{A}|\ensuremath{\uparrow}{〉}_{B})$ (Bell basis) are given, respectively, by $(1\ensuremath{-}x)/4,$ $(1\ensuremath{-}y)/4,$ $(1\ensuremath{-}z)/4,$ and $(1+x+y+z)/4.$ Following Abe and Rajagopal, we use the nonextensive entropic form ${S}_{q}\ensuremath{\equiv}(1\ensuremath{-}\mathrm{Tr}{\ensuremath{\rho}}^{q})/(q\ensuremath{-}1)$$(q\ensuremath{\in}\mathcal{R}{;S}_{1}=\ensuremath{-}\mathrm{Tr}\ensuremath{\rho}\mathrm{ln}\ensuremath{\rho})$ 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.