A bipartite spin-1/2 system having the probabilities $\frac{ 1+3x}{ 4}$ of being in the Einstein-Podolsky-Rosen entangled state $|\Psi^-$$> \equiv \frac{ 1}{ \sqrt 2}(|$$\uparrow>_A|$$\downarrow>_B$$-|$$\downarrow>_A|$$\uparrow>_B)$ and $\frac{ 3(1-x)}{ 4}$ 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 $|\Phi^{ \pm}$$> \equiv \frac{ 1}{ \sqrt 2}(|$$\uparrow>_A|$$\uparrow>_B \pm |$$\downarrow>_A|$$\downarrow>_B)$ and $|\Psi^{ \pm}$$> \equiv \frac{ 1}{ \sqrt 2}(|$$\uparrow>_A|$$\downarrow>_B \pm |$$\downarrow>_A|$$\uparrow>_B)$ (Bell basis) are given respectively by $\frac{ 1-x}{ 4}$, $\frac{ 1-y}{ 4}$, $\frac{ 1-z}{ 4}$ and $\frac{ 1+x+y+z}{ 4}$. Following Abe and Rajagopal, we use the nonextensive entropic form $S_q \equiv \frac{ 1- Tr \rho^q}{ q-1} (q \in \cal{ R}; $$S_1$$= -$ $Tr$ $ \rho \ln \rho)$ which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the $(x,y,z)$ space where local realism is admissible. For instance, in the vicinity of the EPR state, classical realism is possible 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. A critical-phenomenon-like scenario emerges. These results illustrate the computational power of this new nonextensive-quantum-information procedure.
Categories: quant-ph