  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.
