Generalization of the Peres criterion for local realism through nonextensive entropy

A bipartite spin-1/2 system having the probabilities 1+3x 4 of being in the Einstein-Podolsky-Rosen 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 Sq ≡ 1−Trρ q q−1 (q ∈ R; S1= − Tr ρ ln ρ) 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-quantuminformation procedure. 03.65.Bz, 03.67.-a, 05.20.-y, 05.30.-d Typeset using REVTEX