Sufficient and necessary condition of separability for generalized Werner states

Abstract

In a celebrated paper [Optics Communications 179, 447, 2000], A.O. Pittenger and M.H. Rubin presented for the first time a sufficient and necessary condition of separability for the generalized Werner states. Inspired by their ideas, we generalized their method to a more general case. We obtain a sufficient and necessary condition for the separability of a specific class of N d-dimensional system (qudits) states, namely special generalized Werner state (SGWS): W [ d N ] ( v ) = ( 1 - v ) I ( N ) d N + v | ψ d N 〉 〈 ψ d N | , where | ψ d N 〉 = ∑ i = 0 d - 1 α i | i ⋯ i 〉 is an entangled pure state of N qudits system and α i satisfies two restrictions: (i) ∑ i = 0 d - 1 α i α i ∗ = 1 ; (ii) Matrix 1 d ( I ( 1 ) + T ∑ i ≠ j α i | i 〉 〈 j | α j ∗ ) , where T = Min i ≠ j { 1 / | α i α j | } , is a density matrix. Our condition gives quite a simple and efficiently computable way to judge whether a given SGWS is separable or not and previously known separable conditions are shown to be special cases of our approach.