Sufficient and necessary condition of separability for generalized Werner states

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[dN](v)=(1-v)I(N)dN+v|ψdN〉〈ψdN|, where |ψdN〉=∑i=0d-1αi|i⋯i〉 is an entangled pure state of N   qudits system and αiαi satisfies two restrictions: (i) ∑i=0d-1αiαi∗=1; (ii) Matrix 1d(I(1)+T∑i≠jαi|i〉〈j|αj∗), where T=Mini≠j{1/|αiαj|}, 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.