Sufficient and Necessary Condition of Separability for Generalized Werner States

We introduce 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): $W^{[d^N]}(v)=(1-v)\frac{I^{(N)}}{d^N}+v|ψ_d^N><ψ_d^N|$, where $|ψ_d^N>=\sum_{i=0}^{d-1}α_i|i... i>$ is an entangled pure state of $N$ qudits system and $α_i$ satisfys two restrictions: (i) $\sum_{i=0}^{d-1}α_iα_i^*=1$; (ii) Matrix $\frac{1}{d}(I^{(1)}+\mathcal{T}\sum_{i\neq j}α_i|i>< j|α_j^*)$, where $\mathcal{T}=\texttt{Min}_{i\neq j}\{1/|α_iα_j|\}$, 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.

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