Sufficient and Necessary Condition of Separability for Generalized Werner States

TL;DR

This paper establishes a simple, necessary and sufficient criterion for determining the separability of a class of generalized Werner states in multi-qudit systems, improving computational efficiency over previous methods.

Contribution

It introduces a new, unified condition for separability of special generalized Werner states, encompassing and extending prior criteria.

Findings

The condition is simple and computationally efficient.

It generalizes and unifies previous separability criteria.

The approach applies to a broad class of multi-qudit states.

Abstract

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