Necessary and sufficient condition of separability for D-symmetric diagonal states

Abstract

For multipartite states we consider a notion of D-symmetry. For a system of $N$ qubits it concides with usual permutational symmetry. In case of $N$ qudits ($d\ge 3$) the D-symmetry is stronger than the permutational one. For the space of all D-symmetric vectors in $({\mathbb{C}}^d{)}^{\otimes N}$ we define a basis composed of vectors $\{|R_{N,d;k}⟩:0\le k\le N(d-1)\}$ which are analog for Dicke states. The aim of this paper is to discuss the problem of separability of D-symmetric states which are diagonal in the basis $\{|R_{N,d;k}⟩\}$. We show that if $N$ is even and $d\ge 2$ is arbitrary then a PPT property is necessary and sufficient condition of separability for D-invariant diagonal states. In this way we generalize results obtained by Yu for qubits. Our strategy is to use some classical mathematical results on a moment problem.

8 pages