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

TL;DR

This paper characterizes when D-symmetric diagonal states in multipartite quantum systems are separable, establishing that for even N and any dimension d, PPT condition is both necessary and sufficient, generalizing previous qubit results.

Contribution

It provides a complete separability criterion for D-symmetric diagonal states in multipartite systems, extending known qubit results to higher dimensions and even N.

Findings

PPT condition is necessary and sufficient for separability of D-invariant diagonal states when N is even.

The work generalizes Yu's results from qubits to higher dimensions.

Uses classical moment problem results to analyze quantum state separability.

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 ($d\geq 3$) 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 $\{|R_{N,d;k}\rangle: \,0\leq k\leq N(d-1)\}$ 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 $\{|R_{N,d;k}\rangle\}$. We show that if $N$ is even and $d\geq 2$ 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.