Necessary and sufficient condition of separability for D -symmetric diagonal states
Adam J. Rutkowski, Michał Banacki, Marcin Marciniak · Physical Review A · 2019
For multipartite states, we consider a notion of $D$ symmetry. For a system of $N$ qubits, it coincides with the usual permutational symmetry. In the case of $N$ qudits ($d\ensuremath{\ge}3$), the $D$ symmetry is stronger than the permutational one. For the space of all $\mathrm{D}$-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of $D$-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if $N$ is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for $D$-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.