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

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 >= 3), the D symmetry is stronger than the permutational one. For the space of all D-symmetric vectors in (C-d)(circle times N), we define a basis composed of vectors {vertical bar R-N,R-d;k > : 0 <= k <= N(d - 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 {vertical bar R-N,R-d;k >}. We show that if N is even and d >= 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.