Necessary and sufficient condition of separability for $\mathrm{D}$-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 ($d\ge 3$), the $D$ symmetry is stronger than the permutational one. For the space of all $\mathrm{D}$-symmetric vectors in ${\left({\mathbb{C}}^d\right)}^{\otimes N}$, we define a basis composed of vectors $\left\{\right|R_{N,d;k}\rangle :0\le k\le N(d-1)\}$ 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 $\left\{\right|R_{N,d;k}\rangle \}$. We show that if $N$ is even and $d\ge 2$ 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.