On the hierarchy of classicality and symmetry of quantum states

The interrelation between classicality/quantumness and symmetry of states is discussed within the phase-space formulation of finite-dimensional quantum systems. We derive representations for classicality measures $\mathcal{Q}_N[H_{\varrho}]$ of states from the stratum of given symmetry type $[H_{\varrho}]$ for the Hilbert-Schmidt ensemble of qudits. The expressions for measures are given in terms of the permanents of matrices constructed from the vertices of the special Wigner function's positivity polytope. The supposition about the partial order of classicality indicators $\mathcal{Q}_N[H_{\varrho}]$ in accordance with the symmetry type of stratum is formulated.