The global indicator of classicality of an arbitrary $N$-level quantum system

It is commonly accepted that a deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce the global indicator $\mathcal{Q}_N$ for quantification of “classicality-quantumness” correspondence in the form of the functional on the orbit space $\mathcal{O}[\mathfrak{P}_N]$ of the $SU(N)$ group adjoint action on the state space $\mathfrak{P}_N$ of an $N$-dimensional quantum system. The indicator $\mathcal{Q}_{N}$ is defined as a relative volume of a subspace $\mathcal{O}[\mathfrak{P}^{(+)}_N] \subset \mathcal{O}[\mathfrak{P}_N],$ where the Wigner quasiprobability distribution is positive. An algebraic structure of $\mathcal{O}[\mathfrak{P}^{(+)}_N]$ is revealed and exemplified by a single qubit $(N=2)$ and single qutrit $(N=3)$. For the Hilbert-Schmidt ensemble of qutrits the dependence of the global indicator on the moduli parameter of the Wigner quasiprobability distribution has been found.