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

It is commonly accepted that the deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce a global indicator QN for quantifying the “classicality-quantumness” correspondence in the form of a functional on the orbit space O[𝔓N] of the adjoint action of the group SU(N) on the state space 𝔓N of an N-dimensional quantum system. The indicator QN is defined as the relative volume of the subspace $$ \mathrm{O}\left[{\mathfrak{P}}_N^{\left(+\right)}\right]\subset \mathrm{O}\left[{\mathfrak{P}}_N\right] $$ where the Wigner quasiprobability distribution is positive. The algebraic structure of $$ \mathrm{O}\left[{\mathfrak{P}}_N^{\left(+\right)}\right] $$ is revealed and exemplified by the case of a single qubit (N = 2) and a 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 is found. Bibliography: 18 titles.