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

Abstract

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.