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

TL;DR

This paper introduces a global indicator $\\mathcal{Q}_N$ to quantify the classicality of quantum states based on the positivity of the Wigner quasiprobability distribution, providing a new geometric measure for arbitrary $N$-level systems.

Contribution

The paper defines a novel global indicator of classicality for $N$-level quantum systems using the volume of states with positive Wigner distributions, and analyzes its algebraic structure and dependence on parameters.

Findings

The indicator $\\mathcal{Q}_N$ is defined as a relative volume in the state space.

The algebraic structure of the positive Wigner distribution states is characterized.

Dependence of the indicator on parameters is analyzed for qutrits.

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.