On overall measure of non-classicality of $N$-level quantum system and its universality in the large $N$ limit

TL;DR

This paper introduces a global measure of non-classicality for $N$-level quantum systems based on Wigner function negativity, and analyzes its behavior as the system size grows large, showing universality in the limit.

Contribution

It defines a new global non-classicality measure using Wigner function negativity and proves its exact value in the large $N$ limit.

Findings

The measure converges to a universal value as $N$ approaches infinity.

Numerical simulations support the theoretical results.

The measure provides a quantitative way to compare non-classicality across states.

Abstract

In this report we are aiming at introducing a global measure of non-classicality of the state space of $N$-level quantum systems and estimating it in the limit of large $N$. For this purpose we employ the Wigner function negativity as a non-classicality criteria. Thus, the specific volume of the support of negative values of Wigner function is treated as a measure of non-classicality of an individual state. Assuming that the states of an $N$-level quantum system are distributed by Hilbert-Schmidt measure (Hilbert-Schmidt ensemble), we define the global measure as the average non-classicality of the individual states over the Hilbert-Schmidt ensemble. We present the numerical estimate of this quantity as a result of random generation of states, and prove a proposition claiming its exact value in the limit of $N\to \infty$