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In this report we are aiming at introducing a global measure ofnon-classicality of the state space of $N$-level quantum systems and estimatingit in the limit of large $N$. For this purpose we employ the Wigner functionnegativity as a non-classicality criteria. Thus, the specific volume of thesupport of negative values of Wigner function is treated as a measure ofnon-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 averagenon-classicality of the individual states over the Hilbert-Schmidt ensemble. Wepresent the numerical estimate of this quantity as a result of randomgeneration of states, and prove a proposition claiming its exact value in thelimit of $N\to \infty$

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