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In the report we analyze the indicator/measure of classicality of quantumstates defined as the probability to find a state with a positive Wignerfunction within a unitary invariant random ensemble. The indicators ofclassicality of three ensembles associated with the Hilbert-Schmidt, Bures andBogoliubov-Kubo-Mori metrics on the space of quantum states of 3-level systemare computed. Their dependence on a moduli parameter of the Wigner function isstudied for all strata of a qutrit state space stratified in accordance withthe unitary group action.

Comparing classicality of qutrits from Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori ensembles