The Global Indicator of Classicality of an Arbitrary N-Level Quantum System

It is commonly accepted that the deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce a global indicator Q N for quantifying the “classicality-quantumness” correspondence in the form of a functional on the orbit space O[P N ] of the adjoint action of the group SU(N) on the state space P N of an N-dimensional quantum system. The indicator Q N is defined as the relative volume of the subspace O P N + ⊂ O P N where the Wigner quasiprobability distribution is positive. The algebraic structure of O P N + is revealed and exemplified by the case of a single qubit (N = 2) and a 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 is found. Bibliography: 18 titles.