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On the moduli space of Wigner quasiprobability distributions for \(N\)-dimensional quantum systems. (English) Zbl 1435.81110

J. Math. Sci., New York 240, No. 5, 617-633 (2019) and Zap. Nauchn. Semin. POMI 468, 177-201 (2018).
Summary: A mapping between operators on the Hilbert space of an \(N\)-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of Stratonovich-Weyl kernels is given by the intersection of the coadjoint orbit space of the group \(\operatorname{SU}(N)\) and a unit \((N - 2)\)-dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems.

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
14J15 Moduli, classification: analytic theory; relations with modular forms
53D05 Symplectic manifolds (general theory)
81P16 Quantum state spaces, operational and probabilistic concepts
81S10 Geometry and quantization, symplectic methods
57R10 Smoothing in differential topology
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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