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The Wigner distribution of \(n\) arbitrary observables. (English) Zbl 1454.81126

Summary: We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators \(A_1\), …, \(A_n\) and any quantum state, there is a unique joint distribution on \(\mathbb{R}^n\) with the property that the marginals of all linear combinations of the \(A_k\) coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
©2020 American Institute of Physics

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
44A12 Radon transform
22E70 Applications of Lie groups to the sciences; explicit representations
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