The Wigner distribution of n arbitrary observables

authored by: René Schwonnek, Reinhard F. Werner
Abstract: We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, ..., An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak 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.
Organisation(s): Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
Type: Article
Journal: Journal of mathematical physics
Volume: 61
ISSN: 0022-2488
Publication date: 04.08.2020
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Mathematical Physics
Electronic version(s): https://doi.org/10.1063/1.5140632 (Access: Closed)

BibTeX

@article{a9b84154d10d4ad08401059cf0f3c1ad,
title = "The Wigner distribution of n arbitrary observables",
abstract = "We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, ..., An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak 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.",
author = "Ren{\'e} Schwonnek and Werner, {Reinhard F.}",
note = "Funding information: We acknowledge support from the DFG (Grant Nos. RTG 1991 and CRC 1227 DQ-mat) and the BMBF project Q.Link.X. R.S. gratefully acknowledges A. Ketterer and A. Asadian for inspiring discussions.",
year = "2020",
month = aug,
day = "4",
doi = "10.1063/1.5140632",
language = "English",
volume = "61",
journal = "Journal of mathematical physics",
issn = "0022-2488",
publisher = "American Institute of Physics",
number = "8",
}