A1 Refereed original research article in a scientific journal
Characterization of informational completeness for covariant phase space observables
Authors: Kiukas J, Lahti P, Schultz J, Werner RF
Publisher: AMER INST PHYSICS
Publication year: 2012
Journal: Journal of Mathematical Physics
Journal name in source: JOURNAL OF MATHEMATICAL PHYSICS
Journal acronym: J MATH PHYS
Article number: ARTN 102103
Number in series: 10
Volume: 53
Issue: 10
Number of pages: 11
ISSN: 0022-2488
DOI: https://doi.org/10.1063/1.4754278(external)
Self-archived copy’s web address: https://arxiv.org/pdf/1204.3188.pdf(external)
Abstract
In the nonrelativistic setting with finitely many canonical degrees of freedom, a shift-covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for p = 1, 2, infinity of a more general notion of p-regularity defined as the norm density of the span of translates of the operator in the Schatten-p class. We show that the relation between zero sets and p-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4754278]
In the nonrelativistic setting with finitely many canonical degrees of freedom, a shift-covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for p = 1, 2, infinity of a more general notion of p-regularity defined as the norm density of the span of translates of the operator in the Schatten-p class. We show that the relation between zero sets and p-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4754278]
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