Characterization of informational completeness for covariant phase space observables

TL;DR

This paper characterizes the informational completeness and regularity of covariant phase space observables using the zero set of their Fourier-Weyl transform, revealing distinctions among different completeness conditions.

Contribution

It provides a comprehensive characterization of three properties of covariant phase space observables in terms of zero sets and introduces a general framework linking these to harmonic analysis.

Findings

Informational completeness requires the zero set to have dense complement.

Zero set of measure zero corresponds to Hilbert-Schmidt class informational completeness.

Regularity is equivalent to an empty zero set.

Abstract

A 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 $p=1,2,\infty$ 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.