TY - JOUR

T1 - Characterization of informational completeness for covariant phase space observables

AU - Kiukas, J.

AU - Lahti, P.

AU - Schultz, J.

AU - Werner, R. F.

N1 - Funding Information:
This work was partially supported by the Academy of Finland Grant No. 138135. J.S. was supported by the Finnish Cultural Foundation. J.K. was supported by Emil Aaltonen Foundation.
Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/9/27

Y1 - 2012/9/27

N2 - 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, ∞ 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.

AB - 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, ∞ 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.

UR - http://www.scopus.com/inward/record.url?scp=84868358080&partnerID=8YFLogxK

U2 - 10.1063/1.4754278

DO - 10.1063/1.4754278

M3 - Article

AN - SCOPUS:84868358080

SN - 0022-2488

VL - 53

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 10

M1 - 102103

ER -