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.
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October 2012
Research Article|
September 27 2012
Characterization of informational completeness for covariant phase space observables Available to Purchase
J. Kiukas;
J. Kiukas
a)
1Zentrum Mathematik, M5,
Technische Universität München
, Boltzmannstr. 3, 85748 Garching, Germany
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P. Lahti;
P. Lahti
b)
2Turku Centre for Quantum Physics, Department of Physics and Astronomy,
University of Turku
, FI-20014 Turku, Finland
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J. Schultz;
J. Schultz
c)
2Turku Centre for Quantum Physics, Department of Physics and Astronomy,
University of Turku
, FI-20014 Turku, Finland
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R. F. Werner
R. F. Werner
d)
3Institut für Theoretische Physik,
Leibniz Universität Hannover
, Appelstr. 2, 30167 Hannover, Germany
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J. Kiukas
1,a)
P. Lahti
2,b)
J. Schultz
2,c)
R. F. Werner
3,d)
1Zentrum Mathematik, M5,
Technische Universität München
, Boltzmannstr. 3, 85748 Garching, Germany
2Turku Centre for Quantum Physics, Department of Physics and Astronomy,
University of Turku
, FI-20014 Turku, Finland
3Institut für Theoretische Physik,
Leibniz Universität Hannover
, Appelstr. 2, 30167 Hannover, Germany
a)
E-mail: [email protected].
b)
E-mail: [email protected].
c)
E-mail: [email protected].
d)
E-mail: [email protected].
J. Math. Phys. 53, 102103 (2012)
Article history
Received:
April 17 2012
Accepted:
September 06 2012
Citation
J. Kiukas, P. Lahti, J. Schultz, R. F. Werner; Characterization of informational completeness for covariant phase space observables. J. Math. Phys. 1 October 2012; 53 (10): 102103. https://doi.org/10.1063/1.4754278
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