A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Characterization of informational completeness for covariant phase space observables
Tekijät: Kiukas J, Lahti P, Schultz J, Werner RF
Kustantaja: AMER INST PHYSICS
Julkaisuvuosi: 2012
Journal: Journal of Mathematical Physics
Tietokannassa oleva lehden nimi: JOURNAL OF MATHEMATICAL PHYSICS
Lehden akronyymi: J MATH PHYS
Artikkelin numero: ARTN 102103
Numero sarjassa: 10
Vuosikerta: 53
Numero: 10
Sivujen määrä: 11
ISSN: 0022-2488
DOI: https://doi.org/10.1063/1.4754278
Rinnakkaistallenteen osoite: https://arxiv.org/pdf/1204.3188.pdf
Tiivistelmä
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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