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Journal of Physics A Mathematical and General

Gauge covariant observables and phase operators

Daniel A. Dubin, Mark A. Hennings

Published Jan 3, 2001• 4 citations
Publisher

Abstract

We characterize those Hilbert space operators which are first moments of the gauge covariant positive operator-valued measures introduced by Lahti and Pellonpää.
DOI
10.1088/0305-4470/34/2/305

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The questions studied in this thesis are centered around the moment operators of a quantum observable, the latter being represented by a normalized positive operator measure. The moment operators of an observable are physically relevant, in the sense that these operators give, as averages, the moments of the outcome statistics for the measurement of the observable. The main questions under consideration in this work arise from the fact that, unlike a projection valued observable of the von Neumann formulation, a general positive operator measure cannot be characterized by its first moment operator. The possibility of characterizing certain observables by also involving higher moment operators is investigated and utilized in three different cases: a characterization of projection valued measures among all the observables is given, a quantization scheme for unbounded classical variables using translation covariant phase space operator measures is presented, and, finally, a mathematically rigorous description is obtained for the measurements of rotated quadratures and phase space observables via the high amplitude limit in the balanced homodyne and eight-port homodyne detectors, respectively. In addition, the structure of the covariant phase space operator measures, which is essential for the above quantization, is analyzed in detail in the context of a (not necessarily unimodular) locally compact group as the phase space.
A quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al. [“Clean positive operator valued measures,” J. Math. Phys. 46(8), 082109 (2005)10.1063/1.2008996] for finite-dimensional Hilbert space, and was studied subsequently by Kahn [“Clean positive operator-valued measures for qubits and similar cases,” J. Phys. A 40(18), 4817–4832 (2007)10.1088/1751-8113/40/18/009] and Pellonpää [“Complete characterization of extreme quantum observables in finite dimensions,” J. Phys. A 44(8), 085304 (2011)10.1088/1751-8113/44/8/085304]. The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and characterise this property for measures whose range determines a finite-dimensional operator system.
Positive-operator-valued observables in a complex Hilbert space H based on a topological homogeneous space G/H and G-covariant with respect to a strongly continuous unitary representation on H of a locally compact group G are investigated. The goal is to determine whether they are the weak integrals of some positive-operator-valued densities relative to a G-quasi-invariant measure μ on G/H. To this end, the notion of a kernel Hilbert space is generalized to spaces of equivalence classes with respect to μ of mappings of G/H into a complex Hilbert space K.