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arXiv:quant-ph/0001109

Covariance Systems on the Projective Line

Nicola Pinamonti, M. Toller

Published Jan 29, 2000• 0 citations
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Abstract

We study the positive-operator-valued measures on the projective real line covariant with respect to the projective group, assuming that the energy is a positive operator. This problem is similar to the more complicated problem of finding the positive-operator-valued measures on the compactified Minkowski space-time covariant with respect to the conformal group. It also describes a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations.

DOI
10.48550/arxiv.quant-ph/0001109

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We study the positive-operator-valued measures on the projective real line covariant with respect to the projective group, assuming that the energy is a positive operator. This problem is similar to the more complicated problem of finding the positive-operator-valued measures on the compactified Minkowski space-time covariant with respect to the conformal group. It also describes a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations.
We study the positive-operator-valued measures (POVMs) on the projective real line covariant with respect to the projective group. We interpret the projective line as a compactified time axis and we assume that the energy is a positive operator. This formalism may describe a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations. The problem is similar to the more complicated and physically more relevant problem of finding the POVMs on the compactified Minkowski space-time covariant with respect to the conformal group.
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.
© The Author(s) 2024. We present an operationally motivated treatment of quantum reference frames in the setting that the frame is a covariant positive operator valued measure (POVM) on a finite homogeneous space, generalising the principal homogeneous spaces studied in previous work. We focus on the case that the reference observable is the canonical covariant projection valued measure on the given space, and show that this gives rise to a rank-one covariant POVM on the group, which can be seen as a system of coherent states.
We construct a family of bounded, finitely additive, covariant positive operator-valued measures (POVMs) on the real line by applying generalised Banach-Cesaro asymptotic averaging to locally finite covariant POVMs. In contrast to the strictly shift-invariant case, where countably additive operator measures are restricted to scalar multiples of a single fixed operator, the introduced covariant setting allows for non-commuting averaged operator values while successfully normalising the total mass. We develop an integration theory for complex-valued functions by taking the completion of simple integrable functions with respect to the continuous operator norm induced by the positive integral of their absolute values. The integration spaces for the original and the averaged measures are shown to possess distinct null classes. Finally, we introduce the operator-valued Fourier transform on these integration spaces and analyze the core algebraic constraints imposed by the covariance property on the exponential multipliers.