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arXiv:math-ph/0512053

Accardi complementarity in μμμ-deformed quantum mechanics

Claudio Pita-Ruiz, Stephen Bruce Sontz

Published Dec 15, 2005• 0 citations
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Abstract

In this note we show that the momentum and position operators of $μ$-deformed quantum mechanics for $μ> 0$ are not Accardi complementary in a sense that we will define. We conjecture that this is also true if $-1/2 < μ< 0$.

DOI
10.48550/arxiv.math-ph/0512053

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In this note we show that the momentum and position operators of μμμ-deformed quantum mechanics for μ>0μ> 0μ>0 are not Accardi complementary in a sense that we will define. We conjecture that this is also true if −1/2<μ<0-1/2 < μ< 0−1/2<μ<0.
We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity—some of which new—in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position–momentum, position–energy, number–phase, as well as periodic observables relevant to spatial interferometry. We close the paper with some considerations of complementarity in a noisy setting, focusing especially on the case of convolutions of position and momentum, which was a recurring topic in Paul’s work on operational formulation of quantum measurements and central to his philosophy of unsharp reality.
We examine basic properties of complementarity by using the most general description of quantum observables as positive-operator measures. We show that, in general, two observables can be complementary or not depending on the measure of fluctuations adopted and that complementarity is not a symmetric relation. This occurs because the states that determine the measured statistics do not necessarily coincide with the minimum uncertainty states for the same observable. We also show that there are observables without a complementary observable and that complementarity is not preserved by the Neumark extensions.