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Uniform Continuity of POVMs

Abstract

Recently a characterization of uniformly continuous POVMs and a necessary condition for a uniformly continuous POVM F to have the norm-1 property have been provided. Moreover it was proved that in the commutative case, uniform continuity corresponds to the existence of a Feller Markov kernel. We apply such results to the analysis of some relevant physical examples; i.e., the phase space localization observables, the unsharp phase observable and the unsharp number observable of which we study the uniform continuity, the norm-1 property and the existence of a Feller Markov kernel.

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Notes

  1. Since

    $$ \lim_{m\to\infty} \begin{pmatrix} m\\ n \end{pmatrix} \epsilon^n (1-\epsilon)^{m-n}=\lim _{m\to\infty}\frac{\epsilon^n}{n!} \bigl[m(1-\epsilon)^{m-n/n} \bigr]\cdots\bigl[(m-n) (1-\epsilon)^{m-n/n} \bigr]=0,$$

    Notice that \(F_{n}^{\epsilon}\) is compact for every \(n\in\mathbb{N}\) (see pp. 234–235 in [33]).

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Beneduci, R. Uniform Continuity of POVMs. Int J Theor Phys 53, 3531–3545 (2014). https://doi.org/10.1007/s10773-013-1883-x

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