A note on the relationship between localization and the norm-1 property

Abstract

Abstract. The paper focuses on the problem of localization in quantum mechanics. It is well known that it is not possible to define a localization observable for the photon by means of projection valued measures. Conversely, that is possible by using positive operator valued measures. On the other hand, projection valued measures imply a kind of localization which is stronger than the one implied by positive operator valued measure. It has been claimed that the norm-1 property would in some sense reduce the gap between the two kind of localizations. We give a necessary condition for the norm-1 property and show that it is not satisfied by several important localization observables.

Uniform continuity of POVMs

“…In particular, we analyze the phase space localization observables, the unsharp phase observable and the unsharp number observable. We also show [6] that, in several cases, the uniform continuity of a POVM F allows a direct analysis of the norm-1 property. That is tested on the previous examples.…”

“…The aim of the present paper is to describe the results in Ref.s [5,6] that we have just outlined and to analyze some physically relevant examples in the light of such results.…”

“…Anyway, a sharp characterization is possible in the commutative case. Indeed, it was pointed out [1,22,28,5,6,7,8,9,10,11,12,13,14,15,24] that there is a strong relationship between commutative POVMs and PVMs. In particular, it was recently proved [5] that a POVM F is commutative if and only if there exist a spectral measure E and a Feller Markov kernel µ…”

“…A second mathematical question of physical relevance is connected to the continuity property of the POVMs. Indeed, there are relevant physical examples [6,16,17,33] of observables described by POVMs that are continuous with respect to the uniform operator topology. Recently, necessary and sufficient conditions for the uniform continuity have been given and the connections with the concepts of absolute continuity and norm-1 property have been analyzed [6].…”

“…Indeed, there are relevant physical examples [6,16,17,33] of observables described by POVMs that are continuous with respect to the uniform operator topology. Recently, necessary and sufficient conditions for the uniform continuity have been given and the connections with the concepts of absolute continuity and norm-1 property have been analyzed [6]. Moreover, it was proved [5] that a commutative POVM F is uniformly continuous if and only if the Markov kernel in equation ( 1) can be chosen to be a strong Feller Markov kernel.…”