A note on the Relationship between Localization and Norm-1 Property

TL;DR

This paper investigates the relationship between localization in quantum mechanics and the norm-1 property, showing that important localization observables do not satisfy this property, thus highlighting limitations in defining localization operators.

Contribution

It provides a necessary condition for the norm-1 property and demonstrates its failure in key localization observables, clarifying the connection between different localization measures.

Findings

Norm-1 property is not satisfied by several important localization observables.

A necessary condition for the norm-1 property is established.

The gap between projection valued and positive operator valued measures remains significant.

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 measures. 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.