A Positive Operator-Valued Measure for an Iterated Function System

TL;DR

This paper extends the concept of Hutchinson measures to positive operator-valued measures for arbitrary iterated function systems, including those with essential overlap, using a generalized Kantorovich metric.

Contribution

It introduces a new positive operator-valued measure for IFSs with overlaps, generalizing previous projection-valued measures and utilizing a generalized Kantorovich metric.

Findings

Established a metric space completion for the classical Kantorovich metric.

Generalized Hutchinson measure to positive operator-valued measures for all IFSs.

Connected the generalized metric to quantum observables studied by R.F. Werner.

Abstract

Given an iterated function system (IFS) on a complete and separable metric space $Y$, there exists a unique compact subset $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap. In previous work, we developed an alternative approach to proving the existence of this projection-valued measure. The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in. We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory. We conclude with a discussion of Naimark's dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.