Ergodicity of Kusuoka measures on quantum trajectories

TL;DR

This paper investigates the ergodic properties of Kusuoka measures generated by quantum measurement processes, providing simplified conditions for ergodicity and establishing their necessity and reversibility in specific measurement scenarios.

Contribution

It simplifies the ergodicity condition for Kusuoka measures in quantum systems and proves its necessity and reversibility in certain measurement configurations.

Findings

Simplified ergodicity condition for scaled projection measurements.

Necessity of the condition for rank-1 POVMs and two-projection measurements.

Reversibility of the Kusuoka measure in specific measurement cases.

Abstract

In 1989 Kusuoka started the study of probability measures on the shift space that are defined with the help of products of matrices. In particular, he derived a sufficient condition for the ergodicity of such measures, which have since been referred to as Kusuoka measures. We observe that repeated measurements of a unitarily evolving quantum system generate a Kusuoka measure on the space of sequences of measurement outcomes. We show that if the measurement consists of scaled projections, then Kusuoka's sufficient ergodicity condition can be significantly simplified. We then prove that this condition is also necessary for ergodicity if the measurement consists of uniformly scaled rank-1 projections (i.e., it is a rank-1 POVM), or of exactly two projections, one of which is rank-1. For the latter class of measurements we also show that the Kusuoka measure is reversible in the sense that every string of outcomes has the same probability of being emitted by the system as its reverse.