Ergodicity of Kusuoka measures on quantum trajectories

Understand

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.