Iterates of quantum operations

TL;DR

This paper explores the convergence properties of iterated quantum operations within the framework of mean ergodic theory, emphasizing the role of the uniform ergodic theorem and providing examples across finite and infinite-dimensional Hilbert spaces.

Contribution

It introduces new general theorems on the convergence of quantum operation iterates, connecting ergodic theory with operator algebras and quantum information.

Findings

Convergence of quantum operation iterates established

Role of uniform ergodic theorem demonstrated

Examples in finite and infinite-dimensional spaces provided

Abstract

Iterates of quantum operations and their convergence are investigated in the context of mean ergodic theory. We discuss in detail the convergence of the iterates and show that the uniform ergodic theorem plays an essential role. Our results will follow from some general theorems concerning completely positive maps, mean ergodic operators, and operator algebras on Hilbert spaces. A few examples of both finite and infinite dimensional Hilbert spaces are presented as well.