# On period, cycles and fixed points of a quantum channel

## TL;DR

This paper investigates the cyclic properties, fixed points, and invariant states of quantum channels on infinite-dimensional operator algebras, providing structural insights into their behavior and representations.

## Contribution

It introduces a detailed analysis of the atomic structure of fixed points and decoherence free algebra for quantum channels with invariant states, enhancing understanding of their Kraus forms.

## Key findings

- Fixed points are images of a normal conditional expectation.
- Atomic structure of fixed point spaces is characterized.
- Provides a better description of the channel's action and invariant densities.

## Abstract

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.