# Topological order, mixed states and open systems

## TL;DR

This paper investigates the nature and physical significance of topological phases in mixed quantum states, especially their robustness and behavior under environmental influences that preserve system symmetries.

## Contribution

It provides a general analysis of topological phases in mixed states and explores their physical relevance and invariance in open quantum systems.

## Key findings

- Topological phases in mixed states can be robust under certain conditions.
- The Uhlmann phase's physical role in quantum systems remains unclear.
- Environmental effects preserving symmetries influence the behavior of topological phases.

## Abstract

The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states had received only quite recently attention in the literature. In particular, it is still unclear whether the generalisation of the Aharonov-Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse from a general viewpoint topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with a periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.