Mixed state topological order: operator algebraic approach

TL;DR

This paper develops an operator algebraic framework to classify mixed states in 2D quantum spin systems, analyzing how their topological order, represented by braided tensor categories, evolves under decoherence.

Contribution

It introduces a braided $C^*$-tensor category approach for classifying mixed states and studies the stability of topological order under quantum channels.

Findings

The braided $C^*$-tensor category of a state is preserved as a subcategory under finite depth quantum channels.

The framework extends topological order classification to mixed states.

Decoherence modeled by quantum channels affects the categorical structure in a controlled manner.

Abstract

We study the classification problem of mixed states in two-dimensional quantum spin systems in the operator algebraic framework of quantum statistical mechanics. We associate a braided $C^*$-tensor category to each state satisfying a mixed-state version of the approximate Haag duality. We study how this category behaves under decoherence: suppose the state is acted by a finite depth quantum channel. We prove that the braided $C^*$-tensor category of the final state is a braided $C^*$-tensor subcategory of the initial state.