Mixed state topological order: operator algebraic approach

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 $C^{\ast }$-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 $C^{\ast }$-tensor category of the final state is a braided $C^{\ast }$-tensor subcategory of the initial state.