Mixed state topological order: operator algebraic approach

TL;DR

The paper develops an operator-algebraic framework for mixed-state topological order in 2D quantum spin systems by associating a braided $C^*$-tensor category $C_{\omega,\Lambda_0}$ to states satisfying a mixed-state version of approximate Haag duality. It shows that cone von Neumann algebras can be stabilized to be properly infinite, enabling a robust superselection sector structure and a coherent category of localized endomorphisms; stabilization by pure infinite tensor products leaves the category unchanged, and subsystems embed via a strict braided tensor functor. The central result demonstrates that when a system interacts with an environment for a finite time or undergoes a finite-depth quantum channel, the final category $C_{\omega_2\otimes\psi,\Lambda_0}$ embeds as a braided subcategory of the initial $C_{\omega_1\otimes\psi,\Lambda_0}$, with a faithful braided tensor functor connecting stabilized categories under approximately-factorizable automorphisms. This provides a rigorous, category-theoretic account of how anyonic content is preserved or reduced under decoherence, linking topological order, stabilization, and noise in a unified algebraic formalism.

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.

Theorems & Definitions (41)

• Definition 1.1
• Lemma 1.2
• proof
• Definition 1.3
• Remark 1.4
• Remark 1.5
• Definition 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9