Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States

TL;DR

This paper investigates the classification of phases in one-dimensional open quantum systems using matrix product density operators, establishing conditions for trivial and non-trivial phases through local transformations.

Contribution

It introduces a framework for classifying phases of mixed states in 1D quantum systems and constructs explicit renormalization channels based on C*-weak Hopf algebras.

Findings

States from C*-Hopf algebras are in the trivial phase.

Provides explicit local channels for state renormalization.

Connects topological order boundaries to matrix product density operators.

Abstract

The classification of topological phases of matter is fundamental to understand and characterize the properties of quantum materials. In this paper we study phases of matter in one-dimensional open quantum systems. We define two mixed states to be in the same phase if both states can be transformed into the other by a shallow circuit of local quantum channels. We aim to understand the phase diagram of matrix product density operators that are renormalization fixed points. These states arise, for example, as boundaries of two-dimensional topologically ordered states. We first construct families of such states based on C*-weak Hopf algebras, the algebras whose representations form a fusion category. More concretely, we provide explicit local fine-graining and local coarse-graining quantum channels for the renormalization procedure of these states. Finally, we prove that those arising from C*-Hopf algebras are in the trivial phase.