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

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.

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Cited by in corpus (9)

• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Mixed-state quantum anomaly and multipartite entanglement
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise
• Hopf algebras and solvable unitary circuits
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Parent Lindbladians for Matrix Product Density Operators
• Quantum Circuits for Matrix-Product Unitaries