Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States

Abstract

We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological (SPT) phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in $(1+1)D$ and $(2+1)D$ systems, unveiling two distinct types of injectivity conditions inherent in short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. We first illustrate our framework with ASPT phases protected by fermion parity symmetry, then extend the classification of ASPT phases to a general group extension. We demonstrate examples of explicit construction of fixed-point LPDOs for ASPT phases including intrinsic ASPTs in both $(1+1)D$ and $(2+1)D$ systems.

32+4 pages, 6 figures, to appear in Physical Review X

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• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
• Holographic duality between bulk topological order and boundary mixed-state order
• Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Locally Purified Maximally Mixed States At Scale: Entanglement Pruning and Symmetries
• Low-rank optimal control of quantum devices
• Robust Mixed-State Cluster States and Spurious Topological Entanglement Negativity