Symmetry Protected Topological Phases of Mixed States in the Doubled Space

TL;DR

This paper develops a systematic classification of symmetry protected topological phases in mixed quantum states using a doubled Hilbert space approach, revealing new invariants, robustness properties, and symmetry-breaking phenomena.

Contribution

It introduces a doubled space framework to classify mixed-state SPT phases, addressing hermiticity and positivity constraints, and explores their robustness and symmetry-breaking behaviors.

Findings

Classified mixed-state SPT phases systematically.

Analyzed robustness of MSPT invariants under quantum channels.

Studied spontaneous symmetry breaking and related order parameters.

Abstract

The interplay of symmetry and topology in quantum many-body mixed states has recently garnered significant interest. In a phenomenon not seen in pure states, mixed states can exhibit average symmetries -- symmetries that act on component states while leaving the ensemble invariant. In this work, we systematically characterize symmetry protected topological (SPT) phases of short-range entangled (SRE) mixed states of spin systems -- protected by both average and exact symmetries -- by studying their pure Choi states in a doubled Hilbert space, where the familiar notions and tools for SRE and SPT pure states apply. This advantage of the doubled space comes with a price: extra symmetries as well as subtleties around how hermiticity and positivity of the original density matrix constrain the possible SPT invariants. Nevertheless, the doubled space perspective allows us to obtain a systematic classification of mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the consequences of the MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. In addition to MSPT phases, we study the patterns of spontaneous symmetry breaking (SSB) of mixed states, including the phenomenon of exact-to-average SSB, and the order parameters that detect them. Mixed state SSB is related to an ingappability constraint on symmetric Lindbladian dynamics.