Order in the Open: Symmetries and Entanglement of Many-Body Mixed States

Abstract

Real-world quantum systems are open and interact with their environments, requiring a statistical description via mixed states. This thesis investigates the interplay between global symmetries and quantum entanglement in open many-body systems, asking whether symmetries can robustly enforce long-range entanglement and correlation patterns, even under severe decoherence or high temperatures.

In the first half, we extend quantum anomalies to mixed states and establish the anomaly-nonseparability correspondence: mixed states that are strongly symmetric --- where every state in the statistical ensemble possesses the same symmetry charge --- exhibit long-range multipartite entanglement. We show that the unique multipartite structure of this anomalous entanglement gives rise to entirely new phases of matter that are intrinsically mixed, i.e., lacking any pure state representative. Conversely, we demonstrate that strong-weak mixed anomalies, such as Lieb-Schultz-Mattis anomalies, imply long-range correlations without strictly requiring quantum entanglement. Broadening this correspondence to higher-form symmetries, we introduce a definition of mixed-state phases of matter that is insensitive to long-range classical correlations, thereby only capturing distinct patterns of long-range entanglement. We argue that strong symmetries and their anomalies are the defining features of such phases.

In the second half, we shift focus to non-anomalous symmetries and show when they alone suffice to enforce entanglement. We investigate maximally mixed states invariant under on-site symmetries, which naturally emerge as steady states of generic quantum evolutions that preserve these symmetries strongly. We exactly calculate the values of several entanglement measures that are notoriously difficult to tackle analytically or numerically, such as the entanglement of formation and distillation. For continuous non-Abelian symmetries, we find high amounts of long-range entanglement, despite the states being maximally mixed within the symmetric subspace. Finally, we prove that the same strong symmetry conditions and superselection rules prevent the sudden death of entanglement at finite temperatures, even for Abelian symmetries. This explains previously observed behavior in canonical ensembles with Ising symmetry and in fermionic systems.