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arXiv:2401.17357 · Physical Review X

Mixed-State Quantum Anomaly and Multipartite Entanglement

Leonardo A. Lessa, Meng Cheng, Chong Wang

Published Mar 24, 2025• 29 citations
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Abstract

Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in d space dimensions with anomalous symmetry G where the anomaly is characterized by an invariant in the group cohomology H d + 2 [ G , U ( 1 ) ] . We show that any mixed state ρ that is strongly symmetric under G , in the sense that G ρ ∝ ρ is necessarily ( d + 2 )-nonseparable, i.e., is not the mixture of tensor products of d + 2 states in the Hilbert space. Furthermore, such states cannot be prepared from any ( d + 2 )-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in d ≤ 1 and plausibility arguments in d > 1 . The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in d = 1 we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.

DOI
10.1103/physrevx.15.011069

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Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in d space dimensions with anomalous symmetry G where the anomaly is characterized by an invariant in the group cohomology H d + 2 [ G , U ( 1 ) ] . We show that any mixed state ρ that is strongly symmetric under G , in the sense that G ρ ∝ ρ is necessarily ( d + 2 )-nonseparable, i.e., is not the mixture of tensor products of d + 2 states in the Hilbert space. Furthermore, such states cannot be prepared from any ( d + 2 )-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in d ≤ 1 and plausibility arguments in d > 1 . The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in d = 1 we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states.
Generalized symmetries have emerged as a powerful organizing principle for exotic quantum phases. However, their role in open quantum systems, especially for non-invertible cases, remains largely unexplored. We address this by applying a unified tensor-network framework for mixed states with fusion categorical symmetry, which encompasses both invertible and non-invertible ones represented as matrix product operators, and reveals novel quantum phases unique to the open-system setting through the lens of quantum anomalies. In contrast to pure states, where anomalies forbid symmetric short-range correlated phases in one dimension, we construct a broad class of renormalization fixed-point mixed states with zero correlation length given arbitrary strong anomalous fusion categorical symmetry. These states, representing nontrivial mixed-state phases of matter, cannot be efficient prepared via local quantum channels, indicating anomaly-enforced long-range entanglement in the absence of local correlations. Despite this obstruction, we further provide constructions of measurement-enhanced quantum circuits to prepare all these constructed states, offering a practical way to realize and probe anomalous generalized symmetries in open quantum systems.
We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak Z2\mathbb{Z}_2Z2​ symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.
In this paper, we develop a systematic approach to characterize the ′ t Hooft anomaly in open quantum systems. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. In the case where the full symmetry group Γ arises as a central extension of the strong-symmetry group K with the weak-symmetry group G , we find that anomalies of bosonic systems are classified by H d + 2 ( Γ , U ( 1 ) ) / H d + 2 ( G , U ( 1 ) ) in d spatial dimensions. To illustrate the power of anomalies in open quantum systems, we generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed. Analogous to the “anomaly matching” condition ensuring nontrivial low-energy physics in closed systems, anomaly also guarantees nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Notably, we identify a novel ( 1 + 1 ) D mixed-state quantum phase that has no counterpart in closed systems, where the steady state shows no nontrivial correlation function in the bulk, but displays long-range correlation on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Moreover, we explore the generalization of the “anomaly inflow” mechanism in open quantum systems. We construct ( 1 + 1 ) D and ( 2 + 1 ) D Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies. Finally, we generalize our results to the cases including lattice translation symmetries, which serves as the Lieb-Schulz-Mattis (LSM) theorem for mixed states.
Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.
Quantum anomalies strongly constrain the possible behavior of many-body systems. A prime example is the Lieb-Schultz-Mattis (LSM) theorem, which relates UV symmetry and filling constraints to IR features of the energy spectrum and ground-state structure. Here, we ask how LSM constraints shape dynamical signatures and temporal correlations in open quantum systems. Using a spacetime duality, we show that the Liouvillian of a ddd-dimensional repeated quantum channel with a mixed anomaly between strong SSS and weak GGG symmetry can be mapped to a (d+1)(d{+}1)(d+1)-dimensional mixed-state symmetry-protected topological (mSPT) phase. Under this correspondence, the initial and steady states of the channel are identified with boundary states of the higher-dimensional mSPT in the presence of bulk projection. We further introduce the twisted Renyi-NNN correlator (TRNC) as a probe of temporal correlations in the channel and demonstrate that it is dual to the mSPT strange correlator, providing a direct bulk-boundary route to diagnose long-range temporal order implied by the LSM anomaly. Finally, we identify the \textit{Liouvillian singular spectrum}, rather than the Liouvillian spectrum itself, as a more fundamental diagnostic of quantum anomalies, and show that it is dual to the operator entanglement spectrum of the mSPT.
We present a new mechanism for long-range entanglement (LRE) in strongly symmetric many-body mixed states that does not rely on symmetry anomalies or long-range correlations. Our primary example is the maximally mixed state in the translation-invariant subspace on a one-dimensional ring. This state is LRE because translationally symmetric short-range entangled states span a subspace whose dimension grows only polynomially with system size, whereas the full translation-invariant subspace grows exponentially. We further discuss certain unconventional properties of this state, including logarithmically growing conditional mutual information, strong-to-weak spontaneous symmetry-breaking, and Rényi-index-dependent operator-space entanglement. We also construct a geometrically non-local Lindbladian to stabilize this state as the steady state. Our results identify dimensional mismatch as a novel route to LRE that is intrinsic to many-body mixed states.