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arXiv:2504.10569

Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement

Po-Shen Hsin, Ryohei Kobayashi, Abhinav Prem

Published Apr 14, 2025• 0 citations
arXivHTMLPDF

Abstract

We show that generic gapped quantum many-body states which respect an anomalous finite higher-form symmetry have an exponentially small overlap with any short-range entangled (SRE) state. Hence, anomalies of higher-form symmetries enforce $intrinsic$ long-range entanglement, which is in contrast with anomalies of ordinary (0-form) symmetries which are compatible with symmetric SRE states (specifically, symmetric cat states). As an application, we show that the anomalies of strong higher-form symmetries provide a diagnostic for mixed-state topological order in $d \geq 2$ spatial dimensions. We also identify a new (3+1)D intrinsic mixed-state topological order that does not obey remote-detectability by local decoherence of the (3+1)D Toric Code with fermionic loop excitations. This breakdown of remote detectability, as encoded in anomalies of strong higher-form symmetries, provides a partial characterization of intrinsically mixed-state topological order.

DOI
10.48550/arxiv.2504.10569

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We show that generic gapped quantum many-body states which respect an anomalous finite higher-form symmetry have an exponentially small overlap with any short-range entangled (SRE) state. Hence, anomalies of higher-form symmetries enforce intrinsicintrinsicintrinsic long-range entanglement, which is in contrast with anomalies of ordinary (0-form) symmetries which are compatible with symmetric SRE states (specifically, symmetric cat states). As an application, we show that the anomalies of strong higher-form symmetries provide a diagnostic for mixed-state topological order in d≥2d \geq 2d≥2 spatial dimensions. We also identify a new (3+1)D intrinsic mixed-state topological order that does not obey remote-detectability by local decoherence of the (3+1)D Toric Code with fermionic loop excitations. This breakdown of remote detectability, as encoded in anomalies of strong higher-form symmetries, provides a partial characterization of intrinsically mixed-state topological order.
It is known that particles with exotic properties can emerge in systems made of simple constituents such as qubits, due to long-range quantum entanglement. In this paper, we provide quantitative characterizations of entanglement necessary for emergent anyons and fermions by using the geometric entanglement measure (GEM), which quantifies the maximal overlap between a given state and any short-range-entangled states. For systems with emergent anyons, based on the braiding statistics, we show that the GEM scales linearly in the system size regardless of microscopic details. The phenomenon of emergent anyons can also be understood within the framework of quantum error correction (QEC). Specifically, we show that the GEM of any 2D stabilizer codes must be at least quadratic in the code distance. Our proof is based on a generic prescription for constructing string operators, establishing a rigorous and direct connection between emergent anyons and QEC. For systems with emergent fermions, despite that the ground state subspaces could be exponentially huge and their coding properties could be rather poor, we show that the GEM also scales linearly in the system size. Our analysis establishes an intriguing link between quantum anomaly and entanglement: A quantum state respecting anomalous 1-form symmetries must be long-range entangled and have large GEM. Our results also extend to mixed states, such as the Z X -dephased toric code, providing a provably nontrivial class of intrinsically mixed-state phases.
Decoherence is a major obstacle to the preparation of topological order in noisy intermediate-scale quantum devices. Here, we show that decoherence can also give rise to new types of topological order. Specifically, we construct concrete examples by proliferating fermionic anyons in the toric code via local quantum channels. The resulting mixed states retain long-range entanglement, which manifests in the nonzero topological entanglement negativity, though the topological quantum memory is destroyed by decoherence. By comparison with the gapless spin liquid in pure states, we show that the identified states represent a novel intrinsic mixed-state topological order, which has no counterpart in pure states. Through the lens of quantum anomalies of 1-form symmetries, we then provide general constructions of intrinsic mixed-state topological order and reveal the existence of nonbosonic deconfined anyons as another key feature of these novel phases. The extended meaning and characterization of deconfined excitations and their statistics in mixed states are clarified. Moreover, when these deconfined anyons have nontrivial braiding statistics, we prove that the mixed states cannot be prepared via finite-depth local quantum channels from any bipartite separable states. We further demonstrate our construction using the decohered Kitaev honeycomb model and the decohered double-semion model. In the latter case, a surprising scenario arises where decoherence gives rise to additional types of deconfined anyons.
We construct large families of mixed-state topological orders in (3+1)D, including intrinsically mixed-state topological orders. We realize these theories by relating the data parameterizing braided fusion 2-categories to the strong higher-form symmetries and their anomalies that characterize mixed-state topological order. We focus here on bosonic theories, which include theories which admit an emergent fermion in the spectrum of line operators. Our general framework is exemplified on the lattice, where we present concrete models and identify how the data of braided fusion 2-categories are physically encoded. We then formalize this data and show that every braided fusion 2-category arises from a general construction (decoherence followed by strong and weak gauging) as the category of topological operators of a (3+1)D mixed state, with intrinsically mixed-state topological orders distinguished by the presence of strong higher-form symmetries whose anomalies cannot be realized in any (3+1)D gapped ground state. Notably, we show that (3+1)D admits a new mechanism with no (2+1)D counterpart: intrinsically mixed-state order whose topological operators braid non-degenerately, but which carries a gravitational anomaly.
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.
We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature-it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.
Abstract Symmetry-protected topological phases are short-range entangled quantum states characterized by anomalous edge behavior, a manifestation of the bulk-boundary correspondence for topological phases. Moreover, the Li-Haldane conjecture posits that the entanglement spectrum exhibits the same anomaly as the physical edge spectrum, thereby serving as an entanglement-based fingerprint for identifying topological phases. In this work, we extend the entanglement-based diagnostic tools by demonstrating that the edge anomaly is manifested not only in the entanglement spectrum but also in the reduced density matrix itself, a phenomenon we refer to as the mixed-state anomaly. For the two-dimensional SPT phase, we derive a topological contribution in the disorder parameter as a consequence of this mixed-state anomaly. Furthermore, with additional time-reversal symmetry, the anomalous reduced density matrix induces a spontaneous-symmetry-breaking type long-range order in the presence of the disorder operator.