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arXiv:2301.05687 · Physical Review Letters

Mixed-State Topological Order and the Errorfield Double Formulation of Decoherence-Induced Transitions

Yimu Bao, Ruihua Fan, Ashvin Vishwanath, Ehud Altman

Published Apr 28, 2026• 13 citations
arXivHTMLPDF

Abstract

We develop an effective field theory characterizing the impact of decoherence on states with Abelian topological order and on their capacity to protect quantum information. The decoherence appears as a temporal defect in the double topological quantum field theory that describes the pure density matrix of the uncorrupted state, and it drives a boundary phase transition involving anyon condensation at a critical coupling strength. The ensuing decoherence-induced phases and the loss of quantum information are classified by the Lagrangian subgroups of the double topological order. Our framework generalizes the error recovery transitions, previously derived for certain stabilizer codes, to generic topologically ordered states and shows that they stem from phase transitions in the intrinsic topological order characterizing the mixed state.

DOI
10.1103/6f98-tvb8

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We develop an effective field theory characterizing the impact of decoherence on states with Abelian topological order and on their capacity to protect quantum information. The decoherence appears as a temporal defect in the double topological quantum field theory that describes the pure density matrix of the uncorrupted state, and it drives a boundary phase transition involving anyon condensation at a critical coupling strength. The ensuing decoherence-induced phases and the loss of quantum information are classified by the Lagrangian subgroups of the double topological order. Our framework generalizes the error recovery transitions, previously derived for certain stabilizer codes, to generic topologically ordered states and shows that they stem from phase transitions in the intrinsic topological order characterizing the mixed state.
Quantum error correction protects quantum information against decoherence, provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here, we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error-correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry-protected fermion matter.
Quantum error correction protects quantum information against decoherence provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry protected fermion matter.
We investigate universal entanglement signatures of mixed-state phases obtained by decohering pure-state topological order (TO), focusing on topological corrections to logarithmic entanglement negativity and mutual information: topological entanglement negativity (TEN) and topological mutual information (TMI). For Abelian TOs under decoherence, we develop a replica field-theory framework based on a doubled-state construction that relates TEN and TMI to the quantum dimensions of domain-wall defects between decoherence-induced topological boundary conditions, yielding general expressions in the strong-decoherence regime. We further compute TEN and TMI exactly for decohered GGG-graded string-net states, including cases with non-Abelian anyons. We interpret the results within the strong one-form-symmetry framework for mixed-state TOs: TMI probes the total quantum dimension of the emergent premodular anyon theory, whereas TEN detects only its modular part.
Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve nnn copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.
We propose a general framework for studying two-dimensional (2D) topologically ordered states subject to local correlated errors and show that the resulting mixed state can display (imTO)—topological order that is not expected to occur in the ground state of 2D local gapped Hamiltonians. Specifically, we show that decoherence, previously interpreted as anyon condensation in a doubled Hilbert space, is more naturally phrased as, and provides a physical mechanism for, “gauging out” anyons in the original Hilbert space. We find that gauging out anyons generically results in imTO, with the decohered mixed state strongly symmetric under certain anomalous 1-form symmetries. This framework lays bare a striking connection between the decohered density matrix and , which can appear as anomalous surface states of three-dimensional topological orders. Through a series of examples, we show that the decohered state can display a classical memory, encode logical qubits (i.e., exhibit a quantum memory), and even host chiral or nonmodular topological order. We argue that a partial classification of imTO is given in terms of nonmodular braided-fusion categories.
This paper presents a formal framework framing quantum decoherence as an information-theoretic phase transition at the quantum-to-classical boundary. We demonstrate that environment-induced superselection (einselection) can be characterized as a true dynamical phase transition where the system-environment coupling strength acts as a thermodynamic control parameter, and the suppression of off-diagonal density matrix elements functions as an order parameter analogous to magnetization in the Ising model. Using quantum mutual information, von Neumann entropy, and entanglement entropy, we track the operational flow of information from a quantum system into environmental degrees of freedom, demonstrating how a singular classical reality emerges via the redundant copying of pointer states into environmental fragments—a process governed by Quantum Darwinism. We analyze a simulated spin-boson model and random unitary circuits with tunable dissipation, extracting clear signatures of criticality including power-law scaling and diverging correlation times at a critical threshold . Crucially, we connect this behavior to recent advancements (2023–2025) in mixed-state topological order, establishing that decoherence-induced transitions map onto anyon condensation transitions within an errorfield double formulation. Finally, we explore the foundational implications of this framework: it sharpens the measurement problem via informational symmetry breaking, reframes observation as a high-decoherence thermodynamic limit, and introduces a Gödelian perspective by demonstrating that determining the critical threshold for arbitrary many-body environments is formally undecidable due to the non-computability of the spectral gap in the thermodynamic limit.