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arXiv:2502.14034

Decoherence-induced self-dual criticality in topological states of matter

Q. Q. Wang, Romain Vasseur, Simon Trebst, Andreas W. W. Ludwig, Guoyi Zhu

Published Feb 19, 2025• 0 citations
arXivHTMLPDF

Abstract

Quantum measurements performed on a subsystem of a quantum many-body state can generate entanglement for its remaining constituents. The whole system including the measurement record is described by a hybrid mixed state, which can exhibit exotic phase transitions and critical phenomena. We demonstrate that generic measurement-induced phase transitions (MIPTs) can be cast as decoherence-induced critical mixed states in one higher dimension, by constructing a projected entangled pair state (PEPS) prior to decoherence or measurement. In this context, a deeper conceptual understanding of such mixed-state criticality is called for, particularly with regard to algebraic symmetry as an advanced organizing principle for such entangled states of matter. Integrating these connections we investigate the role of self-dual symmetry -- a fundamental notion in theoretical physics -- in mixed states, showing that the decoherence of electric (e) and magnetic (m) vortices from the 2D bulk of the toric code, or equivalently, a 2D cluster state with symmetry-protected topological order, can leave a (1+1)D quantum critical mixed state protected by a weak Kramers-Wannier self-dual symmetry. The corresponding self-dual critical bulk is described by the N->1 limit of the 2D Non-linear Sigma Model in symmetry class D with target space SO(2N)/U(N) at $Θ$-angle $π$, and represents a "measurement-version" of the Cho-Fisher network model subjected to Born-rule randomness...

DOI
10.48550/arxiv.2502.14034

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Quantum measurements performed on a subsystem of a quantum many-body state can generate entanglement for its remaining constituents. The whole system including the measurement record is described by a hybrid mixed state, which can exhibit exotic phase transitions and critical phenomena. We demonstrate that generic measurement-induced phase transitions (MIPTs) can be cast as decoherence-induced critical mixed states in one higher dimension, by constructing a projected entangled pair state (PEPS) prior to decoherence or measurement. In this context, a deeper conceptual understanding of such mixed-state criticality is called for, particularly with regard to algebraic symmetry as an advanced organizing principle for such entangled states of matter. Integrating these connections we investigate the role of self-dual symmetry -- a fundamental notion in theoretical physics -- in mixed states, showing that the decoherence of electric (e) and magnetic (m) vortices from the 2D bulk of the toric code, or equivalently, a 2D cluster state with symmetry-protected topological order, can leave a (1+1)D quantum critical mixed state protected by a weak Kramers-Wannier self-dual symmetry. The corresponding self-dual critical bulk is described by the N->1 limit of the 2D Non-linear Sigma Model in symmetry class D with target space SO(2N)/U(N) at ΘΘΘ-angle πππ, and represents a "measurement-version" of the Cho-Fisher network model subjected to Born-rule randomness...
In this work, we study quantum many-body systems which are self-dual under duality transformation connecting different symmetry protected topological (SPT) phases. We provide a geometric explanation of the criticality of these self-dual models. More precisely, we show a ground state (quasi)degeneracy under the periodic boundary conditions, i.e., the ingappability of the bulk spectrum. Equivalently, the symmetry group at criticality, including the duality symmetry, has a mixed 't Hooft anomaly. This approach can not only predict the spectrum of the self-dual model with ordinary 0-form symmetry but also be applied to that with generalized symmetry, such as higher form and subsystem symmetry. As an application, we illustrate our results with several examples in one and two dimensions, which separate two different SPTs.
A topological phase can undergo a phase transition driven by anyon condensation. A potential obstruction to such a mechanism could arise if there exists a symmetry between anyons that have non-trivial mutual statistics. Here we consider toric code subjected to errors that tend to proliferate anyons with non-trivial mutual statistics. Using triangle inequality, we show that in the presence of electromagnetic duality and a partial-transpose symmetry, a decoherence induced phase transition out of the topological phase must be rather unconventional and lie beyond standard rules of anyon condensation. To explore such physics, we first subject toric code to a self-dual quantum channel where Kraus operators are proportional to X+Z. We find that the topological phase is stable up to the maximal error rate, when viewing density matrix as a pure state in the double Hilbert space. To access an unconventional transition, we then consider a perturbed toric code subjected to the self-dual channel, and find numerical evidence that beyond a critical error rate, the topological phase is destroyed resulting in a critical phase where anyons are only power-law condensed.
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.
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Decoherence on quantum many-body systems induces nontrivial mixed states with specific properties that have no counterparts in pure states. The authors clarify here the emergence of ``intrinsic mixed state topological order'' in a toric code state under decoherence, which accompanies exotic anyon proliferation. Furthermore, the authors find that the decoherence-induced phase transition in the stabilizer formalism is described by a percolation picture in two dimensions. The viewpoint of one-form symmetry and its spontaneous symmetry breaking plays an important role in this study.
Decoherence in realistic quantum platforms motivates a mixed-state notion of topological phases of matter, including average symmetry-protected topological (ASPT) phases. Alongside this progress, generalized symmetries--notably noninvertible and dipole symmetries--have become powerful organizing principles for exotic quantum phases, yet their implications for mixed states remain less explored. In this work, we bridge these directions through a gauging correspondence between mixed-state phases with generalized symmetries and mixed-state phases with ordinary group symmetries, recasting the classification of noninvertible and dipole ASPT phases into familiar classifications of symmetry breaking and ASPT phases with dual symmetries. Using this approach, we classify and construct a subclass of ASPT phases with non-invertible and dipole symmetries in (1+1)d(1+1)d(1+1)d, including phases that are intrinsic to mixed states, and characterize them via string order parameters and protected edge modes.