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Physical review. B./Physical review. B

Intrinsic mixed-state topological order in a stabilizer system under stochastic decoherence: Strong-to-weak spontaneous symmetry breaking from a percolation point of view

Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Published Feb 19, 2025• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.111.064111

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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.
Discovering quantum orders in mixed many-body systems is an ongoing issue. Very recently, the notion of an intrinsic mixed state topologically-ordered (IMTO) state was proposed. As a concrete example, we observe the emergence of IMTO by studying the toric code system under stochastic maximal decoherence by ZXZXZX-diagonal type projective measurement without monitoring. We study how the toric code state changes to an IMTO state at the level of the averaged quantum trajectories. This phase transition is understood from the viewpoint of spontaneous symmetry breaking (SSB) of 1-form weak symmetry, that is, the IMTO is characterized by the symmetry restoration from the SSB, which comes from the proliferation of anyons. To understand the emergent IMTO, order and disorder parameters of 1-form symmetry are numerically studied by stabilizer simulation. The present study clarifies the existence of two distinct microscopic string operators for the fermionic anyons, that leads to distinct fermionic strong and weak 1-form symmetries, and also the obtained critical exponents indicate strong relation between IMTO and percolation.
We study how the color code under decoherence gives rise to an intrinsic mixed-state topological order (imTO), which has no counterpart in pure ground states of local gapped Hamiltonians. For decoherence induced by XX-type operators on red links of the honeycomb lattice, we show that the resulting mixed state inherits half of the topological properties of the color code, including anyon content, logical operators, and topological entanglement structure. Using a gauging procedure for mixed stabilizer states, we identify the emergent phase as closely related to a single toric code. We characterize this phase by topological entanglement negativity (TEN) and perform efficient stabilizer-formalism simulations. While the pure color code has TEN=2ln⁡2{\rm TEN} = 2 \ln 2TEN=2ln2, the maximally decohered state has TEN=ln⁡2{\rm TEN} = \ln 2TEN=ln2, indicating emergence of a single toric code. By tuning the decoherence strength, we find a smooth crossover in TEN accompanied by a pronounced, nearly system-size-independent peak in its variance. We further show that the negativity exhibits characteristic scaling only for subsystem partitions commensurate with the triangular lattice of the emergent toric code. Our results demonstrate that negativity-based quantities provide powerful probes of mixed-state topological order generated by decoherence.
Discovering mixed state quantum orders is an ongoing issue. Recently, it has been recognized that there are (at least) two kinds of symmetries in the mixed state: strong and weak symmetries. Under symmetry-respective decoherence, spontaneous strong-to-weak symmetry breaking (SSSB) can occur. This paper provides a scheme to describe SSSB and other decoherence phenomena in the mixed state by employing the stabilizer formalism and the efficient numerical algorithm of Clifford circuits. We present two systematic numerical studies. In a two-dimensional circuit with a stochastic Ising type decoherence, an SSSB phase transition is clearly observed and its criticality is elucidated by the numerical methods. In particular, we calculate R\'enyi-2 correlations and estimate critical exponents of the SSSB transition. For the second system, we introduce an idea of subgroup SSSB. As an example, we study a system with symmetry-protected-topological order provided by both one-form and zero-form symmetries, and observe how the system evolves under decoherence. After displaying numerical results, we show that viewpoint of percolation is quite useful to understand the SSSB transition, which is applicable for a wide range of decohered states. Finally, we comment on SSSB of one-form-symmetry exemplifying toric code.
We investigate microscopic mechanisms underlying decoherence-induced transitions between topologically ordered states. As a first case study, we analyze the color code using topological entanglement negativity (TEN) and a disorder parameter associated with 1-form symmetry. We interpret these quantities as first- and zeroth-dimensional simplicial homological objects, respectively, and show that the transition can be understood in terms of decoherence percolation. To resolve its local structure, we introduce quasi-local TEN (QLTEN), which visualizes the spatial distribution of local topological properties and the growth of decohered regions. We further introduce explosive percolation (EP), corresponding here to biased decoherence that suppresses the formation of large decohered clusters. As a second case study, we consider the toric code on a triangular lattice under external-field-type decoherence. Numerical results show that QLTEN faithfully captures the emergence of Higgs regions and the survival of logical qubits. Although global TEN, QLTEN clusters, and string operators are strongly correlated in both models, the color code and toric code respond differently to EP patterns. This difference indicates that a universal microscopic description of decoherence-induced topological phase transitions remains challenging even for closely related systems.
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.
Topological orders in (2+12+12+1) dimensions are spontaneous symmetry-breaking (SSB) phases of 1-form symmetries in pure states. The notion of symmetry is further enriched in the context of mixed states, where a symmetry can be either ``strong'' or ``weak.'' In this work, we apply a R\'enyi-2 version of the proposed equivalence relation by [S. Sang et al. (unpublished)] on density matrices that is slightly finer than two-way channel connectivity. This equivalence relation distinguishes general 1-form strong-to-weak SSB (SW-SSB) states from phases containing pure states, and therefore labels SW-SSB states as ``intrinsically mixed.'' According to our equivalence relation, two states are equivalent if and only if they are connected to each other by finite Lindbladian evolution that maintains analytically varying, finite R\'enyi-2 Markov length. We then examine a natural setting for finding such density matrices: disordered ensembles. Specifically, we study the toric code with various types of disorders and show that in each case, the ensemble of ground states corresponding to different disorder realizations forms a density matrix with different strong and weak SSB patterns of 1-form symmetries, including SW-SSB. Furthermore, we show by perturbative calculations that these disordered ensembles form stable ``phases'' in the sense that they exist over a finite parameter range, according to our equivalence relation.