Towards a classification of mixed-state topological orders in two dimensions

Abstract

The classification and characterization of topological phases of matter is well understood for ground states of gapped Hamiltonians that are well isolated from the environment. However, decoherence due to interactions with the environment is inevitable -- thus motivating the investigation of topological orders in the context of mixed states. Here, we take a step toward classifying mixed-state topological orders in two spatial dimensions by considering their (emergent) generalized symmetries. We argue that their 1-form symmetries and the associated anyon theories lead to a partial classification under two-way connectivity by quasi-local quantum channels. This allows us to establish mixed-state topological orders that are intrinsically mixed, i.e., that have no ground state counterpart. We provide a wide range of examples based on topological subsystem codes, decohering $G$-graded string-net models, and "classically gauging" symmetry-enriched topological orders. One of our main examples is an Ising string-net model under the influence of dephasing noise. We study the resulting space of locally-indistinguishable states and compute the modular transformations within a particular coherent space. Based on our examples, we identify two possible effects of quasi-local quantum channels on anyon theories: (1) anyons can be incoherently proliferated -- thus reducing to a commutant of the proliferated anyons, or (2) the system can be "classically gauged", resulting in the symmetrization of anyons and an extension by transparent bosons. Given these two mechanisms, we conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate.

34+10 pages, published version

References in corpus (76)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Topological quantum memory
• Generalized Global Symmetries
• String-net condensation: A physical mechanism for topological phases
• A Theory of Quantum Error-Correcting Codes
• Zoo of quantum-topological phases of matter
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Symmetries and conserved quantities in Lindblad master equations
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Topological quantum order: stability under local perturbations
• A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains
• Stabilizer Formalism for Operator Quantum Error Correction
• Quantum tensor product structures are observable-induced
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• A symmetry principle for Topological Quantum Order
• Quantum memories at finite temperature
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Fermions, strings, and gauge fields in lattice spin models
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• On the classification of topological orders
• Three-dimensional topological lattice models with surface anyons
• Entanglement and topological entropy of the toric code at finite temperature
• Topological Order at Non-zero Temperature
• Quantum criticality under decoherence or weak measurement
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• A theory of 2+1D bosonic topological orders
• Anomalous Symmetry Fractionalization and Surface Topological Order
• Fermionic Modular Categories and the 16-fold Way
• Symmetry fractionalization and twist defects
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Symmetry breaking and error correction in open quantum systems
• Average Symmetry-Protected Topological Phases
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Diagnostics of mixed-state topological order and breakdown of quantum memory
• Generalized string-net models: A thorough exposition
• Universal topological phase of 2D stabilizer codes
• Detecting Topological Order at Finite Temperature Using Entanglement Negativity
• How to Build the Thermofield Double State
• Measurements conspire nonlocally to restructure critical quantum states
• Twisted gauge theories in 3D Walker-Wang models
• Exact bosonization in two spatial dimensions and a new class of lattice gauge theories
• Structure of 2D Topological Stabilizer Codes
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• Intrinsic Mixed-state Topological Order
• Fracton phases via exotic higher-form symmetry-breaking
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Pauli stabilizer models of twisted quantum doubles
• Local topological order inhibits thermal stability in 2D
• Classification of phases for mixed states via fast dissipative evolution
• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Topological order and topological entropy in classical systems
• Microscopic definitions of anyon data
• Entanglement in one-dimensional critical state after measurements
• Separability transitions in topological states induced by local decoherence
• Pauli topological subsystem codes from Abelian anyon theories
• Minimal nondegenerate extensions
• Quantum Approximate Markov Chains are Thermal
• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Channeling quantum criticality
• Exact emergent higher-form symmetries in bosonic lattice models
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Interacting Anyonic Fermions in a Two-Body Color Code Model
• Realizing anomalous anyonic symmetries at the surfaces of 3d gauge theories
• Universal lower bound on topological entanglement entropy
• Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics
• Topological phase transitions at finite temperature
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Verlinde formula from entanglement
• Thermalization in Kitaev's quantum double models via Tensor Network techniques

Cited by in corpus (41)

• Intrinsic Mixed-state Topological Order
• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Stability of mixed-state quantum phases via finite Markov length
• Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions
• Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Efficient Detection of Strong-To-Weak Spontaneous Symmetry Breaking via the Rényi-1 Correlator
• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak
• Average-exact mixed anomalies and compatible phases
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order
• Mixed-state phase transitions in spin-Holstein models
• Reflected multi-entropy and its holographic dual
• Finite-temperature quantum topological order in three dimensions
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Gauge theory and mixed state criticality
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Hardness of observing strong-to-weak symmetry breaking
• Mixed-State Topological Order under Coherent Noise
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Many-body quantum catalysts for transforming between phases of matter
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation
• Anyon condensation in mixed-state topological order
• Quantum Strong-to-Weak Spontaneous Symmetry Breaking in Decohered One Dimensional Critical States
• Revisiting Nishimori multicriticality through the lens of information measures
• Simulation of bilayer Hamiltonians based on monitored quantum trajectories
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems
• Quantum circuit complexity and unsupervised machine learning of topological order
• Parent Lindbladians for Matrix Product Density Operators
• Simulating Topological Order on Quantum Processors
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
• Holographic duality between bulk topological order and boundary mixed-state order
• Robust Mixed-State Cluster States and Spurious Topological Entanglement Negativity
• Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases
• Phases of Floquet code under local decoherence
• Locally Purified Maximally Mixed States At Scale: Entanglement Pruning and Symmetries
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter