Perturbative analysis of topological entanglement entropy from conditional independence

Abstract

We use the structure of conditionally independent states to analyze the stability of topological entanglement entropy. For the ground state of quantum double or Levin-Wen model, we obtain a bound on the first order perturbation of topological entanglement entropy in terms of its energy gap and subsystem size. The bound decreases superpolynomially with the size of the subsystem, provided the energy gap is nonzero. We also study the finite temperature stability of stabilizer models, for which we prove a stronger statement than the strong subadditivity of entropy. Using this statement and assuming i) finite correlation length ii) small conditional mutual information of certain configurations, first order perturbation effect for arbitrary local perturbation can be bounded. We discuss the technical obstacles in generalizing these results.

9 pages, 5 figures, added references, clarified Lemma 1, some other minor changes

References in corpus (16)

• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid
• Ground state entanglement and geometric entropy in the Kitaev's model
• Entanglement and topological entropy of the toric code at finite temperature
• Topological Order at Non-zero Temperature
• Topological order in a 3D toric code at finite temperature
• Quantum Graphical Models and Belief Propagation
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Entanglement, fidelity and topological entropy in a quantum phase transition to topological order
• Quantum Belief Propagation
• On the energy landscape of 3D spin Hamiltonians with topological order
• Markov entropy decomposition: a variational dual for quantum belief propagation
• Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs

Cited by in corpus (29)

• Magic-state resource theory for the ground state of the transverse-field Ising model
• Renyi generalizations of the conditional quantum mutual information
• Spurious topological entanglement entropy from subsystem symmetries
• Deciphering the nonlocal entanglement entropy of fracton topological orders
• An invariant of topologically ordered states under local unitary transformations
• Topological Rényi entropy after a quantum quench
• Quantum Approximate Markov Chains are Thermal
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• Quantum many-body systems in thermal equilibrium
• Probing topological order with Rényi entropy
• Conditional quantum one-time pad
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• Conditional Decoupling of Quantum Information
• A lower bound of quantum conditional mutual information
• Jones index, secret sharing and total quantum dimension
• Localized Virtual Purification
• Determining the structure of real-space entanglement spectrum from approximate conditional independence
• Entropic topological invariant for a gapped one-dimensional system
• Stability of topological purity under random local unitaries
• From decay of correlations to locality and stability of the Gibbs state
• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Evolved Quantum Boltzmann Machines
• Strong decay of correlations for Gibbs states in any dimension
• Stability of thermal equilibrium in long-range quantum systems
• Gibbs state sampling via cluster expansions
• Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states
• Natural gradient and parameter estimation for quantum Boltzmann machines
• Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems