An analog of topological entanglement entropy for mixed states

Abstract

We propose the convex-roof extension of quantum conditional mutual information ("co(QCMI)") as a diagnostic of topological order in a mixed state. We focus primarily on topological states subjected to local decoherence, and employ the Levin-Wen scheme to define co(QCMI), so that for a pure state, co(QCMI) equals topological entanglement entropy (TEE). By construction, co(QCMI) is zero if and only if a mixed state can be decomposed as a convex sum of pure states with zero TEE. We show that co(QCMI) is non-increasing with increasing decoherence when Kraus operators are proportional to the product of onsite unitaries. This implies that unlike a pure state transition between a topologically trivial and a non-trivial phase, the long-range entanglement at a decoherence-induced topological phase transition as quantified by co(QCMI) is less than or equal to that in the proximate topological phase. For the 2d toric code decohered by onsite bit/phase-flip noise, we show that co(QCMI) is non-zero below the error-recovery threshold and zero above it. Relatedly, the decohered state cannot be written as a convex sum of short-range entangled pure states below the threshold. We conjecture and provide evidence that in this example, co(QCMI) equals TEE of a recently introduced pure state. In particular, we develop a tensor-assisted Monte Carlo (TMC) computation method to efficiently evaluate the Rényi TEE for the aforementioned pure state and provide non-trivial consistency checks for our conjecture. We use TMC to also calculate the universal scaling dimension of the anyon-condensation order parameter at this transition.

(Accepted version) 16 pages main text, 3 pages of appendices, 7 figures, minor generalizations added to Sec.III, typos fixed

References in corpus (14)

• Topological Quantum Distillation
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Identifying Topological Order by Entanglement Entropy
• Seeing a c-theorem with holography
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Entanglement Detection in the Stabilizer Formalism
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• A c-theorem for the entanglement entropy
• Creation, manipulation, and detection of Abelian and non-Abelian anyons in optical lattices
• Strong disorder fixed points in the two-dimensional random-bond Ising model
• Multicritical Nishimori point in the phase diagram of the +- J Ising model on a square lattice
• Extracting subleading corrections in entanglement entropy at quantum phase transitions
• Incremental SWAP Operator for Entanglement Entropy: Application for Exponential Observables in Quantum Monte Carlo Simulation
• An integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulation

Cited by in corpus (5)

• Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order
• Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems
• Symmetrizing the Constraints -- Density Matrix Renormalization Group for Constrained Lattice Models
• Multiparty Entanglement Microscopy of Quantum Ising models in 1d, 2d and 3d
• Robust Mixed-State Cluster States and Spurious Topological Entanglement Negativity