Analog of Topological Entanglement Entropy for Mixed States

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 nonincreasing 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 nontrivial 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 two-dimensional toric code decohered by onsite bit- and phase-flip noise, we show that co(QCMI) is nonzero 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 nontrivial consistency checks for our conjecture. We use TMC to also calculate the universal scaling dimension of the anyon-condensation order parameter at this transition.