An analog of topological entanglement entropy for mixed states

• URL: http://arxiv.org/abs/2407.20500v1
• Date: Tue, 30 Jul 2024 02:26:45 GMT
• Title: An analog of topological entanglement entropy for mixed states
• Authors: Ting-Tung Wang, Menghan Song, Zi Yang Meng, Tarun Grover,
• Abstract summary: We show that co(QCMI) is non-increasing with increasing decoherence when Kraus operators are proportional to the product of onsite unitaries. 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. We conjecture and provide evidence that in this example, co(QCMI) equals TEE of a recently introduced pure state.
• Score: 0.3749861135832073
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose the convex-roof extension of quantum conditional mutual information ("co(QCMI)") as a diagnostic of long-range entanglement 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 R\'enyi 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.

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