Diagnostics of mixed-state topological order and breakdown of quantum memory

Abstract

Topological quantum memory can protect information against local errors up to finite error thresholds. Such thresholds are usually determined based on the success of decoding algorithms rather than the intrinsic properties of the mixed states describing corrupted memories. Here we provide an intrinsic characterization of the breakdown of topological quantum memory, which both gives a bound on the performance of decoding algorithms and provides examples of topologically distinct mixed states. We employ three information-theoretical quantities that can be regarded as generalizations of the diagnostics of ground-state topological order, and serve as a definition for topological order in error-corrupted mixed states. We consider the topological contribution to entanglement negativity and two other metrics based on quantum relative entropy and coherent information. In the concrete example of the 2D Toric code with local bit-flip and phase errors, we map three quantities to observables in 2D classical spin models and analytically show they all undergo a transition at the same error threshold. This threshold is an upper bound on that achieved in any decoding algorithm and is indeed saturated by that in the optimal decoding algorithm for the Toric code.

Mixed State Entanglement Measures in Topological Orders

“…-Upon completion of this manuscript, we became aware of a few related works: The use of PT negativity for diagnosing error-corrupted topological orders is discussed in Ref. 36. Refs.…”

Channeling quantum criticality

“…One important question is which properties of quantum matter survive in a noisy environment, and how to understand any residual order in the mixed state? Recently, substantial progress has been made in understanding mixed state versions of gapped quantum systems including symmetry protected topological phases and topological order [16][17][18][19][20][21][22][23].…”