From the breakdown of topological quantum memory to mixed-state topological order


Topological quantum memory can protect information against local errors up to finite error thresholds. Such thresholds are usually determined via the decoding algorithms rather than the intrinsic properties of the mixed states describing corrupted memories. Such properties should be naturally related to the concept of topological order in mixed states. In this talk, I will discuss three information-theoretical diagnostics, the topological 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 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. Time permitted, I will also discuss a so-called error-field double formalism, that provides a more general perspective by mapping the error-induced phases to boundary anyon condensation.