Diagnostics of mixed-state topological order and breakdown of quantum memory
PRX Quantum(2023)
摘要
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.
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