Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory

Author:

Fan Ruihua1,Bao Yimu2ORCID,Altman Ehud23,Vishwanath Ashvin1

Affiliation:

1. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

2. Department of Physics, University of California, Berkeley, California 94720, USA

3. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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 two-dimensional (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. Published by the American Physical Society 2024

Funder

Simons Foundation

U.S. Department of Energy

Office of Science

Gordon and Betty Moore Foundation

NSF-DMR

National Quantum Information Science Research Centers

Quantum Systems Accelerator

NSF QLCI

ICAM

Publisher

American Physical Society (APS)

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