Quantum error correction from complexity in Brownian SYK

doi:10.1007/JHEP08(2023)071

2023-08-14

Vijay Balasubramanian *

DOI：10.1007/JHEP08(2023)071

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摘要

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We study the robustness of quantum error correction in a one-parameter ensemble of codes generated by the Brownian SYK model, where the parameter quantifies the encoding complexity. The robustness of error correction by a quantum code is upper bounded by the mutual purity of a certain entangled state between the code subspace and environment in the isometric extension of the error channel, where the mutual purity of a density matrix rho(AB) is the difference F rho(A : B) equivalent to Tr rho(2)(AB) - Tr rho(2)(A) Tr rho(2)(B). We show that when the encoding complexity is small, the mutual purity is O (1) for the erasure of a small number of qubits (i.e., the encoding is fragile). However, this quantity decays exponentially, becoming O(1/N) for O(log N) encoding complexity. Further, at polynomial encoding complexity, the mutual purity saturates to a plateau of O(e(-N)). We also find a hierarchy of complexity scales associated to a tower of subleading contributions to the mutual purity that quantitatively, but not qualitatively, adjust our error correction bound as encoding complexity increases. In the AdS/CFT context, our results suggest that any portion of the entanglement wedge of a general boundary subregion A with sufficiently high encoding complexity is robustly protected against low-rank errors acting on A with no prior access to the encoding map. From the bulk point of view, we expect such bulk degrees of freedom to be causally inaccessible from the region A despite being encoded in it.

Keyword:

1/N Expansion

AdS-CFT Correspondence

Field Theories in Lower Dimensions