Abstract
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 ρAB is the difference \( {\mathcal{F}}_{\rho}\left(A:B\right)\equiv \textrm{Tr}\ {\rho}_{AB}^2-\textrm{Tr}\ {\rho}_A^2\ \textrm{Tr}\ {\rho}_B^2 \). 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.
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Acknowledgments
We thank Bartek Czech, Abhijit Gadde, Issac Kim, Gautam Mandal, Shiraz Minwalla, Pranab Sen and Sandip Trivedi for helpful discussions. VB is supported in part by the Department of Energy through grant DE-SC0013528 and grant QuantISED DE-SC0020360, as well as the Simons Foundation through the It From Qubit Collaboration (Grant No. 38559). AK is supported by the Simons Foundation through the It from Qubit Collaboration. CL is supported by the Department of Energy through QuantISED grant DE-SC0020360. OP and HR are supported by the Department of Atomic Energy, Government of India, under project identification number RTI 4002.
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Balasubramanian, V., Kar, A., Li, C. et al. Quantum error correction from complexity in Brownian SYK. J. High Energ. Phys. 2023, 71 (2023). https://doi.org/10.1007/JHEP08(2023)071
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DOI: https://doi.org/10.1007/JHEP08(2023)071