Quantum error correction from complexity in Brownian SYK

Journal Article · Mon Aug 14 00:00:00 EDT 2023 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/jhep08(2023)071· OSTI ID:2420293

Balasubramanian, Vijay ^[1]; Kar, Arjun ^[2]; Li, Cathy ^[3]; Parrikar, Onkar ^[4]; Rajgadia, Harshit ^[4]

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1. Univ. of Pennsylvania, Philadelphia, PA (United States); Vrije Universiteit Brussel (VUB), Brussels (Belgium); Univ. of Pennsylvania, Philadelphia, PA (United States)
2. Univ. of British Columbia, Vancouver, BC (Canada)
3. Univ. of Pennsylvania, Philadelphia, PA (United States)
4. Tata Inst. of Fundamental Research, Bombay (India)

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 $$\mathcal{F}$$_p ($A : B$) ≡ $$\mathrm{T}$$r $$p^{2}_{AB}$$ - $$\mathrm{T}$$r $$p^{2}_{A}$$ $$\mathrm{T}$$r $$p^{2}_{B}$$. 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.

• Research Organization: Univ. of Pennsylvania, Philadelphia, PA (United States)
• Sponsoring Organization: Simons Foundation; USDOE Office of Science (SC)
• Grant/Contract Number: SC0013528; SC0020360
• OSTI ID: 2420293
• Journal Information: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 8 Vol. 2023; ISSN 1029-8479
• Publisher: Springer Nature
• Country of Publication: United States
• Language: English

Related Subjects

1/N expansion
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
AdS-CFT correspondence
field theories in lower dimensions