One-loop universality of holographic codes

TL;DR

This paper extends the understanding of holographic error correcting codes to one-loop order, demonstrating their robustness and detailed quantum properties beyond leading order, with implications for bulk-boundary relations and tensor network models.

Contribution

It provides a detailed analysis of one-loop corrections in holographic codes, connecting bulk path integrals with quantum codes and confirming the stability of holographic features beyond leading order.

Findings

Holographic codes maintain universal features at one-loop order.

Bulk and boundary modular Hamiltonians are related as an operator equation.

The code subspace remains invariant under modular flow.

Abstract

Recent work showed holographic error correcting codes to have simple universal features at $O(1/G)$. In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order ($O(1)$ corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in $G$, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.