Holographic Renyi Entropy from Quantum Error Correction

TL;DR

The paper shows that refined Renyi entropies in holography can be derived from operator-algebra quantum error correction when the surface (χ_α) degrees of freedom are maximally mixed, leading to an improved area operator and a clear link between Ryu-Takayanagi terms and edge modes. It demonstrates that AdS/CFT’s cosmic brane prescription arises naturally within OQEC and that alpha-blocks correspond to area eigenstates, enabling a brane interpretation of Renyi spectra. It further proposes reinterpreting holographic tensor networks as area eigenstates and argues for constructing holographic networks with the correct Renyi spectrum via direct-sum (super) tensor networks. The work thus strengthens the connection between quantum error correction, edge modes, and gravitational entanglement structure, with implications for holography in general spacetimes and tensor-network models.

Abstract

We study Renyi entropies $S_n$ in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing $\widetilde{S}_n \equiv n^2 \partial_n (\frac{n-1}{n} S_n)$. We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.

Figures (1)

• Figure 1: Decomposing a lattice gauge theory into subregions $a$ and $\bar{a}$ requires the introduction of extra degrees of freedom (denoted as white dots) at the entangling surface (denoted by a dashed red line).