Linearity of Holographic Entanglement Entropy

TL;DR

This work investigates whether the leading holographic entanglement entropy can be captured as the expectation value of a linear area operator, focusing on states dual to superpositions of distinct bulk geometries in AdS/CFT. By deploying the replica trick in 1+1 holographic CFTs and analyzing semi-classical subspaces, the authors show that entropies indeed average linearly over superposed branches as long as the number of branches is sub-exponential in the central charge, with the area operator acting approximately diagonally in the corresponding code subspace. A key finding is that nonlinearity arises from the homology constraint in the bulk, which selects different RT surfaces across states and prevents a single linear operator from reproducing RT entropies for all intervals, especially in highly mixed or wormhole-like states. The paper also develops general constructions of entropy operators in various large-$N$ settings (qubits, thermal states, free fields) and discusses broader implications for quantum error correction, tensor networks, and one-shot information theory, arguing that large-$N$ or thermodynamic limits underlie the linear entropy behavior and its limitations. Overall, the results support a linearized, operator-based interpretation of the RT entanglement term in a broad class of holographic theories while identifying fundamental nonlinearities tied to bulk topology and state counting.

Abstract

We consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.

Figures (8)

• Figure 1: The configuration of operators in the four-point function expression of the replicated density matrix. The blue line represents the subregion $R$ of the CFT. The twist operators are restricted to the unit circle representing a single spatial slice of the cylinder.
• Figure 2: Two different possible channels for computing the OPE between the two twist operators. The identity block contribution depends sensitively on the chosen channel. The identity block in the channel taken in the right diagram is more dominant than that of the left, and well approximates the four-point function. Dominance switches across $l = \pi$.
• Figure 3: The replicated density matrix is represented after uniformization as a 2$n$ point function of $\mathcal{O}$ and $\mathcal{O}^{\dagger}$ on the unit circle. The dark circles represent $\mathcal{O}$ insertions while the hollow circles represent $\mathcal{O}^\dagger$ insertions.
• Figure 4: The dominant identity block OPE channel for computing the replicated density matrix. The blue circles indicates how the OPE expansion is taken. The locations of $\mathcal{O}^{\dagger}$ (the hollow circles) move as $l$ is changed. We see that the operator pairing switches at $l = \pi$.
• Figure 5: Minimal area surfaces which compute the entanglement entropy of various intervals of the boundary CFT. Entropy of intervals smaller than $\pi$ are the same for both a pure and eternal block hole, and are given by green and blue curves. The two cases begin to differ once the interval is larger than $\pi$, as those are given by different bulk surfaces as shown by the red and magenta curves. We note that the difference for intervals that cover almost the entire boundary is exactly the black hole entropy.