Flat entanglement spectra in fixed-area states of quantum gravity

TL;DR

This work shows that semiclassical gravitational states prepared by Euclidean path integrals, when projected to definite RT/HRT area, exhibit a flat entanglement spectrum at leading order in $G$, with all Renyi entropies $S_n$ equal. Using both replica-trick saddles with conical defects and the cosmic-brane prescription, the authors derive $S_n=rac{ ext{A}}{4G}$ (and $ ilde{S}_n=rac{ ext{A}}{4G}$) for fixed-area states, and generalize to fixed induced metric data on the HRT surface. They interpret the result through a quantum-error-correction lens, connecting center algebras in the entanglement wedge to a fixed-$oldsymbol{eta}$ spectrum and showing how Renyi entropies decompose in an operator-algebra code with complementary recovery. They also discuss an exponentiated JLMS relation, noting that the flat spectrum at fixed $eta$ supports a strengthened bulk-boundary modular flow correspondence at leading order. Overall, the paper links bulk geometric projections to boundary entanglement structure, clarifying center-related aspects of holography and informing tensor-network and code-based descriptions of quantum gravity.

Abstract

We use the Einstein-Hilbert gravitational path integral to investigate gravitational entanglement at leading order $O(1/G)$. We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies $S_n$ are independent of $n$ at this order. The $n$-dependence of $S_n$ in more general states then arises from sums over the RT/HRT-area, which are generally dominated by different values of this area for each $n$. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of algebraic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

Figures (3)

• Figure 1: The encoding circuit for a tensor network holographic code (figure borrowed from Harlow:2018fse). $r$/$\overline{r}$ denotes bulk degrees of freedom at the red dots to the left/right of the green HRT surface $\gamma_R$, $|\chi\rangle$ is a tensor product of a set of EPR pairs on each link cut by $\gamma_R$, and $U_R$/$U_{\overline{R}}$ are unitary transformations generated by the pieces of the network to the left/right of $\gamma_R$.
• Figure 2: Using the bulk Euclidean path integral to prepare a state. The red dots are boundary sources we can adjust to change which state we prepare, and the center degrees of freedom $\alpha$ are the induced metric data on the HRT surface $\gamma_R$.
• Figure 3: Cutting open the $n=1$ gravitational path integral (left) and gluing together 3 copies to form a path integral for computing the third boundary Renyi entropy (right). Red dots are the sources which produce the "ket" part of the density matrix, while blue dots are their CPT conjugates which produce the "bra" part. The HRT surface $\gamma_R$ is represented by the black dot in the center, and is shared between all copies in the Renyi computation.