Deriving covariant holographic entanglement

TL;DR

The paper provides a fully covariant, Lorentzian derivation of the holographic entanglement entropy prescription by embedding the field-theory Schwinger-Keldysh replica construction into the bulk. It shows that, under replica symmetry and appropriate regularity, the dominant bulk saddles in the q→1 limit are governed by codimension-2 extremal surfaces, yielding the HRT area law S_A = Area(E_A)/(4 G_N). The authors thoroughly connect the boundary tracing procedure to bulk geometry via the entanglement wedge and homology constraints, and discuss extensions to higher-derivative gravity, quantum corrections, and potential geometric subtleties such as multiple or complex saddles. This framework clarifies how time-dependent entanglement is encoded holographically and provides a principled route to compute Rényi entropies and modular entropies in dynamical spacetimes.

Abstract

We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Renyi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.

Figures (9)

• Figure 1: A schematic representation of the Schwinger-Keldysh contours necessary for the computation of the (a) density matrix and (b) its powers. We have explicitly shown the computation of $\rho^3$ in (b). The dots and lines in red correspond to an entangled initial state prepared in some manner. This picture does not carry the spatial information necessary to ascertain the reduced density matrices themselves, which is better understood from Figs. \ref{['fig:sk1']}. Note that in contrast to the usual depiction of the Schwinger-Keldysh contour we will draw time running vertically.
• Figure 2: The bulk domains of interest in the Lorentzian construction. (a) We show the Wheeler-DeWitt patch associated with a give Cauchy surface on the boundary. (b) Given a separation of the boundary Cauchy surface into regions ${\cal A}$ and ${\cal A}^c$ respectively, any bulk Cauchy surface $\tilde{\Sigma}_t$ in the Wheeler-DeWitt patch admits a decomposition $\tilde{\Sigma}_t = {\cal R}_{{\cal A}} \cup {\cal R}_{{\cal A}}^c$. We also display the bulk codimension-2 fixed point locus $\mathbf{e}$ anchored on the entangling surface which approaches the extremal surface ${\cal E}_{\cal A}$ as $q\to 1$.
• Figure 3: Schwinger-Keldysh construction for $\text{Tr} \rho(t)$ and ${\rho_{{\cal A}}}$. The forward evolution for $\mid\!\! \Psi\rangle$ proceeds up to $\Sigma_{_t}$, while the backwards evolution for $\langle \, \Psi \! \mid\! \ $ starts there. Gluing the two evolutions together at $\Sigma_{_t}$ enables taking the trace of $\rho(t)$. We also depict the situation, where we open out this trace along ${\cal A} \subset \Sigma_{_t}$ to construct ${\rho_{{\cal A}}}(t)$. These cuts are introduced at $t= t^\pm$ as described in the main text.
• Figure 4: Local Rindler coordinates that we will use to describe the geometric construction of $\rho_{{\cal A}}$. We have focused on the neighbourhood of the entangling surface and indicated the causal domains and coordinates used therein (see text for details).
• Figure 5: A schematic view of the replica Schwinger-Keldysh contour for computing $({\rho_{{\cal A}}})^3$ matrix elements. We have restricted attention to the neighourhood of the entangling surface for simplicity.