Holographic Entanglement Entropy

TL;DR

The paper surveys a decade of progress connecting quantum entanglement in field theories to holographic gravity via the RT/HRT prescriptions, the replica trick, and the geometry of extremal surfaces. It systematically builds from fundamental entanglement definitions in QFT to their holographic realization, then develops the bulk–boundary dictionary through derivations and generalizations, including higher-derivative gravities and bulk quantum corrections. A central theme is that spacetime geometry and gravitational dynamics emerge from entanglement structure, encapsulated by concepts such as the entanglement wedge, quantum error correction, and relative entropy constraints. The work also discusses entropy inequalities, phase transitions in entanglement, and the broader program of reconstructing bulk physics from boundary entanglement data, highlighting both established results and open questions in covariant settings and for non-Einstein gravities.

Abstract

We review the developments in the past decade on holographic entanglement entropy, a subject that has garnered much attention owing to its potential to teach us about the emergence of spacetime in holography. We provide an introduction to the concept of entanglement entropy in quantum field theories, review the holographic proposals for computing the same, providing some justification for where these proposals arise from in the first two parts. The final part addresses recent developments linking entanglement and geometry. We provide an overview of the various arguments and technical developments that teach us how to use field theory entanglement to detect geometry. Our discussion is by design eclectic; we have chosen to focus on developments that appear to us most promising for further insights into the holographic map. This is a draft of a few chapters of a book which will appear sometime in the near future, to be published by Springer. The book in addition contains a discussion of application of holographic ideas to computation of entanglement entropy in strongly coupled field theories, and discussion of tensor networks and holography, which we have chosen to exclude from the current manuscript.

Figures (34)

• Figure 1: A discrete latticized quantum system with a Hilbert space ${\cal H}_\alpha$ at every site. We have indicated the region ${\cal A}$ by shading the enclosed sites while the unshaded area indicates ${\cal A}^c$. We take the lattice spacing to be $\epsilon$.
• Figure 2: A continuum QFT which has been spatially bipartitioned into two components on a Cauchy slice $\Sigma$. We have indicated the region ${\cal A}$ and its complement ${\cal A}^c = \Sigma\backslash {\cal A}$. The separatrix is a spacetime codimension-2 surface, called the entangling surface.
• Figure 3: An illustration of the causal domains associated with a region ${\cal A}$, making manifest the decomposition of the spacetime into the four distinct domains indicated in \ref{['eq:bdy4d']}. Two deformations ${\cal A}'$ are also included for illustration in the right panel.
• Figure 4: The Euclidean geometry for computing the matrix elements of the reduced density matrix ${\rho_{{\cal A}}}$. We have sketched the situation in two-dimensional Euclidean space as indicated. The two cuts at ${\cal A}$ have been separated in an exaggerated manner to indicate the boundary conditions we need to impose, cf., \ref{['eq:rhoAE']}.
• Figure 5: The Schwinger-Keldysh geometry for computing the matrix elements of the reduced density matrix ${\rho_{{\cal A}}}$ in time-dependent settings. On the left panel, we show the general contour which involves a time-fold at the Cauchy surface of interest. On the right panel, we illustrate the opening out of the path integral at ${\cal A}$ to allow for the appropriate past/future boundary conditions \ref{['eq:rhoAL']}.