Holographic Entanglement Entropy

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 preliminary draft of a few chapters of a book which will appear sometime in the near future, to be published by Springer, as part of their Lecture Notes in Physics series. 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. i AcknowledgmentsWe have been extremely fortunate to benefit from the wisdom and deep physical intuition of our wonderful collaborators Veronika Hubeny and Shinsei Ryu who played a pivotal role in helping us develop the basic picture relating quantum entanglement and holography. The importance of their role in shaping the story of holography entanglement entropy cannot be overstated.We have also enjoyed many excellent collaborations in our explorations over the past decade on this subject: thanks to

A contour for the entanglement entropies in harmonic lattices

“…where the function s n (x) is given by (13) for the entanglement entropy and by (14) for the Rényi entropies. We call mode participation function the function p k (i) in (16), as done in [4].…”

“…where the analytic functions s n (x) are given by (13) and (14). In the last step we have employed (17), the cyclic property of the trace and the fact that the functions s n (x) are analytic.…”

“…As for the latter class of models, important goals have been achieved in conformal field theories (CFTs) [8,9,10], free quantum field theories [11] and holography [12,13].…”

Holographic complexity in Vaidya spacetimes. Part I

“…(3.7) can be recast in terms of z as 12) JHEP06(2018)046 Figure 2. Penrose-like diagrams for the thin shell collapsing geometries, we represent spherical horizon collapse from global AdS (left) and planar horizon from Poincaré patch (right).…”

“…The novelty of the gauge/gravity duality [7,8] is that the holographic dictionary between the bulk and boundary theories provides a framework where new tools and techniques from quantum information science can be precisely tested in quantum gravity. Much of this discussion has focused on the idea of holographic entanglement entropy [9][10][11][12], which has led to a remarkably rich and varied range of new insights, e.g., [13][14][15][16][17][18][19][20][21][22][23].…”