Quantum information in holographic duality

Abstract

We give a pedagogical review of how concepts from quantum information theory build up the gravitational side of the AdS/CFT correspondence. The review is self-contained in that it only presupposes knowledge of quantum mechanics and general relativity; other tools—including holographic duality itself—are introduced in the text. We have aimed to give researchers interested in entering this field a working knowledge sufficient for initiating original projects. The review begins with the laws of black hole thermodynamics, which form the basis of this subject, then introduces the Ryu-Takayanagi proposal, the JLMS relation, and subregion duality. We discuss tensor networks as a visualization tool and analyze various network architectures in detail. Next, several modern concepts and techniques are discussed: Rényi entropies and the replica trick, differential entropy and kinematic space, modular Berry phases, modular minimal entropy, entanglement wedge cross-sections, bit threads, and others. We discuss the extent to which bulk geometries are fixed by boundary entanglement entropies, and analyze the relations such as the monogamy of mutual information, which boundary entanglement entropies must obey if a state has a semiclassical bulk dual. We close with a discussion of black holes, including holographic complexity, firewalls and the black hole information paradox, islands, and replica wormholes.

Q-curvature and Path Integral Complexity

“…Moreover, the boundary conditions for all the fields are held fixed in this procedure so that the wavefunctions in these two geometries are proportional to each other. 4 For example, in 2dimensions, all metrics g ab can be brought to a conformally-flat diagonal form via a coordinate transformation. Therefore, without any loss of generality we can consider a 2-dimensional metric written in local coordinates (z, x) as…”

“…If we instead focus on the constraint R (4) = −12µ = const., with 0 < µ ≤ 1, we can look for a solution of the differential equation…”

“…While undoubtedly the main quantity in this story has been the notion of entanglement and its entropy [3], the concept of complexity has become increasingly prominent in the field (see e.g. review [4]). The importance of complexity in holography was motivated by the observation [5][6][7][8] that codimension-one boundary-anchored maximal volumes and co-dimension-zero boundary-anchored causal developments, which appear to be natural probes of the black hole interior in holography, share similar properties with linearly growing tensor networks describing states dual to these black-hole spacetimes [9].…”

“…Interestingly, if one takes the analogy between the two solutions seriously, the constant of integration κ should be related to the mass M of the Euclidean planar black hole. The confusing aspect of this analogy is that in the latter case the holographic CFT state should correspond to a TFD state, a fact which did not enter the construction of the R(4) =const solution explicitly 19. Penalty factors in the context of path integral optimization were previously considered in[60].…”

Holographic Complexity in dS$_{d+1}$

“…(1.2) to indicate that this counts the number of fundamental degrees of freedom in the dual theory describing de Sitter spacetime. 2 Quantum information theory has produced astonishing new insights into many core questions in the AdS/CFT correspondence, e.g., see reviews [36][37][38]. This progress has motivated some interesting recent discussions of de Sitter holography [1,[39][40][41][42].…”