Quantum Information in Holographic Duality

TL;DR

This review analyzes how quantum-information ideas encode spacetime geometry in AdS/CFT, starting from black hole thermodynamics and the RT/HRT entanglement-entropy framework. It develops a multi-layered narrative: entanglement wedges and subregion duality, bulk reconstruction via quantum error correction, and tensor-network models that visualize holographic principles; it then extends to Rényi entropies, differential entropy, and modular Berry phases as probes of bulk structure. The work highlights key contributions, including JLMS, the entanglement wedge cross section, bit threads, and islands/replica wormholes, to address how boundary entanglement constrains bulk physics and information flow. The discussion culminates in holographic complexity and black-hole information dynamics, illustrating how entanglement, geometry, and computation intertwine in the gravity-quantum-information interface and offering a blueprint for future explorations beyond AdS/CFT.

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: Renyi 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.

Figures (27)

• Figure 1: Rindler wedges in (left) 1+1-dimensional flat space and (right) 2+1-dimensional AdS; compare with Figure \ref{['fig:cylinder']}. The colored hyperbolae are trajectories of accelerating observers. The Rindler horizons---the origin in 1+1 dimensions and a line in AdS$_3$---are marked in cyan.
• Figure 2: Two ways of slicing the same Euclidean path integral over the lower half-plane in equation (\ref{['rindlerdecomp']}).
• Figure 3: Global AdS space is presented in equation (\ref{['globaladshyperb']}) as a solid cylinder. The dual CFT lives on its asymptotic boundary, which is the hollow cylinder.
• Figure 4: An equal time slice of an asymptotically AdS geometry, assumed static and horizonless. A Cauchy slice of the boundary CFT---the circle---is divided into regions $A$ and $\bar{A}$. Proposal (\ref{['rtsimplest']}) considers surfaces $\Xi$ that asymptote to the border of $A$ and $\bar{A}$, and sets $S(A) = S(\bar{A})$ to the minimal area achieved by such a $\Xi$.
• Figure 5: Two examples of phase transitions in holographic entanglement entropy. Left: the entanglement entropy of a single interval $A$ in a CFT$_2$ thermal state can be realized holographically either as an individual geodesic homologous to $A$ (orange), or as the union of a geodesic homologous to $\bar{A}$ together with the black hole horizon (green). Right: the holographic entanglement entropy of two intervals in the CFT$_2$ vacuum can be in the connected (continuous lines) or disconnected (dashed lines) phase.