Lectures on Gravity and Entanglement

TL;DR

The notes address how gravity and spacetime emerge from quantum entanglement in AdS/CFT, centering on the Ryu–Takayanagi formula $S_A = \frac{1}{4 G_N}\text{Area}(\tilde A)$ and its covariant extension; they demonstrate how the entanglement first law $\frac{d}{d\lambda}S_A = \frac{d}{d\lambda}\langle H_A\rangle$ leads to the linearized Einstein equations for small AdS perturbations. They expand this to include bulk quantum corrections $S_A^{\rm CFT} = \frac{1}{4 G_N}\text{Area}(\tilde A) + S^{\rm bulk}_{\tilde A}$ and develop the entanglement-wedge framework for bulk reconstruction, underpinned by relative entropy, strong subadditivity, and energy conditions. The discussion also addresses which theories and states are holographic, highlighting the role of large $N$, strong coupling, and particular entanglement structures, and connects these ideas to quantum information concepts like thermofield double states and quantum error correction. Overall, the work provides a coherent program linking entanglement constraints to spacetime geometry and gravitational dynamics with potential implications for quantum gravity and holographic duality.

Abstract

The AdS/CFT correspondence provides quantum theories of gravity in which spacetime and gravitational physics emerge from ordinary non-gravitational quantum systems with many degrees of freedom. Recent work in this context has uncovered fascinating connections between quantum information theory and quantum gravity, suggesting that spacetime geometry is directly related to the entanglement structure of the underlying quantum mechanical degrees of freedom and that aspects of spacetime dynamics (gravitation) can be understood from basic quantum information theoretic constraints. In these notes, we provide an elementary introduction to these developments, suitable for readers with some background in general relativity and quantum field theory. The notes are based on lectures given at the CERN Spring School 2014, the Jerusalem Winter School 2014, the TASI Summer School 2015, and the Trieste Spring School 2015.

Figures (23)

• Figure 1: Basic AdS/CFT. States of a CFT on some fixed spacetime ${\cal B}$ correspond to states of a gravitational theory whose spacetimes are asymptotically locally AdS with boundary geometry ${\cal B}$.
• Figure 2: Different CFT states correspond to different asymptotically AdS geometries.
• Figure 3: Depictions of the maximally extended AdS-Schwarzschild black hole: a) Penrose (conformal) diagram for the spacetime, with exterior regions I and II and interior regions III and IV behind the horizon (dashed); b) spatial geometry of the $t=0$ slice (shown in red in a)), showing the horizon as the minimal area surface dividing the space into two parts each with one asymptotically AdS region.
• Figure 4: Gravity interpretations for the thermofield double state in a quantum system defined by a pair of noninteracting CFTs on $S^d$ times time. A particular quantum superposition of disconnected spacetimes gives a connected spacetime.
• Figure 5: Geometrical features relevant to the Ryu-Takayanagi proposal. In the diagram, the time direction has been suppressed. The left side shows a spatial slice $\Sigma_{\cal B}$ of the spacetime ${\cal B}$ on which the CFT lives. The right side shows a spatial slice of the spacetime $M_\Psi$ dual to the state $|\Psi \rangle$, containing $\Sigma_{\cal B}$ and the extremal surface $\tilde{A}$.