---
title: "Holography and quantum gravity"
description: "Learn AdS signals, interval entanglement and quantum error correction through eight chapters with worked examples and three interactive experiments."
canonical_url: "https://sandboxphysics.com/stories/holography/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/holography/"
last_modified: "2026-10-28"
---

[Holography and quantum gravity](https://sandboxphysics.com/stories/holography/)

FR09

# Holography and quantum gravity

Sandbox Physics · 2026-10-28

How can information on a boundary describe a world with depth? Follow a pulse through AdS, calculate the curve selected by an interval, then erase part of a quantum encoding and recover its phase. These are three concrete calculations on the path toward the holographic correspondence.

## Learn the pieces before connecting them

A boundary quantum state can encode a bulk geometry in theories with the right structure. Begin with three separate ingredients: the clocks and light cones of AdS, entanglement in a quantum state, and probabilities in thermal equilibrium. Then calculate a bulk geodesic from a boundary interval and recover an encoded state after losing some boundary sites.

Chapters 1 and 2 are independent foundations. Chapter 3 prepares the field dictionary, Chapters 4–5 develop interval geometry, and Chapter 6 opens the finite-code branch. Chapters 7–8 connect these calculations with quantum corrections and primary papers. Each chapter has a worked example, a prediction task and explained exercises.

An authored coordinate schematic. Both gold anchors lie on the same cutoff. The arc follows the spatial hyperbolic metric; it is not a light ray.

1. L1 · 01

   ## [Why light reaches the AdS boundary in finite coordinate time](https://sandboxphysics.com/stories/holography/clocks-and-curvature/)

   Use clocks and light cones to distinguish finite global travel time, infinite spatial distance and the boundary of a conformal map.

2. L1 · 02

   ## [Why half of a pure quantum state can look random](https://sandboxphysics.com/stories/holography/entanglement-and-heat/)

   Calculate a reduced state and its entropy, then compare entanglement, classical mixtures and thermal states with identical local statistics.

3. L2 · 03

   ## [Reading boundary correlations from a bulk field](https://sandboxphysics.com/stories/holography/fields-at-the-boundary/)

   Derive a scalar operator dimension from its AdS radial falloff and distinguish a boundary source, quantum response and causal propagation.

4. L3 · 04

   ## [Calculate interval entanglement from a geodesic](https://sandboxphysics.com/stories/holography/regulated-geodesics/)

   Derive an AdS3 spatial geodesic, integrate its finite-cutoff length and remove the shared divergence to obtain interval entropy differences.

5. L3 · 05

   ## [How two intervals can share a bulk region](https://sandboxphysics.com/stories/holography/intervals-and-wedges/)

   Compare homologous geodesic pairings, derive the leading mutual-information transition and distinguish a spatial region from causal and entanglement wedges.

6. L3 · 06

   ## [Recovering a superposition after two qubits are lost](https://sandboxphysics.com/stories/holography/encoding-and-recovery/)

   Construct the five-qubit code, calculate erasure and recovery, and see why several boundary regions can represent one logical operator.

7. L4 · 07

   ## [How quantum corrections move an extremal surface](https://sandboxphysics.com/stories/holography/quantum-extremal-surfaces/)

   Follow bulk entanglement into generalized entropy and quantum extremal surfaces, then study a schematic Page transition with explicit assumptions.

8. L4 · 08

   ## [Read a holography paper through a small calculable model](https://sandboxphysics.com/stories/holography/reading-the-models/)

   Recalculate published interval formulas and a five-qubit code, then identify what finite models do and do not supply for a continuous bulk theory.

## Three calculations you can change

- [Drag a source and follow null signals to the boundary](https://sandboxphysics.com/labs/ads-boundary-signals/)
- [Pull an interval wider and compare competing geodesics](https://sandboxphysics.com/labs/ads-interval-entanglement/)
- [Erase boundary sites and recover a coherent quantum state](https://sandboxphysics.com/labs/boundary-quantum-code/)

The first two use fixed pure AdS3. The third is one five-qubit perfect tensor with an exact recovery channel. Geometry, encoding and the theoretical correspondence have separate assumptions. Their calculated results are theoretical model data.

## Background and existing experiments

The black-hole light experiment introduces curved-spacetime ray tracing; quantum teleportation introduces the role of entanglement and classical communication. They retain their original models and ownership. For a longer study path, use the common textbook plan and the relevant field-theory, gravity, conformal and mathematical routes.

[Black-hole light](https://sandboxphysics.com/labs/black-hole-light/) · [Quantum teleportation](https://sandboxphysics.com/labs/quantum-teleportation/) · [Textbook plan](https://github.com/yangzichao/sandboxphysics/issues/27) · [FR01](https://github.com/yangzichao/sandboxphysics/issues/31) · [FR30](https://github.com/yangzichao/sandboxphysics/issues/60) · [FR37](https://github.com/yangzichao/sandboxphysics/issues/67) · [FR41](https://github.com/yangzichao/sandboxphysics/issues/71)

## Sources and recalculated scope

The course derives its worked examples and draws its own schematics. The numerical references recalculate specified analytic geometry and a finite five-qubit encoding. The source locations below distinguish those calculations from theoretical reading topics.

1. [João Penedones, TASI lectures on AdS/CFT, arXiv:1608.04948v1 (2016)](https://arxiv.org/html/1608.04948v1)

   Sections 3.1–3.3: global and Poincaré coordinates, fixed-background fields, source boundary conditions and dynamical gravity; section 4.1: conditions for a CFT with a weakly coupled bulk description. The course derives its radial optical paths and scalar near-boundary exponents from these metrics and field equations.

2. [Shinsei Ryu and Tadashi Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, hep-th/0603001v2; PRL 96, 181602 (2006)](https://arxiv.org/html/hep-th/0603001v2)

   Equation (5), static area prescription; equations (6)–(10) and Figure 1, vacuum AdS3 interval geometry and the circle result. We independently integrate the Poincaré geodesic and compare regulator limits. Our equal-height finite-cutoff anchors are a separately defined geometric diagnostic. Multiple-interval minima use Headrick, not equation (12) of this early paper.

3. [Matthew Headrick, Entanglement Rényi entropies in holographic theories, arXiv:1006.0047v3; PRD 82, 126010 (2010)](https://arxiv.org/html/1006.0047v3)

   Sections 3.1.3 and 4.2, especially Figure 1: the two homologous pairings for two vacuum intervals, their transition and leading mutual information. We recalculate the equal-length specialization as the gap is varied; no numerical replica saddles or source artwork are reproduced.

4. [Fernando Pastawski, Beni Yoshida, Daniel Harlow and John Preskill, Holographic quantum error-correcting codes, arXiv:1503.06237v2; JHEP 06 (2015) 149](https://arxiv.org/html/1503.06237v2)

   Section 2 defines perfect tensors; Appendix A.1, equations (30)–(32), supplies the five-qubit code. We construct one tensor from its stabilizers and recalculate known-erasure recovery. The browser does not build the many-pentagon network or reproduce its greedy thresholds.

5. [Ahmed Almheiri, Xi Dong and Daniel Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, arXiv:1411.7041v3; JHEP 04 (2015) 163](https://arxiv.org/html/1411.7041v3)

   Sections 3–4: the code-subspace interpretation of bulk reconstruction and quantum secret sharing. Used for the conceptual operator discussion, not claimed as a numerical reproduction of a CFT.

6. [Xi Dong, Daniel Harlow and Aron C. Wall, Bulk Reconstruction in the Entanglement Wedge in AdS/CFT, arXiv:1601.05416v2; PRL 117, 021601 (2016)](https://arxiv.org/html/1601.05416v2)

   Sections I–II and Figure 1 define causal and entanglement wedges and a perturbative code subspace. Sections III–IV state the relative-entropy assumptions behind reconstruction. We explain these assumptions; the finite code checks are a separate exact model.

7. [Thomas Faulkner, Aitor Lewkowycz and Juan Maldacena, Quantum corrections to holographic entanglement entropy, arXiv:1307.2892v2; JHEP 11 (2013) 074](https://arxiv.org/html/1307.2892v2)

   Equations (1.1)–(1.3), section 2.2 and equation (2.15): bulk entanglement, area backreaction, Wald-like and counterterm contributions at one loop. This course explains the order of the expansion and keeps the renormalization terms; it does not evaluate a gravitational path integral.

8. [Netta Engelhardt and Aron C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, arXiv:1408.3203v3; JHEP 01 (2015) 073](https://arxiv.org/html/1408.3203v3)

   Equations (1.1), (3.1)–(3.2), section 3: generalized entropy, normal-direction extremization, the least-entropy prescription with its near-degeneracy qualification, and the relation to FLM. The quadratic displacement example is authored here and is not a solution of semiclassical Einstein equations.

9. [Geoffrey Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255v2; JHEP 09 (2020) 002](https://arxiv.org/html/1905.08255v2)

   Sections 2.2–2.3 and 3: evaporating AdS black holes with absorbing boundary conditions, the Page transition and code-subspace/state-dependence conditions. Our two-candidate entropy plot is an explicitly schematic comparison, with no Hawking spectrum or evaporating spacetime simulation.