FR09
Holography and quantum gravity
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.
L1 · 01
Why light reaches the AdS boundary in finite coordinate time
Use clocks and light cones to distinguish finite global travel time, infinite spatial distance and the boundary of a conformal map.
L1 · 02
Why half of a pure quantum state can look random
Calculate a reduced state and its entropy, then compare entanglement, classical mixtures and thermal states with identical local statistics.
L2 · 03
Reading boundary correlations from a bulk field
Derive a scalar operator dimension from its AdS radial falloff and distinguish a boundary source, quantum response and causal propagation.
L3 · 04
Calculate interval entanglement from a geodesic
Derive an AdS3 spatial geodesic, integrate its finite-cutoff length and remove the shared divergence to obtain interval entropy differences.
L3 · 05
How two intervals can share a bulk region
Compare homologous geodesic pairings, derive the leading mutual-information transition and distinguish a spatial region from causal and entanglement wedges.
L3 · 06
Recovering a superposition after two qubits are lost
Construct the five-qubit code, calculate erasure and recovery, and see why several boundary regions can represent one logical operator.
L4 · 07
How quantum corrections move an extremal surface
Follow bulk entanglement into generalized entropy and quantum extremal surfaces, then study a schematic Page transition with explicit assumptions.
L4 · 08
Read a holography paper through a small calculable model
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
- Pull an interval wider and compare competing geodesics
- Erase boundary sites and recover a coherent quantum state
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 · Quantum teleportation · Textbook plan · FR01 · FR30 · FR37 · FR41
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.
- João Penedones, TASI lectures on AdS/CFT, arXiv:1608.04948v1 (2016)
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.
- Shinsei Ryu and Tadashi Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, hep-th/0603001v2; PRL 96, 181602 (2006)
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.
- Matthew Headrick, Entanglement Rényi entropies in holographic theories, arXiv:1006.0047v3; PRD 82, 126010 (2010)
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.
- Fernando Pastawski, Beni Yoshida, Daniel Harlow and John Preskill, Holographic quantum error-correcting codes, arXiv:1503.06237v2; JHEP 06 (2015) 149
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.
- Ahmed Almheiri, Xi Dong and Daniel Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, arXiv:1411.7041v3; JHEP 04 (2015) 163
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.
- Xi Dong, Daniel Harlow and Aron C. Wall, Bulk Reconstruction in the Entanglement Wedge in AdS/CFT, arXiv:1601.05416v2; PRL 117, 021601 (2016)
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.
- Thomas Faulkner, Aitor Lewkowycz and Juan Maldacena, Quantum corrections to holographic entanglement entropy, arXiv:1307.2892v2; JHEP 11 (2013) 074
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.
- Netta Engelhardt and Aron C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, arXiv:1408.3203v3; JHEP 01 (2015) 073
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.
- Geoffrey Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255v2; JHEP 09 (2020) 002
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.