Holography through Quantum Information and Operator Algebras

Written by AI assistants under the scientific and pedagogical supervision of Marcelo S. Guimarães; see ai-authorship for the division of labour and the models involved.

How can the observables of a quantum theory encode a spacetime? We begin with encodings, restricted access, and recoverable observables in small quantum systems. Their calculations motivate operator algebras and modular structure. We then introduce continuum QFT, the geometric dictionary, and selected gravitational models.

The course distinguishes exact finite constructions, stated continuum theorems, and semiclassical gravitational results. Students repeatedly identify the additional input needed in passing from one level to the next.

Audience and prerequisites

The opening block assumes tensor products, density matrices, Hermitian operators, and basic probability. Schmidt decomposition, partial trace, quantum channels, and entropy are reviewed where used. No prior operator-algebra or error-correction course is required.

Later sections require free QFT and basic general relativity. The needed CFT, AdS, and modular constructions are developed within the course. String theory and completion of the parallel AQFT course are not prerequisites.

Learning outcomes

Students should be able to:

1. Specify accessible observables and distinguish an algebra, its commutant, and its center.
2. Build and test an erasure decoder, including reference entanglement.
3. Calculate finite modular data and identify when modular flow is geometric.
4. Derive entropy and relative-entropy identities in algebraic codes.
5. Distinguish regulator removal, split inclusions, and a state-adapted large-$N$ limit.
6. Complete a scalar source–response calculation and an AdS interval-entropy calculation.
7. Explain wedge reconstruction and JLMS within a specified code and approximation.
8. Compare finite Page models with a calculated gravitational entropy saddle.
9. Use a half-sided modular inclusion to understand an affine translation, without assuming it already determines a full spacetime.

Semester I — Accessible information to the geometric dictionary

Four lecture hours per week, plus a separate problem session where available. Lectures 1–4 are two-hour meetings across Weeks 1–2; each later note covers a four-hour weekly unit.

Week	Lecture	Main topic	Lecture hours
1	1	Observables and restricted access	2
1	2	A quantum system hidden in three qutrits	2
2	3	Modular structure of an entangled pair	2
2	4	An entropy formula in a quantum code	2
3	5	Constraints, regional algebras, and electric sectors	4
4	6	Tensor networks as explicit encodings	4
5	7	When an encoding stops protecting information	4
6	8	A field approached through coupled oscillators	4
7	9	Continuum algebras and separated regions	4
8	10	Rindler observers and modular time	4
9	11	A conformal ball and its modular clock	4
10	12	Conformal field theory as organized quantum mechanics	4
11	13	AdS geometry and the radial organization of fields	4
12	14	Large N and the conditions for a semiclassical bulk	4
13	15	A complete scalar source and response calculation	4
14	16	When entropy is a geodesic length	4
15	17	From a recoverable observable to a geometric region	4

Semester II — Reconstruction, black holes, and algebraic spacetime

Three lecture hours per week, plus a separate problem session where available.

Week	Lecture	Main topic	Lecture hours
1	18	Causal access and bulk reconstruction	3
2	19	Entanglement wedges and changing reconstruction regions	3
3	20	Distinguishability and recovery channels	3
4	21	Relative entropy across the holographic dictionary	3
5	22	What an entanglement first law implies for gravity	3
6	23	Thermal states, horizons, and two-sided geometry	3
7	24	Page curves in finite quantum mechanics	3
8	25	A specified JT gravity and bath model	3
9	26	An island saddle calculated explicitly	3
10	27	Replicas, competing saddles, and limits of the model	3
11	28	What algebraic holography establishes on fixed AdS	3
12	29	Large-N algebras and state-dependent representations	3
13	30	Nested algebras and half-sided modular inclusions	3
14	31	An algebraic model of emergent time	3
15	32	The full chain from information to geometry	3

How to use the notes

All listed notes are written drafts. Lectures 8–32 are master dossiers at the AQFT standard, fuller than a single classroom run, and the opening dossier (Lectures 1–7) follows the same structure with its own reading map. Their proposed teaching routes remain provisional.

The instructor guide identifies the classroom core, self-study work, and research extensions for every unit. A theorem may be stated with hypotheses when its full proof belongs in an AQFT or mathematical methods course. An exercise does not replace a calculation claimed to be derived in the text.

The recurrent questions are: what is the system, what is the state, which algebra is accessible, what is calculated, and what additional assumption supports a geometric interpretation? The conventions and source map support that progression.

Scope choices

The final algebraic route is modular inclusions and emergent time. Crossed-product entropy and traversability remain further-reading options. Long conformal-block calculations, a catalogue of Witten diagrams, and Wilson-loop technology are optional extensions.

The JT/bath sequence contains a static analytic island endpoint and a two-sided thermal entropy plateau. It does not claim to solve a complete evaporating geometry. The finite Page models precede that calculation so students can separate unitarity, typicality, and gravitational saddle assumptions.

Problems and projects

Each note includes problems with answers or answer checkpoints. Core exercises use the note’s prerequisites; self-study exercises consolidate the derivation; extensions declare their additional demands. The project guide offers early finite-model projects, and Lecture 32 adds field, scalar-dictionary, island, and modular projects.

The course is a completed written draft, not a classroom-tested release. See the validation record for what has been checked.

Relation to the AQFT course

The canonical AQFT course is the pedagogical reference. It develops the general foundations in their own right. Here the same tools enter to answer questions about information, reconstruction, entropy, and geometry. Its files remain unchanged.

Earlier week-numbered AdS/CFT notes are historical source material. The sequence above defines the active course.

Wiki connections. gravity duality (research area) · holographic dictionary · GKP–Witten formula · large-N factorization · Ryu–Takayanagi formula · subregion–subalgebra duality · quantum extremal surfaces · Page curve · JT gravity · replica trick in gravity · Tomita–Takesaki modular theory · Bisognano–Wichmann theorem · type III₁ von Neumann algebras · crossed product · thermofield double · type III₁ algebras across the research areas (connection) · the crossed product and the island formula (connection)