Information Theory & Physics

Status: Active — mature classical/quantum core; speculative gravitational frontier Last updated: 2026-06-08

This page is the authoritative deep-dive on the mathematical structures linking information and physics, spanning four strata of decreasing certainty: classical (Shannon/Kolmogorov) information, quantum information, the thermodynamics of computation, and the information-theoretic structure of gravity and spacetime. The organizing discipline throughout is to mark exactly where each layer is a theorem, where it is well-motivated inference, and where it is principled speculation. See EPISTEMICS.md for the marker definitions and THEORY_MAP.md for this domain's place in the larger landscape.

Scope

The domain comprises four nested layers:

1. Classical information theory — Shannon entropy, mutual information, channel capacity, the data-processing inequality, and Kolmogorov (algorithmic) complexity. This layer is ESTABLISHED mathematics with no physical domain restriction.
2. Quantum information — von Neumann entropy, qubits, no-cloning, teleportation, entanglement measures, and entanglement entropy in QFT. ESTABLISHED theory; the operational primitives are experimentally confirmed.
3. Thermodynamics of computation — Landauer's principle, reversible computation, the Maxwell-demon resolution, and physical limits on computation (Margolus–Levitin, Bremermann, Bekenstein). A mix of ESTABLISHED bounds and CONTESTED interpretive bookkeeping.
4. Information and gravity — the Bekenstein and Bousso bounds, holography, the Ryu–Takayanagi formula, AdS/CFT as quantum error correction, and the "it-from-bit" / "spacetime-from-entanglement" programs. This layer ranges from INFERENCE (RT within AdS/CFT) to SPECULATIVE/OPEN (entanglement as the fundamental substrate of spacetime).

Core formalism

1. Classical (Shannon) information theory — ESTABLISHED

A discrete random variable $X \sim p(x)$ over alphabet $\mathcal{X}$ has Shannon entropy $H(X) = -\sum_{x \in \mathcal{X}} p(x) \log p(x),$ in bits if $\log = \log_2$, nats if natural. $H$ is the unique function (up to scale) satisfying continuity, monotonicity in $n$ for the uniform distribution, and the grouping/recursion axiom (Shannon 1948; Khinchin's axiomatization) ESTABLISHED. Operationally, $H(X)$ is the asymptotic optimal lossless compression rate: by the Source Coding Theorem, $N$ i.i.d. copies compress to $\approx N H(X)$ bits with vanishing error and no lower rate is achievable. The engine is the Asymptotic Equipartition Property (AEP): $-\tfrac{1}{N}\log p(X_1,\dots,X_N) \to H(X)$ in probability, concentrating measure on $\approx 2^{NH}$ "typical" sequences each of probability $\approx 2^{-NH}$.

Joint, conditional, relative entropy and mutual information: $H(X,Y) = -\sum p(x,y)\log p(x,y), \qquad H(X \mid Y) = H(X,Y) - H(Y),$ $D(p \Vert q) = \sum_x p(x)\log\frac{p(x)}{q(x)} \;\ge\; 0 \quad (\text{Gibbs' inequality}),$ $I(X;Y) = H(X) - H(X \mid Y) = H(X) + H(Y) - H(X,Y) = D\big(p(x,y) \Vert p(x)p(y)\big) \ge 0.$ Mutual information is symmetric, vanishes iff $X \perp Y$, and equals the relative entropy from the joint to the product distribution — hence its nonnegativity.

A discrete memoryless channel $p(y \mid x)$ has capacity $C = \max_{p(x)} I(X;Y)$. The Noisy Channel Coding Theorem makes $C$ a sharp threshold: reliable communication is possible for any rate $R < C$ and impossible for $R > C$ ESTABLISHED. For the additive white Gaussian noise channel of bandwidth $B$, signal power $P$, noise power $N$, the Shannon–Hartley capacity is $C = B \log_2(1 + P/N)$.

The data-processing inequality (DPI): if $X \to Y \to Z$ is a Markov chain, then $I(X;Z) \le I(X;Y)$ ESTABLISHED. Post-processing cannot increase information about the source. Its quantum generalization (monotonicity of relative entropy under CPTP maps) is the structural backbone of the entropy bounds in gravity discussed below.

2. Kolmogorov (algorithmic) complexity — ESTABLISHED math, OPEN computability caveats

For a universal prefix Turing machine $U$, the Kolmogorov complexity of a finite string $x$ is $K(x) = \min\{ |p| : U(p) = x \},$ the length of the shortest program outputting $x$. The invariance theorem gives $|K_U(x) - K_V(x)| \le c_{U,V}$ for any two universal machines, so $K$ is well-defined up to an additive constant ESTABLISHED. $K$ is uncomputable — no algorithm computes it for all $x$ (equivalent to undecidability of halting; the root of Chaitin's incompleteness results) ESTABLISHED. For a computable source, $\mathbb{E}[K(X)] = H(X) + O(1)$ ESTABLISHED. The distinction is conceptual: $H$ measures average uncertainty of an ensemble; $K$ measures the intrinsic descriptive content of a single object. For a single string there is no canonical "amount of information" — see Where it breaks down.

3. Quantum information — ESTABLISHED

A quantum system is a Hilbert space $\mathcal{H}$; a state is a density operator $\rho \succeq 0$, $\mathrm{Tr}\,\rho = 1$. A qubit is $\mathcal{H} = \mathbb{C}^2$, with pure states $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ parameterized on the Bloch sphere. The von Neumann entropy is $S(\rho) = -\mathrm{Tr}(\rho \log \rho) = -\sum_i \lambda_i \log \lambda_i,$ $\lambda_i$ the eigenvalues of $\rho$. $S = 0$ iff $\rho$ is pure; $S = \log d$ for the maximally mixed state. Key inequalities ESTABLISHED:

• Subadditivity: $S(\rho_{AB}) \le S(\rho_A) + S(\rho_B)$.
• Araki–Lieb: $S(\rho_{AB}) \ge |S(\rho_A) - S(\rho_B)|$.
• Strong subadditivity (SSA) (Lieb–Ruskai 1973): $S(\rho_{ABC}) + S(\rho_B) \le S(\rho_{AB}) + S(\rho_{BC})$, equivalently the conditional mutual information $I(A;C \mid B) \ge 0$. SSA is the deep theorem of the subject; nearly every entropy inequality in quantum gravity descends from it.

Unlike the classical case, $S(\rho_A)$ can exceed $S(\rho_{AB})$ when $AB$ is pure and entangled — the diagnostic of entanglement. The quantum mutual information is $I(A;B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}) \ge 0$, and the quantum relative entropy $S(\rho \Vert \sigma) = \mathrm{Tr}\,\rho(\log\rho - \log\sigma)$ is monotone under any CPTP map $\mathcal{N}$: $S(\mathcal{N}\rho \Vert \mathcal{N}\sigma) \le S(\rho \Vert \sigma)$ — the quantum DPI, equivalent in strength to SSA.

The no-cloning theorem (Wootters–Zurek, Dieks 1982): no unitary $U$ satisfies $U(|\psi\rangle|0\rangle) = |\psi\rangle|\psi\rangle$ for all $|\psi\rangle$ ESTABLISHED. Linearity plus inner-product preservation would force $\langle\psi|\phi\rangle = \langle\psi|\phi\rangle^2$, impossible for non-orthogonal, non-identical states. No-cloning underwrites quantum cryptography and the sharpness of the black-hole information puzzle.

Teleportation (Bennett et al. 1993): one shared Bell pair plus two classical bits transmits one unknown qubit exactly ESTABLISHED. The qubit is destroyed at the sender (consistent with no-cloning) and no superluminal signalling occurs (classical bits are required). Superdense coding is its dual: one qubit plus one Bell pair sends two classical bits.

For a pure bipartite state, the canonical entanglement measure is the entanglement entropy $E(|\psi\rangle_{AB}) = S(\rho_A) = S(\rho_B)$ (equal by the Schmidt decomposition $|\psi\rangle = \sum_i \sqrt{\lambda_i}\,|i\rangle_A |i\rangle_B$). For mixed states, inequivalent measures appear — entanglement of formation, distillable entanglement $E_D$, entanglement cost $E_C$, and the relative entropy of entanglement $E_R(\rho) = \min_{\sigma \in \mathrm{SEP}} S(\rho \Vert \sigma)$ — all reducing to $S(\rho_A)$ on pure states and obeying $E_D \le E_C$ with strict gaps (bound entanglement) ESTABLISHED. Asymptotically reversible LOCC manipulation holds for pure states ($E_D = E_C = S(\rho_A)$) but generally fails for mixed states.

Entanglement entropy in QFT. For a spatial region $A$, $S(\rho_A)$ of the reduced state is UV-divergent and obeys an area law: the leading divergence in $d$ spatial dimensions is $S \sim c\,\mathrm{Area}(\partial A)/\epsilon^{d-1}$ with cutoff $\epsilon$ ESTABLISHED. In a $1+1$ CFT, for an interval of length $\ell$, $S = \tfrac{c}{3}\log(\ell/\epsilon)$ with $c$ the central charge (Holzhey–Larsen–Wilczek; Calabrese–Cardy) ESTABLISHED. The area-law scaling is the bridge to the holographic entropy bounds. See domains/quantum-field-theory.md.

4. Thermodynamics of computation — ESTABLISHED core, CONTESTED interpretation

Landauer's principle (1961): erasing one bit in contact with a bath at temperature $T$ dissipates at least $\Delta Q \ge k_B T \ln 2$ of heat, tying logical irreversibility (a many-to-one map on logical states) to thermodynamic irreversibility. The bound is derivable from phase-space contraction and from the Jarzynski/Crooks fluctuation theorems, and has been experimentally confirmed near saturation (colloidal-particle and single-spin experiments, c. 2012 onward) ESTABLISHED. Its logical status as an independent principle is CONTESTED (see breakdowns).

Reversible computation (Bennett 1973): any computation embeds in a logically reversible one (e.g. via the universal Toffoli gate) that, run quasi-statically, costs arbitrarily little energy per step — only erasure costs ESTABLISHED. The Maxwell-demon / Szilard-engine resolution (Szilard 1929; Landauer; Bennett 1982): the demon's apparent extraction of $k_B T \ln 2$ per bit of which-side information is repaid when its memory is reset, restoring the second law. The modern Sagawa–Ueda generalized second law reads $\langle W_{\rm ext}\rangle \le k_B T\,\Delta I$ — information is a thermodynamic resource convertible to work, demonstrated experimentally ESTABLISHED. See domains/thermodynamics.md and domains/statistical-mechanics.md.

Physical limits on computation:

• Margolus–Levitin theorem (1998) ESTABLISHED: a system with mean energy $E$ above its ground state needs time $t_\perp \ge \tfrac{\pi\hbar}{2E}$ to reach an orthogonal state — a maximum rate of distinct operations $\le 2E/(\pi\hbar)$. Combined with the Mandelstam–Tamm bound $t_\perp \ge \tfrac{\pi\hbar}{2\Delta E}$, the operative speed limit is the $\min$ of the two.
• Bremermann's limit (1962) INFERENCE/heuristic: $\sim c^2/h \approx 1.36 \times 10^{50}$ bit s$^{-1}$ kg$^{-1}$, a back-of-envelope ceiling from $E = mc^2$ plus an energy–time argument, not a theorem.
• Bekenstein bound (1981) INFERENCE: $S \le \dfrac{2\pi k_B R E}{\hbar c}$ for a system of radius $R$ and energy $E$.
• Holographic ('t Hooft–Susskind) bound INFERENCE: maximal entropy is the boundary area in Planck units, $S_{\max} = \tfrac{k_B A}{4\ell_P^2}$, matching the Bekenstein–Hawking entropy $S_{\rm BH} = \tfrac{k_B c^3 A}{4 G \hbar} = \tfrac{k_B A}{4\ell_P^2}$. Lloyd's "ultimate laptop" combines Margolus–Levitin with Bekenstein into operations- and memory-bounds.

5. Holography, RT, and emergent spacetime

Bousso covariant entropy bound (1999) INFERENCE: entropy flux through a light-sheet (a null hypersurface of non-positive expansion) from a surface of area $A$ obeys $S \le A/4G\hbar$ (Planck units) — the covariant refinement of the holographic bound, reducing to Bekenstein in the appropriate limit.

Ryu–Takayanagi (RT) formula (2006) INFERENCE, within AdS/CFT: in a holographic CFT with AdS dual, the entanglement entropy of a boundary region $A$ is $S(A) = \frac{\mathrm{Area}(\gamma_A)}{4 G_N},$ $\gamma_A$ the minimal bulk surface homologous to $A$. The covariant generalization is HRT (extremal surface), and the all-orders form is the quantum extremal surface (QES) prescription (Engelhardt–Wall): extremize $\mathrm{Area}/4G_N + S_{\rm bulk}$. RT geometrizes entropy — boundary SSA becomes a statement about minimal surfaces, and bulk geometry is reconstructible from boundary entanglement (entanglement-wedge reconstruction).

Quantum error correction (QEC) and bulk emergence INFERENCE: Almheiri–Dong–Harlow (2015) reframed AdS/CFT as a quantum error-correcting code — bulk interior operators are redundantly (nonlocally) encoded in the boundary, like logical qubits protected against boundary erasure. The HaPPY code (Pastawski–Yoshida–Harlow–Preskill) and random tensor networks realize RT-type entropies exactly in toy models. Subregion duality is sharpened by the JLMS relation (bulk/boundary relative entropies coincide) and by modular-flow / Petz-map reconstruction (Faulkner–Lewkowycz). See domains/general-relativity.md.

Entanglement as the fabric of spacetime SPECULATIVE: Van Raamsdonk argued that reducing the mutual entanglement of two CFTs (the thermofield double) disconnects the dual spacetime, suggesting spatial connectivity is built from entanglement; the ER=EPR conjecture (Maldacena–Susskind) identifies entanglement with an Einstein–Rosen bridge. The linearized Einstein equations have been derived from the first law of entanglement, $\delta S_A = \delta\langle K_A\rangle$ (modular Hamiltonian $K_A$), in holographic setups (Faulkner–Guica–Hartman–Myers–Van Raamsdonk 2014; Jacobson's 1995 "Einstein equation of state" is the precursor) SPECULATIVE as a route to full GR — only the linearized equations, under specific assumptions. These are striking but framework-dependent results, not a complete theory.

Foundational assumptions

See ASSUMPTIONS_LEDGER.md for the cross-domain ledger and UNIFYING_PRINCIPLES.md.

Domain of validity

Layer 1 (Shannon/Kolmogorov) is mathematically exact with no physical domain restriction — it applies to any probabilistic source or string. Two caveats: (a) the operational theorems are asymptotic (i.i.d., large $N$), so finite-blocklength corrections (Polyanskiy–Poor–Verdú) matter for real codes; (b) $K(x)$ is uncomputable and thus a theoretical, not algorithmic, quantity.

Layer 2 (quantum information) is valid wherever non-relativistic or relativistic quantum theory holds; the operational primitives (no-cloning, teleportation, superdense coding, distillation) are experimentally confirmed to high precision. In continuum QFT, $S(\rho_A)$ is UV-divergent and requires a regulator, so only universal/finite pieces (mutual information, differences, central-charge coefficients) are cutoff-independent and physically meaningful.

Layer 3 (thermodynamics of computation): Landauer's bound holds in the quasi-static, weak-coupling, thermal-bath regime; it is a lower bound saturated only in the reversible limit at fixed $T$. Margolus–Levitin and Mandelstam–Tamm are exact theorems of unitary QM. The Bekenstein bound holds for weakly gravitating systems ($R$ larger than the Schwarzschild radius) and is saturated by black holes; where "radius" and "energy" become ambiguous (strong gravity, cosmology) the Bousso covariant bound is the correct statement.

Layer 4 (RT/holography/emergent spacetime) is rigorously formulated and tested essentially only within AdS/CFT: asymptotically anti-de Sitter spacetimes, large central charge / large $N$, strong 't Hooft coupling (Einstein-gravity regime), with a static (or HRT for time-dependent), homology-respecting setting; quantum corrections need QES/islands. Validity for de Sitter space (our accelerating universe), cosmological horizons, and non-holographic theories is unestablished — this is the central domain-of-validity frontier. See domains/cosmology.md.

Where it breaks down

• Continuum QFT entanglement entropy. $S(\rho_A)$ is UV-divergent and the leading area-law term is regulator-dependent; only differences, mutual information, and universal coefficients are physical. More deeply, the local algebra of a QFT region is a Type III$_1$ von Neumann factor, on which no trace — and hence no standard density matrix or finite entropy — exists. The naive "entanglement entropy of a region" is ill-defined without specifying the operator algebra and a split/cutoff. This is structural, not merely calculational. ESTABLISHED (the obstruction); OPEN (the resolution).
• Black-hole information paradox. Hawking's semiclassical calculation predicts thermal (mixed) radiation from a pure collapse, implying non-unitary evolution — a genuine tension between EFT on a curved background and unitarity. QES/island computations now reproduce the unitary Page curve INFERENCE, strongly suggesting EFT was applied outside its domain (missing replica-wormhole contributions), but a complete first-principles mechanism for a realistic, asymptotically-flat black hole is OPEN. See domains/general-relativity.md.
• Bekenstein bound in strong gravity / cosmology. The bound presupposes a well-defined region radius and energy in a weakly-curved background; near horizons and in cosmological spacetimes these become observer-dependent and the bound loses sharp meaning. Bousso's light-sheet construction repairs this covariantly, but even it requires the expansion/light-sheet conditions and a suitable entropy-flux definition, and lacks a fully general first-principles proof. INFERENCE.
• RT/holography outside AdS. De Sitter space, flat-space holography, and finite-$N$/stringy regimes have no established RT analog; applying "spacetime = entanglement" to cosmology is extrapolation, not derived physics. This is a domain-of-validity boundary, not an internal inconsistency. OPEN.
• Landauer — interpretive status. Not a failure of the inequality (which is measured) but a CONTESTED claim about its logical standing: Earman–Norton and Hemmo–Shenker argue it is either a consequence of, or insufficient to independently rescue, the second law, and that the Maxwell-demon resolution admits inequivalent framings. The physics is agreed; the bookkeeping of where the irreducible cost "lives" is disputed.
• Kolmogorov uncomputability / Shannon mismatch. $K(x)$ is uncomputable and defined only up to a constant, so it cannot be operationalized for finite strings; the identity $\mathbb{E}[K] = H + O(1)$ holds only for computable sources, in expectation. There is no canonical "amount of information" for a single object.
• Type III algebras and "a qubit of the field." Because local subsystems are not tensor factors, much "entanglement as fabric of spacetime" discourse implicitly assumes a Type-I tensor-product structure the QFT lacks. Crossed-product / large-$N$ constructions (yielding Type II algebras with well-defined traces) partially repair this, but the foundational gap is real and recent. OPEN.
• ER=EPR and "spacetime from entanglement" as general claims. No demonstration exists that generic entanglement produces a smooth geometric connection, nor that our universe is so constituted. The first-law derivation recovers only the linearized Einstein equations; the full nonlinear, background-independent reconstruction is not achieved. SPECULATIVE.

Open problems (internal)

• BH information recovery in a realistic black hole. OPEN → partially INFERENCE. The Page curve is reproduced via QES/islands and replica wormholes, indicating unitarity, but how information reaches an external observer, the role of the interior, and the firewall/complementarity debate remain unresolved. Islands are computed in controlled models, not yet shown to resolve the paradox in nature.
• Does holography/RT extend beyond AdS — to de Sitter and cosmology? OPEN. No established dS holographic dual or RT analog; dS/CFT proposals are far less constrained than AdS/CFT. Arguably the central open question of the program.
• The correct definition of entanglement entropy in quantum gravity given Type III algebras and no fundamental tensor factorization. OPEN. Crossed-product / "algebra of observables in gravity" constructions are promising but incomplete.
• A physical realization of hypercomputation vs. a Church–Turing law of nature. OPEN/CONTESTED. All proposed loopholes face severe physical-plausibility objections (infinite energy/precision, instability); no consensus, no decisive theorem.
• Reconcile area-extensive (finite, holographic) gravitational degrees of freedom with the volume-extensive, infinite-dimensional Hilbert space of local QFT. OPEN. The cosmological-constant problem, the species bound, and holographic finiteness all point to over-counting in continuum QFT; the precise reorganization (and whether spacetime is discrete) is unknown.
• Derive the FULL nonlinear Einstein equations from entanglement/information principles, background-independently. OPEN/SPECULATIVE. Linearized equations follow from entanglement first laws and Jacobson's thermodynamic derivation gives Einstein's equations as an equation of state, but a complete nonperturbative generic derivation is not in hand.
• Is quantum mechanics itself derivable from information-theoretic axioms? OPEN. Reconstructions (Hardy 2001; Chiribella–D'Ariano–Perinotti; Clifton–Bub–Halvorson "no signalling / no broadcasting / no bit commitment") derive QM from operational postulates — suggestive — but multiple inequivalent axiom sets succeed, so which (if any) is fundamental is open.

See OPEN_PROBLEMS.md, GAPS_AND_CONTRADICTIONS.md, and HYPOTHESES.md.

Connections to other frameworks

• domains/statistical-mechanics.md & domains/thermodynamics.md: a deep structural identity — Gibbs/Boltzmann and Shannon entropy share the $-\sum p\log p$ form; Jaynes recast statistical mechanics as MaxEnt inference. Landauer and the fluctuation theorems (Jarzynski, Crooks, Sagawa–Ueda) make information a thermodynamic resource at $k_B T \ln 2$ per bit. The second law and relative-entropy monotonicity are reflections of the same arrow INFERENCE on the depth of the identity.
• domains/quantum-mechanics.md: no-cloning, no-signalling, and monogamy of entanglement constrain what is operationally possible; the field both uses QM and seeks to reconstruct it. Decoherence and the Born rule are reinterpreted informationally (quantum Darwinism, Zurek).
• domains/general-relativity.md: the strongest and most contested link — Bekenstein–Hawking entropy ties horizon area to information; holography, RT, and "spacetime from entanglement" propose geometry is organized entanglement; Jacobson derived Einstein's equations thermodynamically. Here the domain is most ambitious and least empirically grounded.
• domains/quantum-field-theory.md & string theory / AdS/CFT: the concrete arena where RT, JLMS, entanglement-wedge reconstruction, QEC/HaPPY tensor networks, and the island formula are computable. Strominger–Vafa string microstate counting gave the first statistical-mechanical derivation of $S_{\rm BH}$, anchoring "it-from-bit" in a UV-complete theory. See also domains/particle-physics.md.
• domains/mathematics.md & computer science: Kolmogorov complexity, the Church–Turing thesis, and the extended (efficient) thesis bridge to CS; quantum computing challenges the efficient thesis. The "complexity = volume / complexity = action" conjectures (Susskind) speculatively link computational complexity to black-hole interiors SPECULATIVE.
• Condensed matter / many-body physics: entanglement-entropy area laws, topological entanglement entropy (Kitaev–Preskill, Levin–Wen), and tensor networks (MERA, PEPS) connect the same structures to phases of matter; MERA's geometry inspired discrete holography.
• domains/cosmology.md: a clashing frontier — holographic and Bekenstein/Bousso bounds constrain the observable universe's information content; the de Sitter horizon has an entropy by analogy with black holes, but dS holography is unestablished. Whether "entanglement = geometry" survives in an accelerating, non-AdS universe is a live tension, not a settled connection.

See UNIFICATION_LANDSCAPE.md, CONSTANTS_AND_SCALES.md, and GLOSSARY.md.

Key references

Standard, real sources; consult BIBLIOGRAPHY.md for the consolidated list.

• C. E. Shannon, "A Mathematical Theory of Communication," Bell System Technical Journal (1948). — Founding paper: entropy, mutual information, source/channel coding theorems, capacity.
• T. M. Cover & J. A. Thomas, Elements of Information Theory, 2nd ed. (Wiley, 2006). — Canonical classical text: AEP, DPI, capacity, rate-distortion, Kolmogorov complexity.
• M. Li & P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications (Springer). — Definitive algorithmic information theory: invariance, incomputability, Chaitin.
• M. A. Nielsen & I. L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000/2010). — Standard quantum-information text.
• R. Landauer, IBM J. Res. Dev. 5, 183 (1961); C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982). — Landauer's principle; reversible computation / Maxwell-demon resolution.
• N. Margolus & L. B. Levitin, Physica D 120, 188 (1998); S. Lloyd, Nature 406, 1047 (2000). — Quantum speed limit; ultimate physical limits to computation.
• J. D. Bekenstein, Phys. Rev. D 23, 287 (1981); L. Susskind, J. Math. Phys. 36, 6377 (1995); R. Bousso, Rev. Mod. Phys. 74, 825 (2002). — Bekenstein bound, holographic principle, covariant bound.
• S. Ryu & T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006); V. Hubeny, M. Rangamani, T. Takayanagi, JHEP 0707, 062 (2007). — RT and HRT formulas.
• A. Almheiri, X. Dong, D. Harlow, JHEP 04 (2015) 163; F. Pastawski, B. Yoshida, D. Harlow, J. Preskill, JHEP 06 (2015) 149. — AdS/CFT as QEC; HaPPY code.
• M. Van Raamsdonk, Gen. Rel. Grav. 42, 2323 (2010); T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995); J. Maldacena & L. Susskind, Fortsch. Phys. 61, 781 (2013). — Spacetime from entanglement; Einstein equation of state; ER=EPR.
• S. W. Hawking, Commun. Math. Phys. 43, 199 (1975); D. N. Page, Phys. Rev. Lett. 71, 3743 (1993); G. Penington (2020); Almheiri–Engelhardt–Marolf–Maxfield (2020). — Information paradox, Page curve, islands/QES.
• E. H. Lieb & M. B. Ruskai, J. Math. Phys. 14, 1938 (1973). — Proof of strong subadditivity.
• J. A. Wheeler, "Information, Physics, Quantum: The Search for Links" (1990). — The "it from bit" manifesto.

See also

• README.md · AGENTS.md · THEORY_MAP.md · EPISTEMICS.md
• domains/quantum-mechanics.md · domains/quantum-field-theory.md · domains/general-relativity.md
• domains/thermodynamics.md · domains/statistical-mechanics.md · domains/cosmology.md
• domains/particle-physics.md · domains/mathematics.md · domains/classical-mechanics.md
• UNIFICATION_LANDSCAPE.md · UNIFYING_PRINCIPLES.md · OPEN_PROBLEMS.md · GAPS_AND_CONTRADICTIONS.md · HYPOTHESES.md
• ASSUMPTIONS_LEDGER.md · CONSTANTS_AND_SCALES.md · FINDINGS.md · GLOSSARY.md · ROADMAP.md · CHANGELOG.md

References

See BIBLIOGRAPHY.md for the consolidated, canonical reference list across the wiki.