Fine-Grained Entropy and Page Curves

A Page curve is a statement about the fine-grained entropy of a specified radiation subsystem, not a generic plot of emitted thermodynamic entropy. In an idealized unitary evaporation, radiation begins pure, becomes entangled with the remaining black hole, and returns to purity when the black hole disappears. Leading semiclassical pair creation instead predicts continued entropy growth. This chapter separates the definitions, theorem, model assumptions, and diagnostics behind that contrast.

Helpful background. Semiclassical Gravity supplies black-hole entropy, temperature, and evaporation scales; Quantum Information supplies reduced states, Rényi entropies, typicality, and entropy inequalities; Quantum Fields in Curved Spacetime supplies the regulated Hawking state on a prescribed background.

The entropy being plotted

At time $t$, let $R(t)$ be radiation already transferred to a declared nongravitating collection region and let $B(t)$ contain every remaining purifier, including the black hole. With a finite-dimensional regulator,

$$\rho_R(t)=\operatorname{Tr}_{B(t)}\rho_{RB}(t), \qquad S_R(t)=-\operatorname{Tr}\!\left[\rho_R(t)\ln\rho_R(t)\right].$$

If $RB$ is pure, then $S_R=S_B$. This equality says nothing about the coarse-grained entropy of emitted quanta and nothing about an entropy computed after changing the radiation algebra. In continuum QFT, local von Neumann entropies require a regulator and subtraction; in gravity, factorization and the choice of radiation region are additional inputs.

A useful recurring model assigns effective dimensions $d_R(t)$ and $d_B(t)$. A typical pure state on $\mathcal H_R\otimes\mathcal H_B$ has smaller-subsystem entropy close to

$$S_R(t)\simeq\min\{\ln d_R(t),\ln d_B(t)\},$$

up to the finite-dimensional Page correction. The crossover $d_R\sim d_B$ is the Page transition in this model. It is a statistical benchmark, not an evaporation Hamiltonian.

Six steps through the curve

The chapter proceeds in the order needed to keep definitions separate from dynamical claims.

1. Entropy of Radiation Subsystems defines the trace, algebra, regulator, purification identities, and continuum qualifications behind $S_R$.
2. Page’s Random-State Theorem states the exact Haar average, distinguishes Page’s conjecture from Sen’s proof, and tests its limiting cases.
3. The Unitary Page Curve converts the dimension balance into an idealized rise, turnover, and final purification.
4. The Semiclassical Hawking Curve derives monotonic growth from approximately independent near-horizon pairs.
5. Rényi Entropies and Spectral Information shows what the full moment sequence reveals beyond the von Neumann entropy.
6. Limits of Page-Curve Diagnostics asks which claims survive changes of ensemble, algebra, regulator, bath, and operational access.

The exact mean entropy for a Haar-random bipartite state was presented by Page as a conjecture Page 1993, Eqs. (1)–(11) and proved by Sen Sen 1996, pp. 1–3 of the published article. Keeping that history visible matters: the theorem is exact for its ensemble, while applying it to a black hole adds typicality and dynamical assumptions.

Two curves, two approximation schemes

The unitary benchmark assumes a pure global state and sufficiently mixing dynamics within an allowed Hilbert space. Early on, $d_R\ll d_B$ and $S_R\approx\ln d_R$; late, $d_B\ll d_R$ and purity forces $S_R\approx\ln d_B$ downward. Complete evaporation with no hidden sector gives $d_B\to1$ and $S_R\to0$.

The Hawking calculation treats successive exterior modes as nearly thermal and entangled with independent interior partners. If $b_k$ is a new radiation mode approximately uncorrelated with earlier radiation $R_{k-1}$, then

$$S(R_{k-1}b_k)\approx S(R_{k-1})+S(b_k)\gt S(R_{k-1}).$$

The curve cannot turn until pair independence, factorization, the state, or the entropy prescription changes at order sufficient to compete with this increase. Almheiri and collaborators review how controlled gravitational calculations can change the fine-grained entropy prescription in model settings Almheiri et al. 2021, §§ 3–6. That result must not be silently identified with every thermodynamic or operational notion of information.

Results and open questions as of 2026-08-19

The finite-dimensional entropy theorems are settled. What remains unresolved is how completely their assumptions describe realistic four-dimensional evaporation and which gravitational algebra a calculated entropy belongs to. Competing formulations agree on some curves while assigning them different physical meanings.

Evidence or model	Result	Scope and limitation
Haar-random bipartite states Page 1993, Eqs. (1)–(11) and Sen 1996, published pp. 1–3	Exact mean entropy is nearly maximal for the smaller factor	No energy, locality, charge, or evaporation dynamics is imposed
Leading Hawking pairs Almheiri et al. 2021, §§ 2–4	Semiclassical radiation entropy rises while independent partners remain inside	Relies on semiclassical factorization and does not control the microscopic endpoint
Unitary random-state evaporation	Dimension balance produces a turnover and final purification	Assumes typical mixing and an identified complete finite-dimensional Hilbert space
Fine-grained gravitational entropy in controlled models Almheiri et al. 2021, §§ 5–8	Competing saddles can produce a Page transition for a declared radiation region	Strongest in low-dimensional bath models; single-theory factorization and higher-dimensional completion require separate checks

One position treats a Page curve as necessary evidence for unitary recovery. A more conservative position treats it as an entropy consistency test whose microscopic and operational interpretation remains model-dependent. A third emphasizes that a curve can change when the subsystem or gravitational path-integral prescription changes. The regime and observable must therefore accompany every conclusion.

The conclusions could change after a microscopic four-dimensional model derives the radiation algebra and its unitary dynamics, or after a theorem replaces random-state typicality with controlled evaporation dynamics, or after an operational protocol measures the relevant fine-grained correlations within causal and backreaction limits.

What Page curves leave unresolved

This chapter does not rederive generic quantum-information theory, the fixed-background Hawking spectrum, or stable black-hole thermodynamics. Those belong to the linked preparation and neighboring volumes. It also does not assume a specific string or holographic microstate construction. The goal is narrower: make every Page-curve statement carry its state, subsystem, entropy, regulator, ensemble, and approximation.

Check the dimension model

Suppose $N$ qubits form a pure evaporating system. After $k$ qubits have entered $R$, take $d_R=2^k$ and $d_B=2^{N-k}$. At what $k$ does the Page benchmark turn, and what endpoint entropy does it predict?

Solution

The two dimensions are equal when $2^k=2^{N-k}$, so the crossover is $k=N/2$ for even $N$ and between the two central integers for odd $N$. The typical entropy follows approximately $\min(k,N-k)\ln2$. At $k=N$, the remaining factor has dimension one, so purity requires $S_R=0$. This conclusion uses the pure complete state and cannot be transferred to a radiation-only open channel without its purifier.

References

• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. 2021. “The Entropy of Hawking Radiation.” Reviews of Modern Physics 93: 035002. DOI.
• Page, Don N. 1993. “Average Entropy of a Subsystem.” Physical Review Letters 71: 1291–1294. Open text.
• Sen, Siddhartha. 1996. “Average Entropy of a Quantum Subsystem.” Physical Review Letters 77: 1–3. DOI. Open PDF, preprint titled “Average Entropy of a Subsystem”.