Page Curves and Entropy Bookkeeping

The Page curve is a typicality result for a finite-dimensional bipartite pure state whose subsystem dimensions change during evaporation. Hawking’s leading semiclassical calculation instead continually creates nearly independent entangled pairs and predicts increasing radiation entropy. Comparing them requires the same fine-grained entropy, a declared factorization or algebra, charge sectors, and an endpoint model.

Required background. The Information Problem: Assumptions and Observables supplies the logical setting. Fine-Grained, Coarse-Grained, and Algebraic Entropy fixes the entropy distinctions.

Helpful background. Decoupling and Subsystem Information Loss gives the channel formulation. Typicality, Non-BPS Extensions, and Evidence Limits constrains typical-state arguments. Landauer Cost, Information Engines, and Field Reservoirs separates entropy from energetic cost.

Page’s finite-dimensional calculation

Let a pure state be Haar-random on $\mathcal H_R\otimes\mathcal H_B$ with dimensions $m=\dim\mathcal H_R\le n=\dim\mathcal H_B$. Page found the mean radiation entropy

$$\mathbb E\,S(R) =\sum_{k=n+1}^{mn}\frac1k-\frac{m-1}{2n},$$

using natural logarithms Page 1993. For large dimensions,

$$\mathbb E\,S(R) =\log m-\frac{m}{2n}+O(n^{-2}).$$

Purity gives $S(R)=S(B)$, so when $m>n$ the same result applies with $m$ and $n$ exchanged. The leading curve is therefore

$$S_{\mathrm{Page}}(t)\simeq \min\{\log d_R(t),\log d_B(t)\}.$$

If the total effective dimension is fixed, $d_Rd_B=D$, the maximum occurs at $d_R=d_B=\sqrt D$. This dimension crossing defines the Page time in the model.

Hawking bookkeeping and the comparison

For leading Hawking pairs $b_i\tilde b_i$, tracing each interior partner gives approximately additive entropy,

$$S_{\mathrm{Hawking}}(R_k) \simeq\sum_{i=1}^k S(b_i),$$

so the entropy continues to rise while pairs are emitted. The disagreement with the Page curve begins when unitarity would require new radiation to purify earlier radiation.

The Page model assumes a finite bipartite Hilbert space, a globally pure state, generic scrambling within the relevant sector, and complete transfer of the black-hole factor into radiation. It does not derive a Hawking temperature, flux, greybody factor, or local horizon state. Conversely, the Hawking calculation does not determine the exponentially subtle correlations needed for a pure final state.

Application: an explicit shrinking model

Take $D=e^{100}$ and let $\log d_R=s$, $\log d_B=100-s$. Then

$$S_{\mathrm{Page}}(s)\simeq \begin{cases} s, & 0\le s\le50,\\ 100-s, & 50\le s\le100. \end{cases}$$

The Hawking independent-pair model gives $S_{\mathrm{Hawking}}\simeq s$ throughout. Both agree before $s=50$; their late-time disagreement is not visible in the coarse radiated energy. This illustrates why a Page-like curve is a fine-grained correlation statement rather than an energy-balance curve.

Charge, nonuniformity, and factorization tests

If a conserved charge decomposes the state as $\bigoplus_q\mathcal H_R^q\otimes\mathcal H_B^{Q-q}$, Haar typicality must be applied within charge blocks and combined with the classical entropy of their probabilities. Replacing each block by a uniform full-space state gives the wrong Page time.

Likewise, a dynamically preferred non-Haar ensemble can have the same dimensions but different entropy. And in gravity, a radiation tensor factor may be only approximate because of constraints and dressing. Repeat the bookkeeping with the actual algebra and center; if the curve changes, the original result was regulator- or factorization-dependent.

Scope and handoff

Page typicality is a benchmark, not a microscopic evaporation mechanism. Gravitational replicas can reproduce a similar leading entropy curve in controlled bath models; their saddle statement is developed on Evaporating Replicas, Radiation Entropy, and Entanglement-Wedge Transitions. The final logical gap is assessed on Microscopic Unitarity versus Semiclassical Entropy Calculations.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Page, D. N. “Average Entropy of a Subsystem.” Physical Review Letters 71 (1993): 1291–1294. DOI.
• Page, D. N. “Information in Black Hole Radiation.” Physical Review Letters 71 (1993): 3743–3746. DOI.