Page Curve: Entanglement Dynamics in Black Holes

• Page curve is the time-dependent evolution of entanglement entropy that rises during black hole evaporation, peaks at the Page time, and declines to preserve unitarity.
• Its mathematical foundation is based on Page’s formula for bipartite quantum systems, demonstrating entropic phase transitions in both gravitational and non-gravitational settings.
• Modern approaches use quantum extremal surfaces and the island rule to link gravitational thermodynamics with quantum information theory across diverse models.

The Page curve is the expected time-dependence of the fine-grained entropy of a quantum subsystem during the process of black hole evaporation or related quantum dynamical settings. The classic Page curve scenario captures the rise, peak, and eventual fall of entanglement entropy between the outgoing Hawking radiation and the remaining black hole, consistent with unitary time evolution. The Page curve is central to the modern resolution of the black hole information paradox and, more generally, reflects fundamental aspects of entanglement dynamics in closed and open quantum systems, including quantum chaotic many-body chains and random pure states.

1. Mathematical Foundation of the Page Curve

The standard formulation of the Page curve arises in the context of a bipartite system $AB$ with Hilbert space $$\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B$$, dimensions $d_A$, $d_B$. For a random global pure state $|\psi\rangle_{AB}$, the average entropy of subsystem $A$ (Page’s formula) is

$$\langle S_A \rangle = H_{d_A d_B} - H_{d_B} - \frac{d_A-1}{2d_B}$$

where $H_n = \sum_{k=1}^n 1/k$ and the formula holds for $d_B \geq d_A$ (Dahlsten, 29 May 2025). This entropy rises from zero when $A$ is small, peaks near the “Page time” (at $d_A \approx d_B$), and falls back to zero as $A$ encompasses almost all degrees of freedom. In black hole physics, one identifies $A$ with the radiation and $B$ with the remaining black hole interior.

2. Emergence in Black Hole Evaporation

In semiclassical gravity, the Hawking radiation process—computed without quantum backreaction—yields monotonically increasing radiation entropy, apparently violating unitarity. Page’s argument and subsequent developments showed that if black hole evolution is unitary, the entropy $S_{\rm rad}(t)$ of the radiation must initially increase, reach a maximum at the Page time $t_{\rm Page}$, and then decrease, eventually vanishing as the black hole evaporates completely (Gautason et al., 2020, Mahajan, 4 Feb 2025).

The modern calculation employs the “island rule” or quantum extremal surface (QES) prescription (Mahajan, 4 Feb 2025): $$S_{\text{rad}}(t) = \min_I \left[ \frac{\mathrm{Area}(\partial I)}{4G_N} + S_{\text{bulk}}(R \cup I) \right]$$ where $I$ (“the island”) is a spacetime region inside the black hole, $\partial I$ its boundary, and $S_{\text{bulk}}$ the quantum field theory entropy. For early times, the minimal surface is trivial and the entropy grows linearly; after the Page time, a non-trivial island appears, cutting off the growth of the radiation entropy.

Explicit models, such as solvable 2D dilaton gravity (e.g., Russo–Susskind–Thorlacius model), yield analytical Page curves such as (Gautason et al., 2020): $$S_{\text{rad}}(t) = \begin{cases} \frac{c}{12} t & t < t_{\text{Page}} \ 2M - \frac{c}{24} t & t > t_{\text{Page}} \end{cases}$$ with the Page time $t_{\text{Page}}$ at a fixed fraction of the black hole lifetime.

3. Dynamical Realizations Outside Gravity

Analogues of the Page curve arise in free and interacting fermionic lattice models subjected to system-reservoir coupling or measurement protocols (Kehrein, 2023, Jha et al., 5 Feb 2025, Ganguly et al., 21 Jan 2025). For a finite-sized “system” (filled fermionic chain) coupled to a much larger empty reservoir, the entanglement entropy between system and environment follows a Page-curve-like time dependence: initial linear or diffusive rise, a maximal “Page value” at $t_P$, and decay as the system empties out. Interactions or dephasing can shift the location of the Page point or modify volume- to area-law scalings, but the overall qualitative structure is robust (Ganguly et al., 21 Jan 2025).

In quantum chaotic circuits and operator growth models, the crossover (void formation) in operator support onto the radiation subsystem dynamically yields a Page curve for the second Rényi and von Neumann entropies, without the need to invoke Haar-random typicality or gravity-specific assumptions (Liu et al., 2020). The “entanglement membrane” picture provides a hydrodynamic-type coarse-grained mechanism, with the Page point determined by membrane tension minimization (Blake et al., 2023).

4. Refinements: Symmetries, Charge, and Phase Structure

Symmetry constraints, such as global $U(1)$ charge or spin conservation, modify the standard random-state Page curve. In charge-conserving qubit models, one obtains a sector-resolved Page curve, where each charge sector has its own characteristic Page time and curve shape (Lau et al., 2022, Li et al., 2023). The “refined Page curve” can show delayed or asymmetric turnover compared to the original Page curve. Explicit symmetry breaking, whether spontaneous or dynamical, restores a more generic Page-like profile.

In gravitational contexts with additional phase structure (Hawking–Page transition, van der Waals–Maxwell transitions), the presence or absence of an equilibrium phase transition modifies the existence and details of the Page curve. In deformed Jackiw–Teitelboim or higher-dimensional AdS black holes, first-order transitions can cause the Page time to vanish (immediate appearance of an island plateau) or diverge (no saturation) depending on the stability of black hole branches (Lu et al., 2022). This behavior reflects the interplay between thermodynamic and fine-grained entropy.

5. Algebraic and Holographic Perspectives

In infinite-dimensional Hilbert spaces, or beyond the type-I von Neumann algebra setting, the Page curve can be formulated in terms of continuous dimensions associated with type II$_1$ factors. Here, the curve is the logarithm of the Murray–von Neumann trace ratio of projections (“relative continuous dimension”), leading to a continuous, piecewise-linear Page curve with a phase transition at the algebraic Page time (Gomez, 2024). This operator-algebraic approach clarifies the universality of the Page phenomenon even when finite-dimensional factorization fails.

Holographic duality (AdS/CFT) and the quantum Ryu–Takayanagi prescription make the Page curve an explicit outcome of semiclassical gravitational path integrals, with the island saddle (or replica wormhole contribution) providing a minimal surface that dominates after the Page time (Mahajan, 4 Feb 2025). Numerical and analytical studies confirm that this transition occurs at the expected entropy balance point, with plateaus of order the Bekenstein–Hawking entropy.

6. Further Generalizations and Physical Implications

The appearance of the Page curve is sensitive to dynamical details: measurement protocols (diffusive vs. logarithmic entropy growth), model of evaporation, and the presence of disorder or backreaction (Jain et al., 2023, Ganguly et al., 21 Jan 2025). Backreaction or deformations may shift the Page time and scrambling time, but the plateau value remains determined by the black hole entropy. In rotating Kerr black holes, the Page curve remains, but both the shape and the Page time depend on initial spin, and angular momentum and mass vanish in lock-step at evaporation’s end (Nian, 2019).

The sharp transition at the Page time, corresponding to a change in dominance between extremal surfaces, can be smoothed by finite $G_N$ corrections or higher-order replica computations, resulting in a more gradual crossover and possible early transfer of information (Khodahami et al., 2023). In all settings, the Page curve encodes the fundamental requirement of unitary time evolution and the necessity to include nonlocal entanglement contributions, such as islands or operator voids, to avoid apparent information loss.

The Page curve remains a unifying feature across black hole thermodynamics, quantum information, random matrix theory, many-body quantum chaos, and operator algebra, providing a quantitative link between statistical typicality, gravitational dynamics, and entanglement structure. Its robust appearance in both gravitational and nongravitational models underscores its foundational role in quantum statistical mechanics and quantum gravity research.