Refined Page Curve: Extensions & Applications

• The refined Page curve is an extended model of subsystem entropy that integrates semiclassical saddles, islands, and symmetry-resolved features to capture information recovery beyond naive entropy counting.
• It refines Page’s original formulation by including competing saddle points—such as replica wormholes and island contributions—to provide precise quantification in both gravitational and random-state settings.
• It also applies to many-body systems where different stochastic unravelings yield distinct entropy dynamics, offering new insights into quantum measurement, entanglement transitions, and charge-resolved processes.

The refined Page curve denotes a family of extensions of Page’s original entropy curve in which the standard rise–peak–fall or rise–saturate structure is retained but the entropy notion, dynamical setup, conditioning, or algebraic framework is sharpened. In its canonical form, the Page curve is the average subsystem entropy of a bipartite pure state, $\mathbb{E}[S(\rho_A)]$, plotted against subsystem size, with $\rho_A=\mathrm{tr}_B(\rho_{AB})$ and $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$; it rises from near zero, is maximal near the half-system point, and falls symmetrically by purity and Schmidt-spectrum symmetry (Dahlsten, 29 May 2025). In black-hole evaporation, the same name is used for the time-dependent fine-grained entropy of Hawking radiation, which rises at early times and is later cut off by generalized-entropy minimization with islands or quantum extremal surfaces (Gautason et al., 2020). The qualifier “refined” is used in several distinct but related senses: competition between more than two saddles, trajectory-resolved entanglement dynamics, slice dependence, symmetry resolution under conserved charges, smoothening of the Page-time transition, and operator-algebraic reformulations (Ganguly et al., 21 Jan 2025, Matsuo, 2023, Li et al., 2023, Khodahami et al., 2023, Gomez, 2024).

1. Baseline notion and what is being refined

The baseline Page curve has two principal meanings in the literature represented here. In the random-state setting, it is a typicality statement: for a pure bipartite state $\Psi_{AB}$ chosen from the unitarily invariant distribution, one studies the reduced density matrix of $A$, and the curve of $\mathbb{E}[S(\rho_A)]$ against subsystem size is symmetric around the half-system point because $S(\rho_A)=S(\rho_B)$ for Schmidt-related reductions (Dahlsten, 29 May 2025). In the evaporation setting, it is a dynamical statement: the entropy of radiation or of a subsystem starts from zero, increases, reaches a Page value at a Page time, and then either decreases or saturates depending on the geometry and entropy prescription (Gautason et al., 2020, Mahajan, 4 Feb 2025).

What is refined varies by context. In semiclassical gravity, refinement usually means that the entropy is not the naive semiclassical Hawking entropy but the minimum generalized entropy over competing saddles, including disconnected islands or replica-wormhole saddles (Mahajan, 4 Feb 2025, Gautason et al., 2020). In condensed-matter and transport models, refinement can mean resolving distinct rise and fall laws across different geometries, measurement protocols, or trajectory unravelings of the same Lindblad dynamics (Ganguly et al., 21 Jan 2025). In charged or symmetry-constrained systems, it can mean replacing the unconstrained random-state ensemble by a stage-dependent symmetry-resolved ensemble (Li et al., 2023). In algebraic approaches, it means replacing finite-dimensional tensor-factor entropy by relative continuous dimension for a one-parameter family of type II$_1$ factors (Gomez, 2024).

A recurrent misconception is that there is a single universal “refined Page curve.” The literature instead uses the phrase for multiple sharpenings of the original concept. What unifies them is not a single formula, but the effort to identify the correct entropy functional, effective Hilbert space, or operator-algebraic object that governs information transfer beyond the most naive Page construction.

2. Semiclassical gravity: generalized entropy, islands, and saddle competition

In the island/QES formulation, the refined Page curve is produced by minimizing the generalized entropy rather than following the Hawking-radiation entropy alone. A standard expression used in the lectures on quantum extremal surfaces is

$$S(A)=\min_{R\sim A}\,\mathrm{gen}(R), \qquad \mathrm{gen}(R)=\frac{\mathrm{Area}(\partial R)}{4G}+S_{\text{bulk}}(R),$$

with the dominant saddle changing from a no-island configuration at early times to an island saddle after the Page time (Mahajan, 4 Feb 2025). In the asymptotically flat RST model, this yields an explicit evaporating-black-hole Page curve, with the Page time

$t_{\rm Page}=\frac{1}{3}t_{\rm lifetime}$

at leading semiclassical order (Gautason et al., 2020). The same analysis interprets the early branch as no-island entropy growth and the late branch as island-controlled entropy decrease.

The refined character becomes more pronounced when additional competing surfaces enter. In higher-dimensional double holography with a time-dependent end-of-the-world brane, the entropy of effective Hawking radiation is the minimum of three surfaces: the Hartman–Maldacena surface, a new boundary RT surface ending on the moving ETW brane, and an island RT surface. The physical sequence can be $\rho_A=\mathrm{tr}_B(\rho_{AB})$0 for $\rho_A=\mathrm{tr}_B(\rho_{AB})$1, while for $\rho_A=\mathrm{tr}_B(\rho_{AB})$2 the HM phase disappears and the Page curve becomes a direct $\rho_A=\mathrm{tr}_B(\rho_{AB})$3 transition, with a triple point at $\rho_A=\mathrm{tr}_B(\rho_{AB})$4 and critical temperature $\rho_A=\mathrm{tr}_B(\rho_{AB})$5 (Chou et al., 2021). Here refinement means that the Page transition is not merely a two-branch minimum.

Related modifications appear in alternative gravitational settings. For rotating BTZ black holes with two baths, the late-time island lies just outside the event horizon and the fine-grained radiation entropy saturates at approximately $\rho_A=\mathrm{tr}_B(\rho_{AB})$6; in the extremal limit, divergent Page and scrambling times are regulated by including superradiance, which spins the black hole down and restores a conventional Page-curve regime (Yu et al., 2021). In three-dimensional Einstein–Gauss–Bonnet gravity, the island still sits outside the horizon, but the late-time entropy plateau and Page time acquire explicit $\rho_A=\mathrm{tr}_B(\rho_{AB})$7-dependent corrections through the parameter

$\rho_A=\mathrm{tr}_B(\rho_{AB})$8

so the transition occurs later as $\rho_A=\mathrm{tr}_B(\rho_{AB})$9 increases (Anand, 2022). In the older Kerr evaporation analysis, the Page curve is constructed with Page’s original piecewise prescription but using numerical evolution of both $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$0 and $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$1 under photon emission; the resulting refined Kerr Page curves retain the standard rise and fall but with Kerr-specific timing and simultaneous late-time disappearance of mass and angular momentum (Nian, 2019).

A plausible implication is that “refinement” in gravity most often names the replacement of a naive entropy history by a minimum over a richer set of semiclassical saddles, but the precise meaning depends strongly on geometry, asymptotics, and what is counted as radiation.

3. Slice dependence, mutual information, and dynamical mechanisms

Several works refine the Page curve not by changing the underlying system, but by changing the diagnostic or the mechanism assigned to the turnover. One such refinement concerns the role of mutual information. In the BTZ island setup with subsystems $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$2 and $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$3, late-time saturation is tied to the condition

$S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$4

which is interpreted as the disconnected phase of the entanglement wedge of $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$5. The vanishing occurs at a time separation of order the scrambling time,

$S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$6

and yields a time-independent fine-grained entropy with logarithmic and inverse-power corrections (Saha et al., 2021). In this formulation, the refined Page curve is organized by a mutual-information transition rather than solely by extremization.

A second refinement is slice dependence. The quantum focusing conjecture analysis of evaporating black holes shows that the Page curve is not a universal statement about entropy along every slicing. For timelike surfaces near the horizon, the entropy can rise and then fall. Along an outgoing null surface, however, the entanglement entropy outside the cut remains increasing even after the Page time, because outgoing modes do not cross that surface in the same way. The paper concludes that the Page-time locus is represented by an approximately null surface and that the quantum focusing conjecture is not violated once the entropy is computed with the island rule (Matsuo, 2023). This directly excludes the common identification of “after Page time” with monotonic entropy decrease on all foliations.

A third refinement is dynamical rather than kinematical. In the operator-gas approach to black-hole evaporation, the second Rényi entropy of the radiation takes the form

$S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$7

with one term arising from ordinary operator spreading and the other from “void formation,” namely rare processes in which operators become trivial on the entire remaining black hole. These voids are initially exponentially suppressed, but dominate after the dynamical Page time $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$8, producing the turnover without invoking Haar typicality (Liu et al., 2020). The same paper proposes that void formation may be the microscopic explanation of island contributions.

An analogous but more coarse-grained mechanism appears in the entanglement-membrane approach. There the entropy is a minimum over spacetime membranes, and the Page curve arises from a membrane transition around the Page time, in direct analogy with the switch between competing QES saddles. For an evaporating black-hole model with shrinking region $S(\rho_A)=-\mathrm{tr}(\rho_A\log\rho_A)$9, the result is

$\Psi_{AB}$0

while in the eternal-bath setting the entropy rises and then saturates at $\Psi_{AB}$1 (Blake et al., 2023). This suggests that refined Page-curve behavior can be tied to universal variational principles in chaotic many-body systems, not only to gravitational path integrals.

4. Many-body and transport realizations

A major recent development is the appearance of refined Page curves in unitary and monitored many-body systems with no direct gravitational input. In an exactly solvable free-fermion model, a finite chain $\Psi_{AB}$2 initially full and coupled to a much larger empty environment $\Psi_{AB}$3 exhibits a genuine rise-and-fall Page curve: the von Neumann entropy grows approximately linearly, reaches a maximum at $\Psi_{AB}$4, and then decreases back toward zero because the system empties. The peak is extensive,

$\Psi_{AB}$5

and the late-time limit is $\Psi_{AB}$6 rather than volume-law saturation (Kehrein, 2023). That work also identifies a non-analyticity in the min-entropy at $\Psi_{AB}$7 and a quantum phase transition in the entanglement Hamiltonian.

The interacting extension preserves the Page-curve shape while sharpening the entanglement-spectrum structure. In a spinless fermionic chain with nearest-neighbor interaction $\Psi_{AB}$8, the von Neumann entropy again grows, peaks at $\Psi_{AB}$9, and bends down, but the min-entropy develops a non-analyticity at a critical time $A$0. The paper interprets this as a level crossing in the entanglement Hamiltonian

$A$1

so that the two phases on either side of the non-analyticity are different ground states of $A$2 (Jha et al., 5 Feb 2025). For weak interactions the extrapolated $A$3 can remain non-zero in the thermodynamic limit; for stronger interactions or stronger system–environment coupling it tends toward zero.

A further refinement appears in Lindblad-dephased fermionic transport, where the same dephasing equation is unraveled in different stochastic ways. The model compares a finite initially filled system expanding into an empty reservoir in Probe-Clean (PC) and Probe-Probe (PP) geometries, and contrasts stochastic unitary unraveling (SUU) with quantum state diffusion (QSD). The trajectory entropy

$A$4

shows qualitatively distinct Page-curve laws for different unravelings of the same Lindblad evolution (Ganguly et al., 21 Jan 2025).

Setup	$A$5	$A$6
PC, SUU	$A$7	$A$8
PC, QSD	$A$9	$\mathbb{E}[S(\rho_A)]$0
PP, SUU	$\mathbb{E}[S(\rho_A)]$1	$\mathbb{E}[S(\rho_A)]$2
PP, QSD	$\mathbb{E}[S(\rho_A)]$3	$\mathbb{E}[S(\rho_A)]$4

In the same work, the Page value shows volume-law scaling in PC+SUU and PP+SUU, logarithmic or area-law tendencies in QSD, and the Page time satisfies $\mathbb{E}[S(\rho_A)]$5 in the SUU cases. Moreover, under SUU one finds

$\mathbb{E}[S(\rho_A)]$6

with $\mathbb{E}[S(\rho_A)]$7, whereas this entropy–current proportionality fails in QSD because the current is unraveling-independent but the entropy is not (Ganguly et al., 21 Jan 2025). This is one of the clearest demonstrations that a refined Page curve can depend on conditioning and measurement backaction even when the averaged Lindblad evolution is unchanged.

The free-fermion random-state literature adds a complementary kinematic refinement. For random fermionic Gaussian states at half filling, the average subsystem entropy obeys a “free-fermion Page curve”

$\mathbb{E}[S(\rho_A)]$8

which agrees with interacting Page behavior for microscopic subsystems but differs by an $\mathbb{E}[S(\rho_A)]$9 amount in entropy density for macroscopically large subsystems. The same curve can emerge dynamically in simple tight-binding quenches when all conserved mode occupations satisfy $S(\rho_A)=S(\rho_B)$0 (Yu et al., 2022).

5. Symmetry-resolved, smooth, and algebraic generalizations

One direction of refinement replaces ordinary subsystem entropy by symmetry-resolved entropy under conserved quantities. In a qubit toy model for charged black-hole evaporation with total charge $S(\rho_A)=S(\rho_B)$1, the standard fixed-charge random-state ensemble is replaced by a stage-dependent reduced Hilbert space designed to imitate the expected semiclassical charge profile: before the midpoint $S(\rho_A)=S(\rho_B)$2, no charge is released, while after the midpoint the radiation charge increases linearly. The entropy decomposes as

$S(\rho_A)=S(\rho_B)$3

and the resulting refined Page curve can have a shifted Page time and asymmetric behavior on the two sides of the maximum (Li et al., 2023). Here refinement means imposing microscopic charge-sector constraints that match a nonuniform macroscopic discharge history.

A second direction replaces the sharp Page-time kink by a smooth crossover. In the smooth-curve construction for evaporating Schwarzschild black holes, the entropy is treated as a smooth functional of the boundary, expanded in deformations $S(\rho_A)=S(\rho_B)$4, and constrained by identities such as

$S(\rho_A)=S(\rho_B)$5

At the lowest nontrivial level, this yields

$S(\rho_A)=S(\rho_B)$6

while inclusion of the next replica contribution gives

$S(\rho_A)=S(\rho_B)$7

The purpose is to model the Page transition as gradual rather than abrupt, and the paper argues that the smooth curve carries substantially more informational content than the sharp piecewise curve in the crossover region (Khodahami et al., 2023).

A third direction is operator-algebraic. The algebraic Page curve replaces finite-dimensional Hilbert-space factorization by a one-parameter family of commuting finite type II$S(\rho_A)=S(\rho_B)$8 factors, $S(\rho_A)=S(\rho_B)$9 with $_1$0, associated with black-hole and radiation entanglement wedges. The Page quantity is defined from the minimum relative continuous dimension,

$_1$1

and the Murray–von Neumann information-transfer parameter

$_1$2

acts as an order parameter: it is $_1$3 before the Page time and $_1$4 after it (Gomez, 2024). In this formulation, the refined Page curve measures changing spatial relations between factors rather than ordinary subsystem entropy.

These generalizations are conceptually orthogonal. Symmetry resolution changes the ensemble. Smoothening changes the Page-time transition law. The algebraic construction changes the underlying kinematics of entanglement itself.

6. Conceptual status, limitations, and open directions

The contemporary literature makes clear that the refined Page curve is not a single theorem with one privileged derivation. In the random-state formulation it is not a dynamical law by itself, but a statement about average subsystem entropy under the Haar-uniform ensemble (Dahlsten, 29 May 2025). In island-based gravity it is a semiclassical minimum over generalized-entropy saddles (Mahajan, 4 Feb 2025, Gautason et al., 2020). In transport and monitored systems it can depend on unraveling, geometry, and whether one studies the averaged density matrix or individual trajectories (Ganguly et al., 21 Jan 2025). In chaotic many-body models it can be interpreted as a competition between distinct dynamical mechanisms, such as membrane branches or void formation (Blake et al., 2023, Liu et al., 2020).

Several objective caveats recur. First, a true rise-and-fall Page curve is not guaranteed in arbitrary bipartitions: in finite bipartite geometries the entropy may simply rise and saturate, with a saturation value $_1$5 distinct from the Page value $_1$6 (Ganguly et al., 21 Jan 2025). Second, the turnover can be slice-dependent; along outgoing null geodesics the entropy can continue increasing even after the Page time (Matsuo, 2023). Third, some smooth or symmetry-resolved refinements are constructive models rather than derivations from a complete microscopic gravity theory (Khodahami et al., 2023, Li et al., 2023). Fourth, even in the semiclassical replica program, extending beyond JT gravity remains difficult.

A current attempt to address the last point is the simplicial quantum-gravity formulation based on Quantum Regge Calculus. There the Page transition is reproduced in a proof-of-principle minisuperspace reduction by comparing Hawking and wormhole saddles in a triangulated, matter-coupled replica geometry. The framework is explicitly designed to accommodate complex saddles in Lorentzian signature and to move beyond the usual $_1$7 Euclidean JT setting. At the same time, the work emphasizes unresolved issues: the definition of the discrete configuration space, ambiguities in the gravitational measure, treatment of asymptotic boundaries, the coarseness of the triangulation, and the status of finite-$_1$8 Lorentzian saddles (Padua-Argüelles, 25 Apr 2025).

The refined Page curve is therefore best understood as an umbrella concept for entropy curves in which the original Page logic has been made more faithful to the actual structure of the problem. Depending on context, the relevant refinement may be the inclusion of islands, the resolution of multiple competing saddles, the conditioning on quantum trajectories, the incorporation of conserved charges, the replacement of a sharp Page-time switch by a smooth crossover, or the reformulation of information transfer in operator-algebraic terms. Across these settings, the common theme is that information recovery is not captured by naive entropy bookkeeping alone; it depends on which degrees of freedom, sectors, slices, or algebras are regarded as physically relevant.