In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
A bstract In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
A bstract In recent years, there is a huge progress in understanding the black hole information problem, and the key is that the black hole entropy of radiation should be calculated by the island formula, which describes the Page curve to ensure the unitarity of black hole evaporation. In the paper, we find that the black hole evaporation interpreted by the island formula can be understood in the language of approximate quantum error correction. Furthermore, the Page transition, as a special property of the Page curve, should be understood as the property of approximate quantum error correction itself, i.e. a general class of quantum systems under certain conditions from approximate quantum error correction can also exhibit phenomenon similar to the Page transition.
We develop a computational framework for the quantum relative entropy based on the replica trick and apply it to analyze the evaporation process method of Jackiw-Teitelboim black holes. We show that the relative entropies between the exact radiation state and the Hawking saddle state and the replica wormholes saddle state quantify information-theoretic costs inherent to coarse-grained semiclassical descriptions. During evaporation, the relative entropy for the Hawking saddle increases monotonically, reflecting a progressively higher thermodynamic cost of sustaining the Hawking description. Conversely, the relative entropy associated with the replica wormholes saddle decreases, which indicates that the description of connected geometries becomes progressively more accurate. Moreover, the sign of the difference between the two quantities reverses precisely at the Page time, providing an information-theoretic diagnosis for the transition between competing gravitational saddles. Our results recast the Page transition as a flow of relative entropy costs between competing semi-classical descriptions, revealing an information-theoretic structure underlying the emergence of effective irreversibility in coarse-grained black hole evaporation.
The Page curve describing the process of black hole evaporation is derived in terms of a family, parametrized in terms of the evaporation time, of finite type II_1 factors, associated, respectively, to the entanglement wedges of the black hole and the radiation. The so defined Page curve measures the relative continuous dimension of the black hole and the radiation along the evaporation process. The transfer of information is quantitatively defined in terms of the Murray von Neumann parameter describing the change of the spatial properties of the factors during the evaporation. In the simplest case the generator of the evaporation process is defined in terms of the action of the fundamental group of the hyperfinite type II_1 factor. In this setup the Page curve describes a phase transition with the transfer of information as order parameter. We discuss the limits of either a type I or a type III description of the black hole evaporation.
A bstract If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, without resorting to typicality. Our argument is based on what we refer to as the “operator gas” approach, which allows one to understand the evolution of the microstate of the black hole from generic features of the Heisenberg evolution of operators. One key feature which leads to the Page curve is the possibility of dynamical processes where operators in the “gas” can “jump” outside the black hole, which we refer to as void formation processes. Such processes are initially exponentially suppressed, but dominate after a certain time scale, which can be used as a dynamical definition of the Page time. In the Hayden-Preskill protocol for young and old black holes, we show that void formation is also responsible for the transfer of information from the black hole to the radiation. We conjecture that void formation may provide a microscopic explanation for the recent semi-classical prescription of including islands in the calculation of the entanglement entropy of the radiation.
The Page curve is the standard signature of unitary information recovery in an evaporating quantum system. This work asks a question usually left implicit: whose Page curve? In a solvable charge-locked evaporation model, the radiation observables separate into two classes: the charge-coherent algebra, available to an observer with the appropriate charge reference frame, and the charge-invariant algebra of an observer without such a reference frame. These two observers do not recover the same information curve. For the full charge-coherent algebra, the Page curve follows the standard unitary pattern: the radiation entropy rises and then returns exactly to zero, with all interior microstates becoming mutually orthogonal and perfectly decodable radiation records. For the charge-invariant algebra, the turnover is incomplete. The accessible entropy terminates at the charge-distribution entropy, representing the information deficit caused by the missing reference frame. Microstates related by a global charge-phase transformation remain indistinguishable to this observer, not because information is destroyed, but because it is inaccessible within that observable algebra. The work establishes an operator-class Page theorem: complete purification and full information recovery are properties of the charge-coherent algebra, while charge-invariant observers experience a Page curve truncated by the entropy associated with the missing charge reference frame. The model also reveals a mode-structure law. Multi-channel charged radiation can carry relative-charge information internally and locks only the global charge-phase bit, while a single effective channel can lock the full information content. An external reference frame restores accessibility only when its charge quantum matches the charge quantum of the hidden coherence, demonstrating a precise reference-frame selection rule. This work is a circuit-level quantum information model with no gravity and no dynamical backreaction. It does not claim a black-hole evaporation solution. Instead, it provides an exactly solvable framework showing how information recovery can depend on observable algebra and reference frame in systems with conserved charges. All results are accompanied by reproducibility scripts and verification procedures.