A dynamical mechanism for the Page curve from quantum chaos

TL;DR

The paper provides a dynamical mechanism for the Page curve in black hole evaporation by introducing an operator-gas framework that tracks how chaotic operator growth and rare void-formation events shape entanglement. It shows that the competition between continuous operator spreading and discontinuous void formation reproduces the Page transition without invoking state typicality, and extends the analysis to an eternal black hole coupled to a bath to illustrate information transfer and island-like behavior. By connecting to Hayden-Preskill and semi-classical island computations, the work suggests void formation as a microscopic underpinning of replica-wormhole/island prescriptions and points toward future extensions to more realistic models and higher Renyi entropies. Overall, the results indicate that unitarity and the Page curve can emerge from generic chaotic dynamics without assuming typicality, with broad implications for information recovery in black-hole–bath systems.

Abstract

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.

Figures (15)

• Figure 1: The Page curves of the black hole and the radiation. At $t=0$, the black hole consists of the whole system and is in a pure state. The dotted line is the semi-classical entropy of the black hole from its horizon area. The dashed line is the entropy of the semi-classical radiation from Hawking's calculation. The solid curve is the entanglement entropy for the black hole and the radiation in a full quantum description. It should be seen as two curves, one for the black hole and one for the radiation, which coincide as required by unitarity. The Page time $t_p$ refers to time scale where the solid curve turns around from increasing to decreasing with time.
• Figure 2: Universal features of operator growth in a chaotic system. The region within the black circle represents the space of degrees of freedom, and shaded regions indicate the subsystems where the operators are supported. A given initial operator evolves to a superposition of different final operators. After some time $t$ greater than the scrambling time $t^*$, a typical process is one where the operator becomes supported on the entire space, as shown in term (a). For any subsystem $A$, there is a small probability that the final operator is equal to the identity in $A$, as shown in the term (b). We refer to the presence of terms like (b) as void formation in $A$.
• Figure 3: Evolution of the operator gas in an evaporating black hole. In all cases, the region on the left represents the black hole, and the region on the right represents the radiation. The black hole subsystem $B(t)$ grows smaller as a function of time during the evaporation process, while the radiation subsystem $R(t)$ grows larger. (a) During evolution, an operator which is supported in the full system (i.e. including both $B$ and $R$) remains supported in both subsystems. (b) Operators which are initially supported in the black hole at time $t$ have some probability of "jumping" outside $B(t+\Delta t)$ at a later time, forming a void which includes the black hole.
• Figure 4: (a) A schematic illustration of a two-sided eternal black hole coupled to a one-dimensional bath. (b) The evolution of the entanglement entropies of the black hole and the bath in this setup. The black curve represents the evolution of the entanglement entropy of both the black hole and the bath under unitary evolution, and is hence a counterpart of the Page curve for this setup. The red dashed line shows the evolution of the entanglement entropy of the radiation from naive semiclassical calculations, or without including void formation processes, which is a manifestation of the information loss problem in this setup.
• Figure 5: Time-evolution of the evaporating black hole. In the first half of each time-step, a Haar-random matrix from $U(q^{k-t})$ is applied within $B(t)$, and in the second half, a site is taken out of $B$ and put in $R$ to give $B(t+1)$ and $R(t+1)$.