The Page Transition in the Multi-Time Correlations of Hawking Radiation

Abstract: The Page curve and the island formula describe a reorganization of quantum information in Hawking radiation across the Page time. We study how this reorganization appears in correlations among emissions collected at several retarded times. The late radiation is treated as a covariant multi-time process, or process tensor, and $$χ<em>{\mathrm{island}}=\mathcal{N}</em>{\varnothing}-\mathcal{N}<em>{R}$$ measures the temporal memory removed when the early radiation register $R$ is supplied. Operationally, this quantity is the corresponding change in the optimal distinguishability of the radiation process from a Markovian reference. We exhibit evaporation processes whose single-bin states and all process marginals below a finite order $k</em>\ast\geq 3$ remain fixed and coincide with a CP-divisible reference, while $χ<em>{\mathrm{island}}$ changes by order one at order $k</em>\ast$. The Page transition can therefore appear as a transition in the Markov order of the radiation process. When the interior mediates the late-time memory, entanglement-wedge reconstruction of the predictive interior from $R$ Markovianizes the late process; conversely, when the late process has a minimal predictive memory and its future-affecting information is mediated by that memory, Markovianization by $R$ supplies a simulator of its influence. A Hayden--Preskill evaporation comb and an evaporating-JT calculation realize this mechanism. The late radiation retains temporal memory across the transition, while the early radiation becomes sufficient for predicting its future multi-time statistics.