Microcanonical Energy Sharing and a Page-like Curve for the Capacity of Entanglement

Abstract

We study the capacity of entanglement in the microcanonical ensemble for an effectively additive bipartite system. Using typicality and the block structure of the microcanonical reduced state, we show that in the thermodynamic regime the capacity is controlled by energy-sharing fluctuations and can be expressed purely in terms of standard thermal response data of the subsystems. As an illustration, we apply the result to a toy model consisting of a Schwarzian ``black-hole'' sector coupled to a two-dimensional CFT radiation sector. At fixed total energy, the growth of the radiation sector forces the common temperature to decrease, producing a smooth Page-like single-hump curve for the capacity. We also distinguish the capacity of the reduced microcanonical state from that of a Haar-typical pure state in the same energy shell. A block-Wishart analysis shows that the effective-heat-capacity result holds on both sides of the Page crossover, while a universal folded-normal scaling function resolves a narrow window in which the two effective Hilbert-space dimensions become comparable. The construction is an equilibrium microcanonical mechanism for Page-like capacity curves, rather than a complete dynamical evaporation calculation.