Sharp null-energy lower bounds for Page-like entropy excursions in two-dimensional conformal field theory

We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_{+}$ obeys a sharp lower bound with two branches separated by the threshold $\mathrm{\Delta }S=(c\mathrm{/}6)\ln 2$. We also derive a sharp lower bound on the net energy $E_{\mathrm{n}\mathrm{e}\mathrm{t}}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_{-}$. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize $E_{+}$ for general QNEC states and the one that minimizes $E_{-}$ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.