  We consider the entanglement entropy in 2d conformal field theory in a class
of excited states produced by the insertion of a heavy local operator. These
include both high-energy eigenstates of the Hamiltonian and time-dependent
local quenches. We compute the universal contribution from the stress tensor to
the single interval Renyi entropies and entanglement entropy, and conjecture
that this dominates the answer in theories with a large central charge and a
sparse spectrum of low-dimension operators. The resulting entanglement
entropies agree precisely with holographic calculations in three-dimensional
gravity. High-energy eigenstates are dual to microstates of the BTZ black hole,
so the corresponding holographic calculation is a geodesic length in the black
hole geometry; agreement between these two answers demonstrates that
entanglement entropy thermalizes in individual microstates of holographic CFTs.
For local quenches, the dual geometry is a highly boosted black hole or conical
defect. On the CFT side, the rise in entanglement entropy after a quench is
directly related to the monodromy of a Virasoro conformal block.
