  For holographic CFT states near the vacuum, entanglement entropies for
spatial subsystems can be expressed perturbatively as an expansion in the
one-point functions of local operators dual to light bulk fields. Using the
connection between quantum Fisher information for CFT states and canonical
energy for the dual spacetimes, we describe a general formula for this
expansion up to second-order in the one-point functions, for an arbitrary
ball-shaped region, extending the first-order result given by the entanglement
first law. For two-dimensional CFTs, we use this to derive a completely
explicit formula for the second-order contribution to the entanglement entropy
from the stress tensor. We show that this stress tensor formula can be
reproduced by a direct CFT calculation for states related to the vacuum by a
local conformal transformation. This result can also be reproduced via the
perturbative solution to a non-linear scalar wave equation on an auxiliary de
Sitter spacetime, extending the first-order result in arXiv/1509.00113.
