  We provide a framework for a perturbative evaluation of the reduced density
matrix. The method is based on a path integral in the analytically continued
spacetime. It suggests an alternative to the holographic and `standard' replica
trick calculations of entanglement entropy. We implement this method within
solvable field theory examples to evaluate leading order corrections induced by
small perturbations in the geometry of the background and entangling surface.
Our findings are in accord with Solodukhin's formula for the universal term of
entanglement entropy for four dimensional CFTs.
