  A new approach is presented to compute entropy for massless scalar quantum
fields. By perturbing a skewed correlation matrix composed of field operator
correlation functions, the mutual information is obtained for disjoint
spherical regions of size $r$ at separation $R$, including an expansion to all
orders in $r/R$. This approach also permits a perturbative expansion for the
thermal field entropy difference in the small temperature limit ($T \ll 1/r$).
