We derive the property of strong superadditivity of mutual information arising
from the Markov property of the vacuum state in a conformal field theory and
strong subadditivity of entanglement entropy. We show this inequality encodes
unitarity bounds for different types of fields. These unitarity bounds are
precisely the ones that saturate for free fields. This has a natural
explanation in terms of the possibility of localizing algebras on null
surfaces. A particular continuity property of mutual information characterizes
free fields from the entropic point of view. We derive a general formula for
the leading long distance term of the mutual information for regions of
arbitrary shape which involves the modular flow of these regions. We obtain
the general form of this leading term for two spheres with arbitrary
orientations in spacetime, and for primary fields of any tensor
representation. For free fields we further obtain the explicit form of the
leading term for arbitrary regions with boundaries on null cones.