  The information convex allows us to look into certain information-theoretic
constraints in two-dimensional topological orders. We provide a derivation of
the topological contribution $\ln d_a$ to the von Neumann entropy, where $d_a$
is the quantum dimension of anyon $a$. This value emerges as the only value
consistent with strong subadditivity, assuming a certain topological dependence
of the information convex structure. In particular, it is assumed that the
fusion multiplicities are coherently encoded in a 2-hole disk. A similar
contribution ($\ln d_{\alpha}$) is derived for gapped boundaries. This method
further allows us to identify the fusion probabilities and certain constraints
on the fusion theory. We also derive a linear bound on the circuit depth of
unitary non-Abelian string operators and discuss how it generalizes and changes
in the presence of a gapped boundary.
