  Certain duality of relative entropy can fail for chiral conformal net with
nontrivial representations. In this paper we quantify such statement by
defining a quantity which measures the failure of such duality, and identify
this quantity with relative entropy and global index associated with
multi-interval subfactors for a large class of conformal nets. In particular we
show that the duality holds for a large class of conformal nets if and only if
they are holomorphic. The same argument also applies to CFT in two dimensions.
In particular we show that the duality holds for a large class of CFT in two
dimensions if and only if they are modular invariant. We also obtain various
limiting properties of relative entropies which naturally follow from our
formula.
