On entropy and complexity of coherent states

Consanguinity of entropy and complexity is pointed out through the example of coherent states of the $SL(2,\C)$ group. Both are obtained from the K\"ahler potential of the underlying geometry of the sphere corresponding to the Fubini-Study metric. Entropy is shown to be equal to the K\"ahler potential written in terms of dual symplectic variables as the Guillemin potential for toric manifolds. The logarithm of complexity relating two states is shown to be equal to Calabi's diastasis function. Optimality of the Fubini-Study metric is indicated by considering its deformation.