  In this paper we discuss the asymptotic entropy for ancient solutions to the
Ricci flow. We prove a gap theorem for ancient solutions, which could be
regarded as an entropy counterpart of Yokota's work. In addition, we prove that
under some assumptions on one time slice of a complete ancient solution with
nonnegative curvature operator, finite asymptotic entropy implies
kappa-noncollapsing on all scales. This provides an evidence for Perelman's
more general assertion that on a complete ancient solution with nonnegative
curvature operator, bounded entropy is equivalent to kappa-noncollapsing.
