  In [ZY2], the second author proved Perelman's assertion, namely, for an
ancient solution to the Ricci flow with bounded and nonnegative curvature
operator, bounded entropy is equivalent to noncollapsing on all scales. In this
paper, we continue this discussion. It turns out that the curvature operator
nonnegativity is not a necessary condition, and we need only to assume a
consequence of Hamilton's trace Harnack. Furthermore, we show that this
condition holds for steady Ricci solitons with nonnegative Ricci curvature.
