  We establish a point-wise gradient estimate for $all$ positive solutions of
the conjugate heat equation. This contrasts to Perelman's point-wise gradient
estimate which works mainly for the fundamental solution rather than all
solutions. Like Perelman's estimate, the most general form of our gradient
estimate does not require any curvature assumption. Moreover, assuming only
lower bound on the Ricci curvature, we also prove a localized gradient estimate
similar to the Li-Yau estimate for the linear Schr\"odinger heat equation. The
main difference with the linear case is that no assumptions on the derivatives
of the potential (scalar curvature) are needed.
  A generalization of Perelman's W-entropy is defined in both the Ricci flow
and fixed metric case. We also find a new family of heat kernel estimates.
