Entropy lower bounds of quantum decision tree complexity

We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard the computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log(n)) bits. Let E(f) be the Shannon entropy of the random variable f(X), where X is uniformly random in f's domain. Our main result is that it takes \Omega(E(f)) queries to compute any \emph{total} function f. It is interesting to contrast this bound with the \Omega(E(f)/log(n)) bound, which is tight for \emph{partial} functions. Our approach is the polynomial method.