Estimation of Shannon and R\'enyi entropies of unknown discrete distributions
is a fundamental problem in statistical property testing and an active research
topic in both theoretical computer science and information theory. Tight bounds
on the number of samples to estimate these entropies have been established in
the classical setting, while little is known about their quantum counterparts.
In this paper, we give the first quantum algorithms for estimating
α-R\'enyi entropies (Shannon entropy being 1-Renyi entropy). In
particular, we demonstrate a quadratic quantum speedup for Shannon entropy
estimation and a generic quantum speedup for α-R\'enyi entropy
estimation for all α≥0, including a tight bound for the
collision-entropy (2-R\'enyi entropy). We also provide quantum upper bounds for
extreme cases such as the Hartley entropy (i.e., the logarithm of the support
size of a distribution, corresponding to α=0) and the min-entropy case
(i.e., α=+∞), as well as the Kullback-Leibler divergence between
two distributions. Moreover, we complement our results with quantum lower
bounds on α-R\'enyi entropy estimation for all α≥0.Comment: 43 pages, 1 figur