Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}.
It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language
is accepted by a deterministic finite automaton of size O(n), any one-way
quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was
based on the fact that the evolution of a QFA is required to be reversible.
When arbitrary intermediate measurements are allowed, this intuition breaks
down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n,
thus also improving the previous bound. The improved bound is obtained by
simple entropy arguments based on Holevo's theorem. This method also allows us
to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes
(random access codes) introduced by Ambainis et al. We then turn to Holevo's
theorem, and show that in typical situations, it may be replaced by a tighter
and more transparent in-probability bound.Comment: 8 pages, 1 figure, Latex2e. Extensive modifications have been made to
increase clarity. To appear in FOCS'9
