  We prove tight entropic uncertainty relations for a large number of mutually
unbiased measurements. In particular, we show that a bound derived from the
result by Maassen and Uffink for 2 such measurements can in fact be tight for
up to sqrt{d} measurements in mutually unbiased bases. We then show that using
more mutually unbiased bases does not always lead to a better locking effect.
We prove that the optimal bound for the accessible information using up to
sqrt{d} specific mutually unbiased bases is log d/2, which is the same as can
be achieved by using only two bases. Our result indicates that merely using
mutually unbiased bases is not sufficient to achieve a strong locking effect,
and we need to look for additional properties.
