  We derive explicit bounds for the average entropy characterizing measurements
of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead
to novel entropic uncertainty relations, while upper bounds allow us to
formulate universal certainty relations. For $L=2$ the maximal average entropy
saturates at $\log N$ as there exists a mutually coherent state, but certainty
relations are shown to be nontrivial for $L \ge 3$ measurements. In the case of
a prime power dimension, $N=p^k$, and the number of measurements $L=N+1$, the
upper bound for the average entropy becomes minimal for a collection of
mutually unbiased bases. Analogous approach is used to study entanglement with
respect to $L$ different splittings of a composite system, linked by bi-partite
quantum gates. We show that for any two-qubit unitary gate $U\in
\mathcal{U}(4)$ there exist states being mutually separable or mutually
entangled with respect to both splittings (related by $U$) of the composite
system. The latter statement follows from the fact that the real projective
space $\mathbb{R}P^{3}\subset\mathbb{C}P^{3}$ is non-displacable. For $L=3$
splittings the maximal sum of $L$ entanglement entropies is conjectured to
achieve its minimum for a collection of three mutually entangled bases, formed
by two mutually entangling gates.
