Certainty relations, mutual entanglement and non-displacable manifolds

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.