  One of the essential features of quantum mechanics is that most pairs of
observables cannot be measured simultaneously. This phenomenon is most strongly
manifested when observables are related to mutually unbiased bases. In this
paper, we shed some light on the connection between mutually unbiased bases and
another essential feature of quantum mechanics, quantum entanglement. It is
shown that a complete set of mutually unbiased bases of a bipartite system
contains a fixed amount of entanglement, independently of the choice of the
set. This has implications for entanglement distribution among the states of a
complete set. In prime-squared dimensions we present an explicit
experiment-friendly construction of a complete set with a particularly simple
entanglement distribution. Finally, we describe basic properties of mutually
unbiased bases composed only of product states. The constructions are
illustrated with explicit examples in low dimensions. We believe that
properties of entanglement in mutually unbiased bases might be one of the
ingredients to be taken into account to settle the question of the existence of
complete sets. We also expect that they will be relevant to applications of
bases in the experimental realization of quantum protocols in
higher-dimensional Hilbert spaces.
