  A broad class of informationally complete symmetric measurements is
introduced. It can be understood as a common generalization of symmetric,
informationally complete POVMs and mutually unbiased bases. Additionally, it
provides a natural way to define two new families of mutually unbiased
symmetric measurement operators in any finite dimension. We show a general
method of their construction, together with an example of an optimal
measurement. Finally, we analyze the properties of symmetric measurements and
provide applications in entropic relations and entanglement detection.
