We revisit the connection between the complementarity and uncertainty principles of quantum mechanics within the framework of Mach–Zehnder interferometry. We focus our attention on the trade-off relation between complementary path information and fringe visibility. This relation is equivalent to the uncertainty relation of Schrödinger and Robertson for a suitably chosen pair of observables. We show that it is equivalent as well to the uncertainty inequality provided by Landau and Pollak. We also study the relationship of this trade-off relation with a family of entropic uncertainty relations based on Rényi entropies. There is no equivalence in this case, but the different values of the entropic parameter do define regimes that provides us with a tool to discriminate between non-trivial states of minimum uncertainty. The existence of such regimes agrees with previous results of Luis (2011 Phys. Rev. A 84 034101), although their meaning was not sufficiently clear. We discuss the origin of these regimes with the intention of gaining a deeper understanding of entropic measures.