  An interferometer - no matter how clever the design - cannot reveal both the
wave and particle behavior of a quantum system. This fundamental idea has been
captured by inequalities, so-called wave-particle duality relations (WPDRs),
that upper bound the sum of the fringe visibility (wave behavior) and path
distinguishability (particle behavior). Another fundamental idea is
Heisenberg's uncertainty principle, stating that some pairs of observables
cannot be known simultaneously. Recent work has unified these two principles
for two-path interferometers. Here we extend this unification to $n$-path
interferometers, showing that WPDRs correspond to a modern formulation of the
uncertainty principle stated in terms of entropies. Furthermore, our
unification provides a framework for solving an outstanding problem of how to
formulate universally valid WPDRs for interferometers with more than two paths,
and we employ this framework to derive some novel WPDRs.
