  Quantum coherence is a fundamental feature of quantum mechanics and an
underlying requirement for most quantum information tasks. In the resource
theory of coherence, incoherent states are diagonal with respect to a fixed
orthonormal basis, i.e., they can be seen as arising from a von Neumann
measurement. Here, we introduce and study a generalization to a resource theory
of coherence defined with respect to the most general quantum measurement,
i.e., to an arbitrary positive-operator-valued measure (POVM). We establish
POVM-based coherence measures and POVM-incoherent operations which coincide for
the case of von Neumann measurements with their counterparts in standard
coherence theory. We provide a semidefinite program that allows to characterize
interconversion properties of resource states, and exemplify our framework by
means of the qubit trine POVM, for which we also show analytical results.
