  The brightness theorem---brightness is nonincreasing in passive systems---is
a foundational conservation law, with applications ranging from photovoltaics
to displays, yet it is restricted to the field of ray optics. For general
linear wave scattering, we show that power per scattering channel generalizes
brightness, and we derive power-concentration bounds for systems of arbitrary
coherence. The bounds motivate a concept of "wave \'{e}tendue" as a measure of
incoherence among the scattering-channel amplitudes, and which is given by the
rank of an appropriate density matrix. The bounds apply to nonreciprocal
systems that are of increasing interest, and we demonstrate their applicability
to maximal control in nanophotonics, for metasurfaces and waveguide junctions.
Through inverse design, we discover metasurface elements operating near the
theoretical limits.
