Generalized Interference of Fermions and Bosons

Using tools from representation theory, we derive expressions for the coincidence rate of partially-distinguishable particles in an interferometry experiment. Our expressions are valid for either bosons or fermions, and for any number of particles. In an experiment with $n$ particles the expressions we derive contain a term for each partition of the integer $n$; Gamas's theorem is used to determine which of these terms are automatically zero based on the pairwise level of distinguishability between particles. Most sampling schemes (such as Boson Sampling) are limited to completely indistinguishable particles; our work aids in the understanding of systems where an arbitrary level of distinguishability is permitted. As an application of our work we introduce a sampling scheme with partially-distinguishable fermions, which we call Generalized Fermion Sampling.