Boson-fermion complementarity in a linear interferometer: An identity relating the determinant and permanent of a matrix

Bosonic and fermionic statistics are well known to give rise to antinomic behaviors, most notably boson bunching vs fermion antibunching. Here, we establish a fundamental relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. The bosonic and fermionic transition probabilities appear together in a single equation which constrains their values, hence expressing a boson-fermion complementarity that is independent of the details of the interferometer. For two particles in any interferometer, for example, it implies that the average between the bosonic and fermionic probabilities must coincide with the probability obeyed by classical particles. Crucially, this fundamental relation also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of an arbitrary complex matrix, hence extending an identity by Muir dating from the nineteenth century.