Experimental direct estimation of nonlinear functionals of photonic quantum states via interferometry with a controlled-swap operation.

We propose a multi-photon interferometer with a generalized C-SWAP operation that can estimate Tr[ρ(1)ρ(2)···ρ(n)], a nonlinear functional of n photonic density matrices. The scheme is demonstrated for three single-photon states whose overlap is experimentally measured as the interference visibility of a control qubit encoded into photonic paths. The validity of this method is verified by comparing the visibility with the results of Hong-Ou-Mandel experiments.