Exact characterization of three-terminal fermion counting under one-body scattering

Quantum scattering is a fundamental probe of the scattering region and the quantum dynamics of outgoing particles. In two terminals, the statistics of non-interacting fermions is described by a generalized binomial distribution, that arises from the Pauli exclusion principle, while deviations are a signature of interactions. In this work, we extend this result to a scattering region coupled to three outgoing terminals (or channels), obtaining exact bounds for the statistics of the outgoing fermions. In particular, for two fermions and three terminals, four inequalities involving six measured probabilities determine whether a non-interacting scatterer can explain the observed statistics. Applying these inequalities to recent scattering experiments in cold atoms, we are able to identify a regime where interactions have a significant contribution on the outgoing particles. Therefore, our work opens up the possibility of identifying fermion interactions at the nanoscale in systems that range from mesoscopic conductors to optical traps in cold atoms.