Permutation-parity exchange at a beam splitter: Application to Heisenberg-limited interferometry

Quantum-optical permutation and parity observables are unitarily exchanged by a 50:50 beam splitter. Bosonic coalescence effects are reexamined from this point of view. We show that photon-number resolving counters behind a beam splitter define a permutation detector for the input optical field. With suitable phase encoding, the detector also enables Heisenberg-limited interferometry.