Which role does multiphoton interference play in small phase estimation in quantum Fourier transform interferometers?

Recently, quantum Fourier transform interferometers have been demonstrated to allow a quantum metrological enhancement in phase sensitivity for a small number nn of identical input single photons [J. P. Olson, K. R. Motes, P. M. Birchall, N. M. Studer, M. LaBorde, T. Moulder, P. P. Rohde and J. P. Dowling, Phys. Rev. A 96 (2017) 013810; K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson and P. P. Rohde, Phys. Rev. Lett. 114 (2015) 170802; O. Zimmermann, Bachelor Thesis (Ulm University, 2015) arXiv: 1710.03805.]. However, multiphoton distinguishability at the detectors can play an important role from an experimental point of view [V. Tamma and S. Laibacher, Phys. Rev. Lett. 114 (2015) 243601.]. This raises a fundamental question: How is the phase sensitivity affected when the photons are completely distinguishable at the detectors and therefore do not interfere? In other words, which role does multiphoton interference play in these schemes? Here, we show that for small phase values, the phase sensitivity achievable in the proposed schemes with indistinguishable photons is enhanced only by a constant factor with respect to the case of completely distinguishable photons at the detectors. Interestingly, this enhancement arises from the interference of only a polynomial number (in n) of the total n! multiphoton path amplitudes in the n-port interferometer. These results are independent of the number n of single photons and of the phase weight factors employed at each interferometer channel.