Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shotnoise limit

It was suggested in (Motes et al 2015 Phys. Rev. Lett. 114 170802) that optical networks with relatively inexpensive overheads-single photon Fock states, passive optical elements, and single photon detection-can show significant improvements over classical strategies for single-parameter estimation, when the number of modes in the network is small (n < 7). A similar case was made in (Humphreys et al 2013 Phys. Rev. Lett. 111 070403) for multi-parameter estimation, where measurement is instead made using photon-number resolving detectors. In this paper, we analytically compute the quantum Cramer-Rao bound to show these networks can have a constant-factor quantum advantage in multi-parameter estimation for even large number of modes. Additionally, we provide a simplified measurement scheme using only single-photon (on-off) detectors that is capable of approximately obtaining this sensitivity for a small number of modes.