Linear optical quantum metrology with single photons --- Experimental errors, resource counting, and quantum Cram\'er-Rao bounds

Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such number-path entanglement has thought to have been resource intensive to create in the first place --- typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feed-forward, which are difficult to implement. Recently in [Phys. Rev. Lett. 114, 170802 (2015)] we showed that number-path entanglement from a BosonSampling inspired interferometer can be used to beat the shot-noise limit. In this manuscript we compare and contrast different interferometric schemes, discuss resource counting, calculate exact quantum Cram\'er-Rao bounds, and study details of experimental errors.