Metrology with Unknown Detectors

The best possible precision is one of the key figures in metrology, but this is established by the exact response of the detection apparatus, which is often unknown. There exist techniques for detector characterisation, that have been introduced in the context of quantum technologies, but apply as well for ordinary classical coherence; these techniques, though, rely on intense data processing. Here we show that one can make use of the simpler approach of data fitting patterns in order to obtain an estimate of the Cram\'er-Rao bound allowed by an unknown detector, and present applications in polarimetry. Further, we show how this formalism provide a useful calculation tool in an estimation problem involving a continuous-variable quantum state, i.e. a quantum harmonic oscillator.