Linear combination estimator of multiple-outcome detections with discrete measurement outcomes

Finding the optimal estimator capable of saturating the Cram\'er-Rao bound is important for quantum parameter estimation. The maximum likelihood estimator (MLE) is well known to be asymptotically optimal, but it is usually nonanalytic and hence requires extensive numerical computation. On the other hand, the inversion estimator is widely used in experiments due to its simplicity, but it is usually suboptimal. Here we derive an estimator based on the linear combination of multiple inversion estimators associated with the occurrence probability of different measurement outcomes. This linear combination estimator shares the asymptotic optimality of the MLE and the simplicity of the inversion estimator and is applicable to general multi-outcome measurements, irrespective of the specific system and the noise in the system. We demonstrate this estimator for the intensity-difference measurement to a six-photon twin-Fock input state.