Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss

We present a practical scheme for measuring completely unknown phases with a precision beyond the shot-noise limit even in the presence of loss. Our scheme consists of sending a sequence of unentangled particles and NOON states through an interferometer and analyzing the measurement outcomes using a Bayesian analysis. We compare our results with two recent schemes [L. Pezze and A. Smerzi, Europhys. Lett. 78, 30004 (2007); B. L. Higgins et al., Nature (London) 450, 393 (2007)] that are closely related but operate in the lossless regime. We show that our technique outperforms the previous schemes when even a modest amount of loss is present and so may prove to be a valuable technique for making precision measurements beyond the classical limit in a range of practical scenarios.