The problem of estimating a generic phase-shift experienced by a quantum state is addressed for a generally degenerate phase shift operator. The optimal positive operator-valued measure is derived along with the optimal input state. Two relevant examples are analyzed: i) a multi-mode phase shift operator for multipath interferometry; ii) the two mode heterodyne phase detection.
@article{arxiv.quant-ph/9812034,
title = {On the general problem of quantum phase estimation},
author = {G. M. D'Ariano and C. Macchiavello and M. F. Sacchi},
journal= {arXiv preprint arXiv:quant-ph/9812034},
year = {2009}
}