Protocol for global multiphase estimation

Global estimation strategies allow one to extract information on a phase or a set of phases without any prior knowledge, which is required for local estimation strategies. We devise a global multiphase protocol based on Holevo's estimation theory and apply it to the case of digital estimation, i.e., we estimate the phases in terms of the mutual information between them and the corresponding estimators. We test the protocol in two particular cases, i.e., the single-phase and the double-phase estimation. In the single-phase scenario, the protocol encompasses two specific known optimal strategies. We extend them to the simultaneous estimation of two phases and evaluate their performance. Then we retrieve the ultimate digital bound on precision when a generic number of phases is simultaneously estimated. This bound is again expressed in terms of mutual information and is general for any digital multiphase estimation protocol. We show that in the multiphase strategy there is only a constant quantum advantage with respect to a sequence of independent single-phase estimations. This extends a recent similar result, which settled a controversy on the search for the multiphase enhancement.