We study the issue of simultaneous estimation of several phase shifts induced by commuting operators on a quantum state. We derive the optimal positive-operator-valued measure corresponding to the multiple-phase estimation. In particular, we discuss the explicit case of the optimal detection of double phase for a system of identical qutrits and generalize these results to optimal multiple-phase detection for d-dimensional quantum states.

• Received 31 January 2003

DOI:https://doi.org/10.1103/PhysRevA.67.062302