Optimal partial estimation of multiple phases -: art. no. 012311

We derive the optimal trade-off between the estimation fidelity and the fidelity of the state after the estimation for a single copy of an unknown pure state of a d-level system belonging to a specific subset of the set of all pure states. The set of states to be considered here is formed by all pure states of a d-level system produced by d-independent phase shifts of some reference state. We also propose a measurement scheme realizing such an optimal partial estimation of multiple phases in which the trade-off between the fidelities can be controlled by the state of the ancilla.