Optimal estimation of group transformations using entanglement
Giulio Chiribella, Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi · Physical Review A · 2005
We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. As a first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that $N$ copies of the state are available. In the limit of large $N$, the fidelity of the optimal estimation is shown to be $1\ensuremath{-}3∕(4N)$.