Twirling Projective Measurements in Quantum Estimation

Theoretical investigations of quantum estimation have clarified the achievable bound of the estimation error. However, the optimal estimation requires adaptive or collective schemes, where the preparation cost of experiments is too much or the quantum measurements are beyond the current technology. On the other hand, improving the accuracy of the parameter estimation is often critical in some of quantum computation experiments. Motivated with this, in the simplest case we will discuss how to improve current measurements at a reasonable cost by using the mathematical tool of asymptotic risk expansion. When estimating the expectations of two noncommutative observables, the standard method is to make a large number of repetitions of both the projective measurements corresponding to the two observables. We show that certain choices of four kinds of projective measurements, which we call twirling projective measurements, yield a smaller error when the total repetition number is fixed. This example demonstrates that an elaborate combination of projective measurements can improve accuracy without the need for advanced technology such as quantum entanglement.