Estimation of shift parameters of a quantum state

Estimation of shift parameters such as arrival time, phase, angle of rotation, position of a non-relativistic or relativistic particle is considered. An approach from the point of view of quantum estimation theory enables to give a proper definition of the time observable and the position observables of a massless relativistic particle, i.e. observables to which there do not correspond self-adjoint operators. Some new inequalities for estimates of shift parameters are obtained; in particular a rigorous uncertainty relation for coordinates of the photon is established.     

On the general problem of mean estimation

Mean estimation for a random function is considered in its most general context, including both the case of infinite dimensional mean and exact estimation. The existence theorem for the best linear unbiased estimates is established. The general definition of the pseudobest estimate is given; necessary and sufficient conditions for the efficiency of the pseudobest estimates are obtained. As an application the problem of the periodic trend estimation is considered.