Phase estimation in the presence of phase diffusion: the qubit case

In this paper we address the estimation of the phase shift imposed on a qubit in the presence of phase diffusion. We evaluate the ultimate quantum limits to precision and determine optimal probes and measurements achieving those bounds. We also analyse in detail the performances of spin measurements and have found that although the corresponding Fisher information depends on the unknown value of the phase shift, we may still achieve the ultimate bound using a two-step adaptive method.