Quantum estimation of a two-phase spin rotation

We study the estimation of an infinitesimal rotation of a spin-j system, characterized by two unknown phases, and compare the estimation precision achievable with two different strategies. The first is a standard `joint estimation' strategy, in which a single probe state is used to estimate both parameters, while the second is a `sequential' strategy in which the two phases are estimated separately, each on half of the total number of system copies. In the limit of small angles we show that, although the joint estimation approach yields in general a better performance, the two strategies possess the same scaling of the total phase sensitivity with respect to the spin number j, namely $\Delta\Phi \simeq 1/j$. Finally, we discuss a simple estimation strategy based on spin squeezed states and spin measurements, and compare its performance with the ultimate limits to the estimation precision that we have derived above.