system, characterised 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
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 ΔΦ≃ 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.