  We analyse the phase space representation of the optimal measurement of a
phase shift in an interferometer with equal photon loss in both its arms. In
the local phase estimation scenario with a fixed number of photons, we identify
features of the spin Wigner function that warrant sub-shot noise precision, and
discuss their sensitivity to losses. We derive the asymptotic form of an
integral kernel describing the process of photon loss in the phase space in the
limit of large photon numbers. The analytic form of this kernel allows one to
assess the ultimate precision limit for a lossy interferometer. We also provide
a general lower bound on the quantum Fisher information in terms of spin Wigner
functions.
