  Parity or quadratic spin (e.g., $J_{z}^{2}$) readouts of a Mach-Zehnder (MZ)
interferometer probed with a twin Fock input state allow to saturate the
optimal sensitivity attainable among all mode-separable states with a fixed
total number of particles, but only when the interferometer phase $\theta$ is
near zero. When more general Dicke state probes are used, the parity readout
saturates the quantum Fisher information (QFI) at $\theta=0$, whereas
better-than-standard quantum limit performance of the $J_{z}^{2}$ readout is
restricted to an $o(\sqrt{N})$ occupation imbalance. We show that a method of
moments readout of two quadratic spin observables $J_{z}^{2}$ and
$J_{+}^{2}+J_{-}^{2}$ is globally optimal for Dicke state probes, i.e., the
error saturates the QFI for all $\theta$. In the lossy setting, we derive the
time-inhomogeneous Markov process describing the effect of particle loss on
twin Fock states, showing that method of moments readout of four
at-most-quadratic spin observables is sufficient for globally optimal
estimation of $\theta$ when two or more particles are lost. The analysis
culminates in a numerical calculation of the QFI matrix for distributed MZ
interferometry on the four mode state $\vert {N\over 4},{N\over 4},{N\over
4},{N\over 4}\rangle$ and its lossy counterparts, showing that an advantage for
estimation of any linear function of the local MZ phases $\theta_{1}$,
$\theta_{2}$ (compared to independent probing of the MZ phases by two copies of
$\vert {N\over 4},{N\over 4}\rangle$) appears when more than one particle is
lost.
