Abstract
Parity or quadratic spin (e.g., J_z²) 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 θ is near zero. When more general Dicke state probes are used, the parity readout saturates the quantum Fisher information (QFI) at θ=0, whereas better-than-standard quantum limit performance of the J_z² readout is restricted to an o(√N) occupation imbalance. We show that a method of moments readout of two quadratic spin observables J_z² and J₊²+J₋² is globally optimal for Dicke state probes, i.e., the error saturates the QFI for all θ. 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 θ 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 N 4,N 4,N 4,N 4 and its lossy counterparts, showing that an advantage for estimation of any linear function of the local MZ phases θ₁, θ₂ (compared to independent probing of the MZ phases by two copies of N 4,N 4) appears when more than one particle is lost.