Adaptive phase estimation with two-mode squeezed-vacuum and parity measurement

A proposed phase-estimation protocol based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that the Cram\'er-Rao sensitivity is sub-Heisenberg [P. M. Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)]. However, these measurements are problematic, making it unclear if this sensitivity can be obtained with a finite number of measurements. This sensitivity is only for a phase near zero, and in this region there is a problem with ambiguity because measurements cannot distinguish the sign of the phase. Here, we consider a finite number of parity measurements and show that an adaptive technique gives a highly accurate phase estimate regardless of the phase. We show that the Heisenberg limit is reachable, where the number of trials needed for mean photon number $\overline{n}=1$ is approximately 100. We show that the Cram\'er-Rao sensitivity can be achieved approximately, and the estimation is unambiguous in the interval ($\ensuremath{-}\ensuremath{\pi}/2,\ensuremath{\pi}/2$).