Heisenberg-limited estimation robust to photon losses in a Mach-Zehnder network with squeezed light

We propose a quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an antisqueezing operation at one of its output channels. A simple and intuitive geometrical picture of the state evolution is provided by the marginal Wigner functions of the state at each interferometer output channel. The protocol allows us to detect the values of the sum β=12(φ1+φ2)+θin-θout, of the relative phase θin-θout between the two squeezers, and of the average of the phase delays φ1 and φ2 in the two arms of the interferometer. The detection sensitivity scales at the Heisenberg limit and, remarkably, is robust not only to detector inefficiencies but also to any photon losses occurring before the antisqueezing operation. Interestingly, we demonstrate that in the latter case an increase of sensitivity can even occur by increasing the losses in a suitable range.