Heisenberg-limited estimation robust to detector inefficiency in a multi-parameter Mach-Zehnder network with squeezed light

We propose a multi-parameter quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an anti-squeezing 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 to detect the value of the sum $\beta =\frac{1}{2}({\phi }_1+{\phi }_2)+{\theta }_{\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{o}\mathrm{u}\mathrm{t}}$, of the relative phase ${\theta }_{\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ between the two squeezers, and the average of the phase delays ${\phi }_1,{\phi }_2$ in the two arms of the interferometer. The detection sensitivity scales at the Heisenberg limit and, remarkably, is robust to the detector inefficiency.