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

We propose a multi-parameter quantum metrological protocol based on Mach-Zehnder interferometer with squeezed vacuum input state and an anti-squeezing operation at one of its output channels. A simple intuitive geometrical picture the evolution is provided by marginal Wigner functions each channel. The allows to detect value sum $\beta=\frac{1}{2}(\varphi_1+\varphi_2)+\theta_\mathrm{in}-\theta_\mathrm{out}$, relative phase $\theta_\mathrm{in}-\theta_\mathrm{out}$ between two squeezers, average delays $\varphi_1,\varphi_2$ in arms interferometer. detection sensitivity scales Heisenberg limit and, remarkably, robust detector inefficiency.