Multiparameter quantum metrology at Heisenberg scaling for an arbitrary two-channel linear interferometer with squeezed light

TL;DR

This work tackles the problem of estimating all four real parameters of an arbitrary two-channel interferometer $U(2)$ with Heisenberg-limited precision. By deriving the quantum Fisher information matrix for Gaussian probes, it analyzes two practically feasible inputs—a two-mode squeezed state (TMSS) and two single-mode squeezed states (SMSS)—and shows that Heisenberg scaling, $ ext{ΔΦ} \,\propto\, 1/N$, can be achieved for all parameters when appropriate displacement and phase-matching are employed. The authors provide explicit QCRB expressions for each parameter and demonstrate a unifying route—via a 50:50 beamsplitter rotation—that maps SMSS results to TMSS results, facilitating experimental design. This framework advances multiparameter quantum metrology in optical networks, offering concrete benchmarks and guidance for distributed quantum sensing using Gaussian resources and highlighting the essential role of displacement in achieving full parameter saturation.

Abstract

We present a framework for simultaneously estimating all four real parameters of a general two-channel unitary U(2) with Heisenberg-scaling precision. We derive analytical expressions for the quantum Fisher information matrix and show that all parameters attain the 1/N scaling in the precision by using experimentally feasible Gaussian probes such as two-mode squeezed states or two single-mode squeezed states. Our results extend multiparameter metrology to its most general two-mode setting and establish concrete design principles for experimental implementations of Heisenberg-scaling, multi-parameter optical interferometry with experimentally feasible resources. It not only sheds light on the fundamental interface between quantum interference of squeezed light and quantum metrological advantage in multiparameter estimation, but it also provides an important stepstone towards the development of a wide range of quantum technologies based on distributed quantum metrology in arbitrary optical networks.

Figures (5)

• Figure 1: Schematic of the estimation setup. A TMSS, in Eq. \ref{['eq:INPUTstate']} is injected into an unknown two-channel unitary $U$.
• Figure 2: Log--log plot of the scalar precision bound $\mathrm{Tr}[\mathcal{H}^{-1}]$ (sum of the QCRBs), as a function of the total average photon number $N$. The scalar bound $\mathrm{Tr}[\mathcal{H}^{-1}]$ scales as $1/N^2$, demonstrating Heisenberg-scaling precision in all parameters. The inset plot shows the dependence of $\mathrm{Tr}[\mathcal{H}^{-1}]$ on the parameter $\omega$ at fixed $N=5$, illustrating the parameter dependency of the total precision bound. Here, we choose the optimal value $\omega=\pi/4$ for the main plot.
• Figure 3: A $50{:}50$ BS transformation $U_{\rm BS}$ is applied to a general unitary operation. While $U_{\rm BS}$ is fixed, the overall unitary remains arbitrary, preserving the generality of the transformation.
• Figure 4: Plot of the scalar bound $\mathrm{Tr}[\mathcal{H}_{\rm TMSS}^{-1}]$ vs displacement weight $\tau$ by considering $N_c=N_s=N/2$. Here, $N=5$ and $\omega=\pi/4$.
• Figure 5: Eigenvalues of the QFIM $\mathcal{H}_{\rm SMSS}^\Gamma$ as a function of the squeezing weight $\eta\in[0,1]$. The squeezing weight $\eta$ determines how the total mean number of squeezed photons $N_s$ is distributed between the two input modes, with $N_{s_1}=\eta N_s$ and $N_{s_2}=(1-\eta)N_s$. The plots illustrate that the first eigenvalue (in black) has a vanishing $O(N_s^2)$-coefficient for all $\eta$; two eigenvalues are maximized at $\eta=1/2$, corresponding to equal squeezing in both modes. Here $N_s=3$, $\phi=0$, and $\omega=\pi/4$.