Simultaneous optical phase and loss estimation revisited: measurement and probe incompatibility

TL;DR

This work investigates simultaneous estimation of phase and loss in optical modes, focusing on probe and measurement incompatibilities that limit multiparameter metrology. It develops and applies quantitative tools—including the iterative see-saw optimization and Holevo-based bounds—to analyze three estimation schemes (single-mode, two-mode with reference, and two-mode with equal loss). The authors show that probe incompatibility can be overcome with carefully engineered non-Gaussian states or two-mode entanglement, while measurement incompatibility persists in all considered settings, even in the large-photon-number limit. The results provide a detailed map of when and how compatibility can be achieved and offer practical guidance for designing quantum optical sensors that approach fundamental precision limits.

Abstract

Quantum multiparameter metrology is hindered by incompatibility issues, such as finding a single probe state (probe incompatibility) and a single measurement (measurement incompatibility) optimal for all parameters. The simultaneous estimation of phase shift and loss in a single optical mode is a paradigmatic multiparameter metrological problem in which such tradeoffs are present. We consider two settings: single-mode or two-mode probes (with a reference lossless mode), and for each setting we consider either Gaussian states or arbitrary quantum states of light restricted only by a maximal number of photons allowed. We find numerically that, as the number of photons increases, there are quantum states of light for which probe incompatibility disappears both in the single- and two-mode scenarios. On the other hand, for Gaussian states, probe incompatibility is present in the single-mode case and may be removed only in the two-mode setting thanks to the entanglement with the reference mode. Finally, we provide strong arguments that the fundamental incompatibility aspect of the model is measurement incompatibility, which persists for all the scenarios considered, and unlike probe-incompatibility cannot be overcome even in the large photon number limit.

Figures (7)

• Figure 1: Three conceptually different simultaneous phase and loss estimation schemes: (1) single-mode scheme, (2) two-mode scheme, where the second mode is not affected by loss, plays the role of a reference mode, and may offer an improved performance thanks to the use of two-mode entangled states of light, (3) two-mode scheme where loss affects both modes equally.
• Figure 2: Interferommetric scheme considered. Probe preparation: Pure state $| \psi \rangle$ enters the interferometer, goes through a beamsplitter with transmissivity $\tau_{\mathrm{in}}$. Parameters encoding: In sequence, a phase shift $\varphi$ is applied, resulting in a unitary evolution $\{ \hat{a}_1 , \hat{a}_2 \} \rightarrow \{ \hat{b}'_1 , \hat{b}_2 \}$; followed by the effect of loss, modelled by an evolution $b'_1 \rightarrow \{\hat{b}_1 , \hat{c}_1 \}$. This stage is discussed in Sec.\ref{['sec_probe_incomp']}, where we investigate the fundamental quantum precision limits and the probe incompatibilities, Measurement: The output state is transformed by the action of an output beamsplitter, leading the evolution $\{ \hat{b}_1 , \hat{b}_2 \} \rightarrow \{ \hat{c}_1 , \hat{c}_2 \}$ and then subjected to a measurement strategy; photon counting and homodyne detection. This stage is discussed in Sec. \ref{['sec_measurement_incomp']}, where we investigate the precision limits including the measurement incompatibilities.
• Figure 3: Panel (a) shows the photon number variance $\langle n_1 \rangle / \Delta^2 n_1$; panel (b) the photon number average $\langle n_1 \rangle / N$ of the optimal states $| \psi^{(1)}_N \rangle$ and $| \psi^{(2)}_N \rangle$ obtained from the ISS optimization method. Panels (c) and (d) show the normalized QFI $\mathcal{F} (\rho_{\bm \lambda})$ for the optimized states $| \psi^{(1)}_N \rangle$, $| \psi^{(2)}_N \rangle$; the Gaussian states with strong displacement $| \psi^{(2)}_\alpha (\chi) \rangle$ and strong squeezing $| \psi^{(2)}_r (\chi) \rangle$. For the Gaussian states we have $\chi=0$ (dashed), $\chi=\pi/4$ (dash-dotted) and $\chi=\pi/2$ (dotted).
• Figure 4: The top panels show the photon-number distribution of the optimal single-mode state $| \psi^{(1)}_{307} \rangle$: (a) for phase estimation only, and (b) for simultaneous phase and loss estimation. The bottom panels show the photon-number distribution of the optimal two-mode state $| \psi^{(2)}_{307} \rangle$: (c) for phase estimation only, and (d) for simultaneous phase and loss estimation. In all the graphs we fix $\eta=0.1$.
• Figure 5: Panels (a) and (b) at the top show the measurement incompatibility quantifiers (MIQ), where we consider the fundamental bound $\overline{\mathcal{R}}^H(\rho_{\bm \lambda})$ (dashed), the bounds $\mathcal{R}_{\{ O_k \}}(\rho_{\bm \lambda})$ for the photon counting (dotted) and homodyne detection (dash-dot). In panel (a) these bounds are plotted in function of $\eta$ with fixed photon number $N=20$ and in panel (b) they are plotted as a function of $N$ with fixed $\eta=0.1$. For photon counting, we consider the half photon strategy with $\tau_{\mathrm{out}}=1/2$ for the phase estimation and $\tau_{\mathrm{out}}=1$ for the loss estimation. For homodyne detection, we consider the simultaneous estimation strategy with $\xi=\pi/4$ and $\tau_{\mathrm{out}}=1$ for both parameters. In the bottom panels (c) and (d), we show the measurement incompatibility for the independent estimation with the figure of merit being $1-F^{-1}_{\varphi \varphi} / \Delta^2 \varphi$ for the phase estimation (crosses) and $1-F^{-1}_{\eta \eta} / \Delta^2 \eta$ for the loss estimation (crosses), considering photon counting in panel (c) and homodyne detection in panel (d). In both graphs (c) and (d) we have $N=94$ and $\eta=0.1$. In all the fours plots, we consider the weight matrix $W=\text{diag}(F^{\text{(max)}}_\varphi, F^{\text{(max)}}_\eta)$ and the energy distributions $\bar{N}_\alpha=\bar{N} - \sqrt{\bar{N}}$, $\bar{N}_r = \sqrt{\bar{N}}$ and $\mu=0$ for Gaussian states.