Measurement compatibility in multiparameter quantum interferometry

TL;DR

The paper addresses the challenge of compatibility in multiparameter quantum metrology by analyzing joint estimation of a phase with nuisance parameters (loss and phase diffusion) in a Mach-Zehnder interferometer using HB and gHB states. It introduces and evaluates practical compatibility figures of merit based on the Holevo bound, quantum CRB, and NHCRB for double-homodyne and photon-counting measurements, highlighting how parameter weighting via a diagonal weight matrix $W(y)$ shapes compatibility. Key findings show strong intrinsic incompatibility between phase and loss (and weaker between phase and diffusion), with measurement compatibility strongly influenced by the measurement choice, probe state, and the loss level; photon counting generally yields better compatibility than double homodyne in these multiparameter scenarios, and low loss improves compatibility overall. The work provides practical guidance for designing quantum sensors by focusing on compatibility as a distinct performance metric, offering insights into when separable measurements can closely approach fundamental bounds and how to choose weights to optimize joint estimation in realistic settings.

Abstract

The Cramér-Rao bound captures completely the performance of single-parameter quantum sensors. On the other hand, its extension to multiple parameters demands more caution. Different aspects need to be captured at once, including, critically, compatibility. In this article we consider compatibility in quantum interferometry for an important class of probe states, measured by double homodyne or photon counters, standard benchmarks for these applications. We include the presence of loss and phase diffusion in the estimation of a phase. Our results illustrate how different weighting of the precision on individual parameters affects their compatibility, adding to the list of considerations for quantum multiparameter estimation.

Figures (10)

• Figure 1: Plot of measurement compatibility measure $r^C_H$ and fundamental compatibility measure $r_{BG}$ versus the weights $y$ for the joint-estimation of phase and losses in one arm of the MZI. HB states with $N=2,4,6,8$ (panel (a)) and gHB states with $N=2,3,4,5,6,7,8$ (panel (b)), and double homodyne measurement are considered. In panel (b) the plotting of $r_{BG}$ employs concentric circles of varying colors to represent the near-perfect overlap of compatibilities for different values of $N$, as indicated by their common center. Note that in the following plots, such overlapping points are depicted in this manner. The curves are plotted at $\eta_a=0.5$ and $\Delta=0$. For HB states, the measurement compatibility decreases significantly as $N$ increases, while the fundamental compatibility decreases only slightly as $N$ increases.
• Figure 2: Plot of measurement compatibility measure $r^C_H$ versus the weights $y$ for photon counting for the joint-estimation of phase and loss in one arm of the MZI. HB states with $N=2,4,6,8$ (panel(a)) and gHB states with $N=2,3,4,5,6,7,8$ (panel(b)) are considered. Note that the measurement compatibility remains relatively high compared to that of the double homodyne measurement, with the highest compatibility observed at $N = 2$. The other known parameters are taken at the same values as those used in the assessment of the double homodyne measurement.
• Figure 3: Plot of measurement compatibility measure $r^C_H$ and fundamental compatibility measure $r_{BG}$ versus the weights $y$ for the joint-estimation of phase and losses in one arm of the MZI with a known amount of loss on the reference arm. HB states with $N=2,4,6,8$ (panel (a)) and gHB states with $N=2,3,4,5,6,7$ (panel (b)), and double homodyne measurement are considered. The curves are plotted at $\eta_a=0.5$, $\eta_b=0.5$, and $\Delta=0$. Note that for HB and gHB states, the measurement compatibility exhibits an intersection at $y=0.6$, causing its behavior with respect to $N$ to change on either side of this point. In contrast, the fundamental compatibility increases slightly as $N$ increases.
• Figure 4: Plot of measurement compatibility measure $r^C_H$ versus the weights $y$ for photon counting for the joint-estimation of phase and losses in one arm of the MZI with a known amount of loss on the reference arm. HB states with $N=2,4,6,8$ (panel(a)) and gHB states with $N=2,3,4,5,6,7$ (panel(b)) are considered. Notably, the measurement compatibility remains high in this case as well, exceeding that of the double homodyne measurement, with optimal compatibility occurring at $N=2$. However, the compatibility strongly decreases as $N$ increases. The other known parameters are taken at the same values as those used in the assessment of the double homodyne measurement.
• Figure 5: Joint-estimation of phase and loss: Three-dimensional plots of double homodyne measurement compatibility $r^C_H$ (panel (a)) and fundamental compatibility $r_{BG}$ (panel (b)) seen as a function of the losses, $\eta_a$ and $\eta_b$, in each arm with equal parameter weights i.e., at $y=0.5$. The probe state considered here is the gHB state $\ket{\Psi_{\mathrm{gHB}}(N,0)}$ with $N=2$ (blue dots) and $N=4$ (red dots) which create smooth surfaces. Note that the points closer to low values of losses exhibit high measurement and fundamental compatibilities. Also, note that the fundamental compatibility does not vary with respect to the chosen values of $N$ for almost all pairs of values of losses resulting in overlapping points.