Quantum Metrology under Coarse-Grained Measurement

TL;DR

This work analyzes quantum metrology under coarse-grained measurements in homodyne detection, using a Mach-Zehnder interferometer fed by a squeezed vacuum and a coherent state. It derives the Fisher information for coarse-grained quadrature measurements, showing that a large fraction of the ideal FI is preserved (quantified by a bin-dependent factor $f_M$) and that Heisenberg scaling can persist under coarse graining. To saturate the Cramér-Rao bound in this setting, the authors introduce a method of moments estimator with optimally weighted bin observables, providing explicit forms for the optimal weights via the pseudoinverse of the covariance matrix; the FI for coarse graining satisfies $F=\sum_k \frac{1}{P_k}\left(\frac{\partial P_k}{\partial\varphi}\right)^2$. The experiments demonstrate quantum-enhanced phase estimation with 3.8 dB squeezing in the ideal case and show 1.2 dB improvement with only two bins, with performance rapidly approaching the ideal as the bin count increases, validating the practicality of quantum metrology under severe measurement imperfections.

Abstract

While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cramér-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.

Figures (7)

• Figure 1: (a) Phase estimation with coherent state $\left\vert{\alpha}\right\rangle$ and squeezed state $\left\vert{r}\right\rangle$ input, first introduced in Ref Caves1981. Homodyne detection measures $\hat{p}$ quadrature at the dark port. $\alpha$: coherent-state amplitude, $r$: squeezing parameter, HD: homodyne detection, LO: local oscillator. (b) Probability density function of a quadrature outcome under ideal fine-grained measurement, given by Eq. (\ref{['eq:Gprob']}). (c) Detection probability in $k$-th bin under coarse-grained measurements ($M$: bin number), given by Eq. (\ref{['eq:meano']}). Both plots show an example of $\alpha=10$ and $r=0.3$ within the quadrature range of $\left\vert p\right\vert \leq 4$. The green and purple curves show the distributions at $\varphi=0$ and $\varphi=2^{\circ}$, respectively. The bin boundary is denoted by $b_{k}$.
• Figure 2: (a) Fisher information ratio $f_M=F_M(0)/F_{\text{id}}(0)$ in Eq. (\ref{['eq:FI/FIx']}) as a function of the bin number $M$. In the analysis, we consider the total range of the four-standard-deviation of a quadrature distribution, as explained in the main text. The purple and green dots represent the Fisher information ratio under the equal-sized bins ($f^{\text{eq}}_M$) and optimal bins ($f^{\text{opt}}_M$), respectively. (b) Phase estimation error as increasing the total average photon number $\langle\hat{n}_{\text{tot}}\rangle$. We consider the equal distribution of the total photons between the squeezed vacuum and the coherent state, which gives a Heisenberg scaling in the case of ideal homodyne detection Pezze2008. HL and SQL represent the Heisenberg limit ($\Delta\varphi = \langle\hat{n}_{\text{tot}}\rangle^{-1}$, the black dashed line) and the standard quantum limit ($\Delta\varphi = \langle\hat{n}_{\text{tot}}\rangle^{-1/2}$, the red solid line), respectively. Coarse-grained measurement for a bin number $M$ exhibits only a vertical shift from the ideal homodyne detection, thus exhibiting the Heisenberg scaling as well.
• Figure 3: Examples of optimal weights for $M=5$ and $\varphi = 0$ under coarse-grained measurement with (a) equal-sized bins and (b) optimal bins, obtained from Eq. (\ref{['eq:optw']}). The weight for $k^{\text{th}}$ bin (between $b_{k}$ and $b_{k+1}$) is denoted by $w_{k}$.
• Figure 4: Experimental setup for measuring the interferometric phase $\varphi$. A squeezed vacuum state (from OPO) and a coherent state (in probe) are used as the input of the interferometer. The output from the dark port of the interferometer is detected by homodyne detector (HD). Quadrature outcomes are obtained with an oscilloscope after demodulating the HD signals at the frequency of 2 MHz. Coarse-graining effects are investigated by binning the quadrature outcomes in various conditions. SHG: second-harmonic generation, OPO: optical parametric oscillator, $\text{BS}_{\text{in(out)}}$: input(output) beam splitter, $\text{PD}_{1(2)}$: photodiode 1(2), LO: local oscillator.
• Figure 5: Experimental determination of the optimal weights and the associated calibration functions. (a) Equal-sized binning and (b) optimal binning, where the panels from left to right correspond to bin numbers $M = 2$ to $M = 7$. For both (a) and (b), the first row shows the detection probabilities $P_k$ ($k=1,\dots,M$, with bins indexed from left to right bars) at the phase $\varphi_{0}=-0.02^{\circ}$; the second shows the partial derivatives of expectation values of multiple observables $\partial_{\varphi} \langle\hat{\mathbf{o}}\rangle_\varphi$, with the increasing bin index illustrated by a gradual color change from blue to green; the third shows the covariance matrices $\boldsymbol{\Gamma}_{\varphi_{0}}$; the fourth shows the optimal weight vectors $\textbf{w}_{\varphi_{0}}$; and the last row shows the calibration functions $g(\varphi)$.