Experimental Joint Estimation of Phase and Phase Diffusion via Deterministic Bell Measurements

TL;DR

This work tackles joint estimation of phase and phase-diffusion amplitude under phase-diffusive noise by implementing deterministic Bell measurements on two-copy quantum states encoded via a four-step quantum walk in a linear-optical setup. The approach achieves approximately a 50% precision improvement over separable measurements and nearly saturates the two-copy Lu–Wang bound in the small-diffusion regime, connecting experimental results to a clear information-theoretic limit. By examining the trade-offs with the figure of merit Tr(Q_2^{-1}F_2) and validating the framework against theory, the paper demonstrates a practical path to robust multi-parameter quantum metrology in noisy environments. It also highlights the role of collective measurements in surpassing single-copy limits and discusses the ultimate Holevo-Cramér-Rao bound as a target for future enhancements with more copies.

Abstract

Accurate phase estimation plays a pivotal role in quantum metrology, yet its precision is significantly affected by noise, particularly phase-diffusive noise caused by phase drift. To address this challenge, the joint estimation of phase and phase diffusion has emerged as an effective approach, transforming the problem into a multi-parameter estimation task. However, the incompatibility between optimal measurements for different parameters prevents single-copy measurements from reaching the fundamental precision limits defined by the quantum Cramer-Rao bound. Meanwhile, collective measurements performed on multiple identical copies can mitigate this incompatibility and thus enhance the precision of joint parameter estimation. This work experimentally demonstrates joint phase and phase-diffusion estimation using deterministic Bell measurements on a two-qubit system. A linear optical network is employed to implement both parameter encoding and deterministic Bell measurements, achieving improved estimation precision compared to any separable measurement strategy. This work proposes a new framework for phase estimation under phase-diffusive noise and underscores the substantial advantages of collective measurements in multi-parameter quantum metrology.

Figures (7)

• Figure 1: Schematic of simultaneous estimation for phase $\phi$ and phase diffusion amplitude $\Delta$ by collective measurements. A random phase shift $\tilde{\phi}\sim\mathcal{N}(\phi, 2\Delta^2)$ is applied to one arm of the interferometer to encode both the parameter $\phi$ and $\Delta$. The final quantum state, $\hat{\rho}_{\phi,\Delta}$, is generated, and collective measurements are performed on two copies of this state, $\hat{\rho}_{\phi,\Delta}^{\otimes 2}$, to jointly estimate the parameters.
• Figure 2: Experimental setup for the joint estimation of phase $\phi$ and phase diffusion $\Delta$ via Bell measurements. The setup utilizes a four-step quantum walk to prepare a two-copy parameterized quantum state and perform deterministic Bell measurements.
• Figure 3: Distribution of the intensities at four output ports as functions of $\tilde{\phi}_1,\tilde{\phi}_2 \in [0,2\pi)$. Different colors indicate varying intensity values. Each surface in the plots is generated by fitting over a grid of $100\times100$ sampled data points within this interval.
• Figure 4: The experimental results of joint estimation of phase $\phi$ and phase diffusion amplitude $\Delta$ via Bell measurements. The joint estimation precision is evaluated across ten data sets with $\Delta \in \{0.1, 0.3\}$, $\phi \in \{\pi/16, \pi/8, \pi/4, 3\pi/8, 7\pi/16\}$ and $\nu \approx 10^{4}$.
• Figure 5: Comparison of the theoretical two-copy Lu--Wang bound in Eq. (\ref{['eq:Ratio_2copy']}) with achievable precision limits in Eq. (\ref{['1.5bound']}) for a two-qubit system. (a) $\Delta=0.1$. (b) $\Delta=0.3$. The plots depict the trade-off between phase and diffusion precision. Although the Lu--Wang uncertainty relation allows for both Regions I and II, only Region II is practically attainable. The 25 experimental data points in each plot confirm this, all lying within Region II.