Bayesian stepwise estimation of qubit rotations

Abstract

This work investigates Bayesian stepwise estimation (Se) for measuring the two parameters of a unitary qubit rotation. While asymptotic analysis predicts a precision advantage for SE over joint estimation (JE) in regimes where the quantum Fisher information matrix is near-singular ("sloppy" models), we demonstrate that this advantage is mitigated within a practical Bayesian framework with limited resources. We experimentally implement a SE protocol using polarisation qubits, achieving uncertainties close to the classical Van Trees bounds. However, comparing the total error to the ultimate quantum Van Trees bound for JE reveals that averaging over prior distributions erases the asymptotic SE advantage. Nevertheless, the stepwise strategy retains a significant practical benefit as it operates effectively with simple, fixed measurements, whereas saturating the JE bound typically requires complex, parameter-dependent operations.

Figures (4)

• Figure 1: Performance of the SE, compared to JE, as captured by the quantity $r_\beta$, as well as its optimised version $r$. The theoretical predictions are reported as a function of the axis $\theta$ and of the phase shift $\gamma$ as the gold surface. The blue surface corresponds to the limit 1. When resources are equally allocated ($\beta =1/2$), an advantage, identified by $r_\beta<1$, can be highlighted, depending on the estimation sequence. This is even more relevant when optimising over the allocation of resources.
• Figure 2: Raw data for the sequential estimation. The counts correspond to measurements along the $Z$ direction: the blue dots correpond to the outcome 0, the golden squares to the outcome 1. The dashed lines are theoretical predictions for $p_0$ and $p_1$ for $\gamma = \pi/9$ as a guide to the eye.
• Figure 3: Stepwise two-parameter estimation with prior distributions of width $\tau=5^\circ$. Top left: estimation of the shift $\hat{\gamma}$ as a function of $\theta$: blue points are for the experimental results, error bars correspond to one standard deviation, while the shaded region indicate the apriori distribution for $\gamma$ (within two stardard deviations). Top right: standard deviation $\Delta\gamma$ as a function of $\theta$. The blue points are the experimental results, the solid line corresponds to the stepwise classical Van Trees limit. Bottom left: results for the estimation of axis $\hat{\theta}$ as a function of the actual angle $\theta$: the dashed line is the diagonal. Bottom right: standard deviation $\Delta\theta$ as a function of $\theta$. The blue points are the experimental results, the solid line corresponds to the stepwise Van Trees limit.
• Figure 4: Comparison of the total error $\Sigma$ for different widths of the prior: $\tau =2.5^\circ$ (upper),$\tau =5^\circ$ (middle), $\tau =10^\circ$ (lower). In all panels, the solid line corresponds to the Van Trees limit for JE. Blue points correspond to estimating $\gamma$ first, yellow points to estimating $\theta$ first.