Saturating the Quantum Cramér--Rao Bound in Prioritised Parameter Estimation

Abstract

Measurement incompatibility is a cornerstone of quantum mechanics. In the context of estimating multiple parameters of a quantum system, this manifests as a fundamental trade-off between the precisions with which different parameters can be estimated. Often, a parameter can be optimally measured, but at the cost of gaining no information about incompatible parameters. Here, we report that there are systems where one parameter's information can be maximised while not completely losing information about the other parameters. In doing so, we find attainable trade-off relations for quantum parameter estimation with a structure that is different to typical Heisenberg-type trade-offs. We demonstrate our findings by implementing an optimal entangling measurement on a Quantinuum trapped-ion quantum computer.

Figures (3)

• Figure 1: Trade-off curves for qubit phase--dephasing estimation, which define the boundary of the accessible region for mean squared errors. The blue, green, purple, and orange lines are the trade-off curves for one-, two-, three-, and four-copy collective measurements, respectively. The mean squared errors are scaled by respective quantum Fisher informations, so that a value of 1 represents optimal estimation. The boundary of the shaded region is the quantum Cramér--Rao bound and is attainable in the infinite-copy limit for this system. The unfilled circles on the curves denote the minimum mean squared errors for phase-prioritised estimation. The black (grey) points are the mean squared errors from the trapped-ion experiment (emulated experiment), with error bars denoting one standard deviation obtained via bootstrapping. The true values of the parameters are $\phi=0$ and $\Delta=1/2$, but these values do not affect the features of the trade-off. Inset: magnified plot centred around the optimal two-copy phase-prioritied mean squared errors.
• Figure 2: Estimation error trade-off curves for displacement sensing with Fock states $\ket{n}$ (curves for $\ket{1}$, $\ket{2}$, $\ket{3}$ in order towards bottom-left corner). The mean squared errors are normalised by the respective quantum Fisher informations (which are different for each $n$) so that the axes represent the closeness to optimal for each parameter and the trade-offs can be compared. The unfilled circles denote the minimum achievable prioritised mean squared errors. The boundary of the shaded region is the quantum Cramér--Rao bound.
• Figure 3: Sum of mean squared errors $V(\hat{\phi})+V(\hat{\Delta})$ of $\phi$-prioritised measurements (blue, filled circles) with different numbers of copies of the probe state. The orange unfilled circles are the Nagaoka--Cramér--Rao bound, a lower bound for all measurements, and the boundary of the pink is the quantum Cramér--Rao bound which here is attainable in the asymptotic limit of the number of copies. The mean squared errors are scaled by the number of probe state copies used. The true values of the parameters are $\phi=0$ and $\Delta=1/2$.

Theorems & Definitions (1)

• Theorem 1