The Most Informative Cramér--Rao Bound for Quantum Two-Parameter Estimation with Pure State Probes

Abstract

Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.

Figures (2)

• Figure 1: Allowed regions for the eigenvalues $\mu$, $\nu$ of the matrix $G(\Pi)$, for sample values of $\beta$. The solid lines represent the boundaries of the allowed regions (above the solid lines is not possible, as are values less than 0 or greater than 1). The dashed curves demonstrate that the boundary of the regions is an arc of an ellipse. For $\beta=1$, the ellipse degenerates to a line.
• Figure 2: Mean squared error lower bounds for displacement sensing with grid states. The lower bounds and mean photon number are calculated numerically for different squeezing levels.