For N copies of a quantum probe, each with a position X and momentum P, we can perform some form of measurement on these probes to arrive at an estimate for X and P. This is a two-parameter estimation problem.
The precision of our estimate on X and P is bounded by the quantum Cramer-Rao bound. This bound depends only on the state of the probe and is independent of the measurement scheme. For a single parameter estimation problem, the quantum Cramer-Rao bound can always be saturated. In particular, when the probe is a squeezed state, a homodyne measurement of the squeezed quadrature is optimal.
However, for a two-parameter estimation of X and Y, we cannot find a measurement scheme that saturates the Cramer-Rao bound. This implies that either: (i) the bound is not tight or (ii) the measurement scheme is not optimal. Or both.
The aim of the project is to calculate tigher bounds on the precision of the estimate.