Quantum Measurement Bounds beyond the Uncertainty Relations

For any pair of conjugate variables, the same reasoning can establish tight bounds on the precision with which they can be estimated.

“However, the remarkable insight of [269] is that for any pair of conjugate variables this reasoning can be used to establish tight bounds on the precision with which they can be estimated.”

While the Cramer-Rao bound is asymptotically tight in the number of measurements, for moderate numbers of measurements the reasoning based on conjugate variables leads to a tighter bound.

“Interestingly, while the Cramer-Rao bound is asymptotically tight (in the number of measurements performed), for moderate numbers of measurements this line of reasoning leads to a tighter bound.”

Giovannetti, Lloyd, and Maccone derived an alternative bound on the root mean squared error for μ, equation (148), using a quantum speed limit that links fidelity to parameter difference.

“In [269] the authors derive an alternative bound on the root mean squared error for μ ∆µ κ M( H -E 0 ) ,(148) where E 0 is the minimum eigenvalue of H and κ is a constant.To establish equation ( 148) the authors use the fact that a 'quantum speed limit' in this instance, which links the fidelity, F, of the states for values of the parameter μ and µ to the difference µ -µ through the mapping U µ ,”