Multiparameter quantum metrology in the Heisenberg limit regime: Many-repetition scenario versus full optimization

We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms: one where the measurement is repeated many times (so the Cramer-Rao bound is guaranteed to be asymptotically saturable). and the second one where all the resources are allocated into one experimental realization (analyzed with the mimimax approach). We investigate the potential advantage of measuring all the parameters simultaneously compared to estimating them individually, while spending the same total amount of resources. We show that in general the existence of such an advantage, its magnitude, and conditions under which it occurs depends on which of the two paradigms has been chosen. In particular, for the problem of magnetic field sensing using N entangled spin 1/2, we show that the predictions based purely on the Cramer-Rao formalism may be overly pessimistic in this matter: the minimax approach reveals the superiority of measuring all the parameters jointly whereas the Cramer-Rao approach indicates lack of such an advantage.

Automated summary

This article discusses the application of the Heisenberg limit in multiparameter metrology within two different paradigms. It investigates the potential advantage of measuring all parameters simultaneously compared to estimating them individually, showing that the magnitude of this advantage and the conditions under which it occurs depend on the chosen paradigm.