Multiparameter quantum metrology in the Heisenberg Limit regime: many repetition scenario vs. full optimization

We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms -- the one, where the measurement is repeated many times (so the Cram\'er-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 parameter 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 Cram\'er-Rao formalism may be overly pessimistic in this matter -- the minimax approach reveals the superiority of measuring all the parameters jointly whereas the Cram\'er-Rao approach indicates lack of such an advantage.