Criticality induced compatibility in multiparameter metrology.

Many-body systems near a quantum phase transition (QPT) exhibit several properties which makes them appealing for metrological purposes. Indeed, it is now well established that the divergences of the quantum Fisher information (QFI) observed near a QPT can be used to increase the precision in the estimation of a parameter. Meanwhile, when it comes to the simultaneous estimation of multiple parameters, the benefits of criticality are much harder to analyze due to possible incompatibilities arising from the Heisenberg uncertainty. This involves the use of quite convoluted quantities, as the Holevo-Cramer-Rao bound, which are generally difficult to evaluate. Here we study the quantumness ($R$), a scalar index, which provides an asymptotic bound on the compatibility of a metrological scheme. The advantage of this approach is that $R$ can be easily evaluated once the QFI and the mean Uhmlann curvature are known. Moreover, a scaling analysis of $R$ reveals that many-body criticalities generally improve the compatibility in a multi-parameter framework. We also evaluate $R$ in different representative systems, such as Ising chain and XY chain, in which we find this positive criticality effects.