Orders matter: tight bounds on the precision of sequential quantum estimation for multiparameter models

Abstract

In multiparameter quantum metrology, the ultimate precision of joint estimation is dictated by the Holevo Cram\'er-Rao bound. In this paper, we discuss and analyze in detail an alternative approach: the stepwise estimation strategy. In this approach, parameters are estimated sequentially, using an optimized fraction of the total available resources allocated to each step. We derive a tight and achievable precision bound for this protocol, the stepwise separable bound, and provide its closed-form analytical expression, revealing a crucial dependence on the chosen measurement ordering. We provide a rigorous comparison with the joint measurement strategy, deriving analytical conditions that determine when the stepwise approach offers superior precision. Through the analysis of several paradigmatic SU(2) unitary encoding models, we demonstrate that the stepwise strategy can indeed outperform joint measurements, particularly in scenarios characterized by non-optimal probes or models with a high degree of sloppiness. Our findings establish stepwise estimation as a powerful alternative to joint and collective measurements, proving that sequential protocols can provide a genuine metrological advantage, especially in resource-constrained or imperfect experimental settings.