Hierarchy of saturation conditions for multiparameter quantum metrology bounds

Abstract

The quantum Cramér-Rao (QCR) bound sets the ultimate local precision limit for unbiased multiparameter estimation. Unlike in the single-parameter case, however, its saturability is not generally guaranteed and is often analyzed through a hierarchy of commutativity-based conditions. Here, we resolve the logical structure of these conditions for unitary parameter-encoding transformations. We identify strict separations among the conditions, reveal previously overlooked gaps in their implications, and construct explicit counterintuitive examples that expose the boundaries among distinct classes. In particular, we show that commutativity of the parameter-encoding generators alone does not guarantee saturation of the QCR bound when realistic noise leads to mixed probe states. Our results provide a systematic classification of saturation conditions in multiparameter quantum metrology and clarify fundamental precision limits of noisy distributed quantum sensing beyond idealized pure-state regimes.