Dimension matters: precision and incompatibility in multi-parameter quantum estimation models

Abstract: We study the role of probe dimension in determining the bounds of precision and the level of incompatibility in multi-parameter quantum estimation problems. In particular, we focus on the paradigmatic case of unitary encoding generated by $\mathfrak{su}(2)$ and compare precision and incompatibility in the estimation of the same parameters across representations of different dimensions. For two- and three-parameter unitary models, we prove that if the dimension of the probe is smaller than the number of parameters, then simultaneous estimation is not possible (the quantum Fisher matrix is singular). If the dimension is equal to the number of parameters, estimation is possible but the model exhibits maximal (asymptotic) incompatibility. However, for larger dimensions, there is always a state for which the incompatibility vanishes, and the symmetric Cram\'er-Rao bound is achievable. We also critically examine the performance of the so-called asymptotic incompatibility (AI) in characterizing the difference between the Holevo-Cram\'er-Rao bound and the Symmetric Logarithmic Derivative (SLD) one, showing that the AI measure alone may fail to adequately quantify this gap. Assessing the determinant of the Quantum Fisher Information Matrix (QFIM) is crucial for a precise characterization of the model's nature. Nonetheless, the AI measure still plays a relevant role since it encapsulates the non-classicality of the model in one scalar quantity rather than in a matrix form (i.e., the Uhlmann curvature).