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 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 Cramér-Rao bound is achievable. We also critically examine the performance of the so-called asymptotic incompatibility (AI) in characterising the difference between the Holevo-Cramér-Rao bound and the Symmetric Logarithmic Derivative one, showing that the AI measure alone may fail to adequately quantify this gap. Assessing the determinant of the Quantum Fisher Information Matrix is crucial for a precise characterisation 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).

中文翻译：

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维度很重要：多参数量子估计模型的精度和不兼容性


我们研究探针维数在确定多参数量子估计问题中的精度范围和不兼容性水平方面的作用。特别是，我们关注由 su(2) 生成的单一编码的范例情况，并比较跨不同维度表示的相同参数估计的精度和不兼容性。对于二参数和三参数酉模型，我们证明如果探针的维数小于参数的数量，则不可能同时估计（量子费希尔矩阵是奇异的）。如果维度等于参数数量，则可以进行估计，但模型表现出最大（渐近）不兼容性。然而，对于更大的维度，总是存在一种不相容性消失的状态，并且可以实现对称的 Cramér-Rao 界。我们还批判性地检查了所谓的渐近不相容性 (AI) 在表征 Holevo-Cramér-Rao 界与对称对数导数之间的差异时的表现，表明仅靠 AI 测量可能无法充分量化这一差距。评估量子费希尔信息矩阵的决定因素对于精确表征模型的性质至关重要。尽管如此，人工智能测量仍然发挥着相关作用，因为它将模型的非经典性封装在一个标量而不是矩阵形式（即乌尔曼曲率）中。