Holevo界的精确刻画：基于量子Fisher信息族

中文摘要 AI 辅助

量子Cramér-Rao界通过对称对数导数量子Fisher信息（SLD QFI）约束参数估计的精度。对于正则单参数估计模型，该界是渐近可达的，但在多参数情形下通常不可达，因为不同参数的最优测量可能不兼容。对于多参数估计，若允许集体测量，相应的渐近可达精度极限即为Holevo界，其标准表述是对辅助厄米算子的优化。我们通过一族在SLD与右对数导数（RLD）QFI之间插值的QFI，给出了Holevo界的精确刻画，从而弥合了这一表述上的表面差异。具体而言，对于任何局部可辨识的有限维估计问题，我们证明标准变分定义中Holevo界的算子可行区域与整个QFI族对应的Cramér-Rao型矩阵约束所定义的区域的交集一致。该刻画同样适用于秩亏态，且无需预先选择权重矩阵。因此，通过在所得公共可行区域上最小化相应的加权代价，即可恢复传统的依赖于权重的Holevo界。

英文摘要

The quantum Cramér-Rao bound constrains the precision of parameter estimation through the symmetric logarithmic derivative quantum Fisher information (SLD QFI). It is asymptotically achievable for regular single-parameter estimation models, but generally not in the multiparameter setting, where optimal measurements for different parameters may be incompatible. For multiparameter estimation, allowing collective measurements, the corresponding asymptotically achievable precision limit is the Holevo bound, whose standard formulation is an optimization over auxiliary Hermitian operators. We bridge this apparent difference in formulation by giving an exact characterization of the Holevo bound in terms of a family of QFIs interpolating between the SLD and right logarithmic derivative (RLD) QFIs. Specifically, for any locally identifiable finite-dimensional estimation problem, we prove that the operator-feasible region in the standard variational definition of the Holevo bound coincides with the intersection of the regions defined by the corresponding Cramér-Rao-type matrix constraints for the entire QFI family. This characterization also applies to rank-deficient states and does not require any prior choice of weight matrix. Consequently, the conventional weight-dependent Holevo bound is recovered by minimizing the corresponding weighted cost over the resulting common feasible region.

发表机构

• University of Waterloo（滑铁卢大学）
• Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
• Kyushu University（九州大学）
• JST, FOREST（日本科学技术振兴机构，FOREST项目）

机构由 AI 辅助整理，请以论文原文为准。

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8 pages, 1 figure