Title: Weight-dependent and weight-independent measures of quantum incompatibility in multiparameter estimation Show Chinese title

Title: 参数估计中量子不相容性的权重相关和权重无关度量

Abstract: Multiparameter quantum estimation faces a fundamental challenge due to the inherent incompatibility of optimal measurements for different parameters, a direct consequence of quantum non-commutativity. This incompatibility is quantified by the gap between the symmetric logarithmic derivative (SLD) quantum Cram\'er-Rao bound, which is not always attainable, and the asymptotically achievable Holevo bound. This work provides a comprehensive analysis of this gap by introducing and contrasting two scalar measures. The first is the weight-independent quantumness measure $R$, which captures the intrinsic incompatibility of the estimation model. The second is a tighter, weight-dependent measure $T[W]$ which explicitly incorporates the cost matrix $W$ assigning relative importance to different parameters. We establish a hierarchy of bounds based on these two measures and derive necessary and sufficient conditions for their saturation. Through analytical and numerical studies of tunable qubit and qutrit models with SU(2) unitary encoding, we demonstrate that the weight-dependent bound $C_{T}[W]$ often provides a significantly tighter approximation to the Holevo bound $C_{H}[W]$ than the $R$-dependent bound, especially in higher-dimensional systems. We also develop an approach based on $C_{T}[W]$ to compute the Holevo bound $C_{H}[W]$ analytically. Our results highlight the critical role of the weight matrix's structure in determining the precision limits of multiparameter quantum metrology. Show Chinese abstract

Abstract: 多参数量子估计面临的根本挑战源于不同参数最优测量的固有不相容性，这是量子非对易性的直接结果。 这种不相容性由对称对数导数（SLD）量子Cramér-Rao界（该界并不总是可达到的）与渐近可达到的Holevo界之间的差距来量化。 本工作通过引入并对比两种标量度量，对这一差距进行了全面分析。 第一种是与权重无关的量子性度量$R$，它捕捉了估计模型的内在不相容性。 第二种是一个更紧致的、与权重相关的度量$T[W]$，它显式地结合了成本矩阵$W$，用于分配不同参数的相对重要性。 我们基于这两种度量建立了界限层次结构，并推导了它们饱和的必要和充分条件。 通过对手动调节的具有SU(2)单位编码的量子比特和量子三态模型进行分析和数值研究，我们证明了与权重相关的界限$C_{T}[W]$通常比与$R$相关的界限更紧密地逼近Holevo界限$C_{H}[W]$，尤其是在高维系统中。 我们还开发了一种基于$C_{T}[W]$的方法，以解析方式计算 Holevo 界$C_{H}[W]$。 我们的结果突出了权重矩阵的结构在确定多参数量子计量精度极限中的关键作用。