We report lowest-order series expansions for primary matrix functions of quantum states based on a perturbation theory for functions of linear operators. Our theory enables efficient computation of functions of perturbed quantum states that assume only knowledge of the eigenspectrum of the zeroth order state and the density matrix elements of a zero-trace, Hermitian perturbation operator, not requiring analysis of the full state or the perturbation term. We develop theories for two classes of quantum state perturbations, perturbations that preserve the vector support of the original state and perturbations that extend the support beyond the support of the original state. We highlight relevant features of the two situations, in particular the fact that functions and measures of perturbed quantum states with preserved support can be elegantly and efficiently represented using Fr\'echet derivatives. We apply our perturbation theories to find simple expressions for four of the most important quantities in quantum information theory that are commonly computed from density matrices: the Von Neumann entropy, the quantum relative entropy, the quantum Chernoff bound, and the quantum fidelity.

翻译：我们基于线性算符函数的微扰理论，报道了量子态初等矩阵函数的低阶级数展开。该理论能够高效计算受扰量子态的函数，仅需已知零阶状态的本征谱和零迹厄米微扰算符的密度矩阵元，无需分析完整状态或微扰项。我们针对两类量子态微扰发展了理论：保持原始状态向量支撑的微扰，以及将支撑扩展到原始状态支撑之外的微扰。我们重点阐述了这两种情况的相关特征，特别是保持支撑的受扰量子态的函数和测度可利用Fréchet导数优雅高效地表示。将该微扰理论应用于量子信息论中四个最常由密度矩阵计算的重要量——冯·诺依曼熵、量子相对熵、量子切尔诺夫界和量子保真度，我们得到了简洁的表达式。