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.

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量子信息的微扰理论

我们报告了基于线性算子函数的微扰理论的量子态主矩阵函数的最低阶级数展开。我们的理论能够有效计算扰动量子态的函数，这些函数假设只知道零阶状态的本征谱和零迹 Hermitian 扰动算子的密度矩阵元素，而不需要分析全态或扰动项。我们开发了两类量子态扰动的理论，即保留原始状态向量支持的扰动和将支持扩展到原始状态支持之外的扰动。我们强调了这两种情况的相关特征，特别是具有保留支持的扰动量子态的函数和度量可以使用 Fr\'echet 导数优雅有效地表示这一事实。我们应用微扰理论来找到量子信息理论中四个最重要的量的简单表达式，这些量通常从密度矩阵计算：冯诺依曼熵、量子相对熵、量子切尔诺夫界和量子保真度。