Hiai--Petz 斜信息与单一可观测量的尖锐不确定性关系
Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable
- Tohoku University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究引入 Hiai--Petz 非可加斜信息类，并证明其满足尖锐的单可观测量不确定性关系，最优系数仅依赖态的最大与最小特征值，同时推广了度量调整斜信息与 SLD Fisher 信息等特例。
AI中文摘要：
我们从斜信息的角度研究单一可观测量所关联的固有量子不确定性。通过放宽普通可加性同时保留其他基本不确定性要求，我们引入了一类 Hiai--Petz 非可加斜信息，其中包含 Hansen 度量调整斜信息作为边界情形 θ=1。我们的主要定量结果是针对完整 Hiai--Petz 类的尖锐单一可观测量不确定性关系。对于固定态，最优系数仅依赖于其最大和最小特征值，并且即使在保留方差的最大经典贡献后仍然保持尖锐。该一般定理作为特例，给出了度量调整斜信息、SLD 量子 Fisher 信息以及幂对易子族的尖锐界。我们进一步证明，对于 1/2≤s<1，K_s(ρ,A)=1/2‖[ρ^s,A]‖_HS^2 在 Hiai--Petz 框架内通过 Stolarsky 算子均值实现，从而确立了其在 1/2<s<1 时的凸性。更一般地，对于每个固定的算子单调函数 f，Hiai--Petz 谱核在度量调整斜信息 I^f 和公共端点 K_1 之间几何插值。所得族满足精确的幂迹加权合成律，在内部 0<θ<1 取代普通可加性。
英文摘要：
We study intrinsic quantum uncertainty associated with a single observable from the viewpoint of skew information. By relaxing ordinary additivity while retaining the other fundamental uncertainty requirements, we introduce a class of Hiai--Petz nonadditive skew informations that contains Hansen's metric-adjusted skew informations as the boundary case $θ=1$. Our main quantitative result is a sharp single-observable uncertainty relation for the full Hiai--Petz class. For a fixed state, the optimal coefficient depends only on its largest and smallest eigenvalues and remains sharp even after the maximal classical contribution to the variance is retained. The general theorem yields, as special cases, sharp bounds for metric-adjusted skew informations, the SLD quantum Fisher information, and the power-commutator family. We further show that $K_s(ρ,A)=\frac12\|[ρ^s,A]\|_{\mathrm{HS}}^2$, $1/2\le s<1$, is realized within the Hiai--Petz framework through Stolarsky operator means, thereby establishing its convexity for $1/2<s<1$. More generally, for each fixed operator monotone function $f$, the Hiai--Petz spectral kernel interpolates geometrically between the metric-adjusted skew information $I^f$ and the common endpoint $K_1$. The resulting family obeys an exact power-trace-weighted composition law, replacing ordinary additivity in the interior $0<θ<1$.
