Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable
Abstract
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 Ks(ρ,A)=1/2|[ρs,A]|HS2, 1/2≤ 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 If and the common endpoint K1. The resulting family obeys an exact power-trace-weighted composition law, replacing ordinary additivity in the interior 0<θ<1.
Related
BibTeX
@misc{kimura2026hiaipetz,
title = {{Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable}},
author = {Gen Kimura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13870},
eprint = {2609.13870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13870},
}
RIS
TY - UNPB
TI - Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable
AU - Gen Kimura
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13870
UR - https://arxiv.org/abs/2609.13870
AB - 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)=1/2|[ρ^s,A]|_HS^2, 1/2≤ 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.
ER -