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&#39;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 $0<θ<1$.