Related papers: Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable

Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable

URL: http://arxiv.org/abs/2609.13870v1
Date: Sat, 12 Sep 2026 10:45:46 GMT
Title: Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable
Abstract summary: We study intrinsic quantum uncertainty associated with a single observable from the viewpoint of skew information.<n>By relaxing ordinary additivity we introduce a class of Hiai--Petz nonadditive skew informations.<n>Our main quantitative result is a sharp single-observable uncertainty relation for the full Hiai--Petz class.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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 $K_s(ρ,A)=\frac12\|[ρ^s,A]\|_{\mathrm{HS}}^2$, $1/2\le s<1$, 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 $0<θ<1$.

Related papers

Global Minimax Readout of a Qubit Direction [5.246986428523558]
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length $$.<n>Every fixed-local architecture, including recorded classical randomization, reduces exactly to a zero-barycenter probability measure on the Bloch ball.<n>For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid.
arXiv Detail & Related papers (2026-08-17T17:26:32Z)
Readout-Rank Laws for Isotropic Quantum Tangents [0.0]
Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond.<n>We show that even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(nk2-n)$ of the full-record information.
arXiv Detail & Related papers (2026-08-07T11:09:51Z)
Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
arXiv Detail & Related papers (2026-07-28T09:29:37Z)
Regularized Online RLHF with Generalized Bilinear Preferences [68.44113000390544]
We consider the problem of contextual online RLHF with general preferences.<n>We adopt the Generalized Bilinear Preference Model to capture preferences via low-rank, skew-symmetric matrices.<n>We prove that the dual gap of the greedy policy is bounded by the square of the estimation error.
arXiv Detail & Related papers (2026-02-26T15:27:53Z)
Optimal Unconstrained Self-Distillation in Ridge Regression: Strict Improvements, Precise Asymptotics, and One-Shot Tuning [61.07540493350384]
Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth and the teacher's own predictions.<n>We show that for any prediction risk, the optimally mixed student improves upon the ridge teacher for every regularization level.<n>We propose a consistent one-shot tuning method to estimate $star$ without grid search, sample splitting, or refitting.
arXiv Detail & Related papers (2026-02-19T17:21:15Z)
Statistical Inference and Learning for Shapley Additive Explanations (SHAP) [20.663970002208846]
The SHAP (short for Shapley additive explanation) framework has become an essential tool for attributing importance to variables in predictive tasks.<n>Despite their ubiquity, there do not exist approaches for performing statistical inference on these quantities.<n>We show that, by treating the SHAP curve as a nuisance function that must be estimated from data, one can reliably constructally normal estimates of the $p$th powers of SHAP.
arXiv Detail & Related papers (2026-02-11T05:01:47Z)
Singular Bayesian Neural Networks [1.2891210250935148]
Bayesian neural networks promise calibrated uncertainty but require $O(mn)$ parameters for standard mean-field Gaussian posteriors.<n>We induce a posterior that is singular with respect to the Lebesgue measure, concentrating on the rank-$r$ manifold.<n>We derive PAC-Bayes generalization bounds whose complexity term scales as $sqrtr(m+n)$ instead of $sqrtm n$, and prove loss bounds that decompose the error into optimization and rank-induced bias.
arXiv Detail & Related papers (2026-01-30T23:06:34Z)
Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond [40.79840141270367]
Given an unnormalized probability density $piproptomathrme-V$, estimating its normalizing constant $Z=int_mathbbRdmathrme-V(x)mathrmdx$ or free energy $F=-log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning.<n>We propose a new algorithm based on reverse diffusion samplers, establish a framework for analyzing its complexity, and empirically demonstrate its efficiency in tackling multimodality.
arXiv Detail & Related papers (2025-02-07T00:05:28Z)
Convergence Rate Analysis of LION [54.28350823319057]
LION converges iterations of $cal(sqrtdK-)$ measured by gradient Karush-Kuhn-T (sqrtdK-)$. We show that LION can achieve lower loss and higher performance compared to standard SGD.
arXiv Detail & Related papers (2024-11-12T11:30:53Z)
Maximal $α$-Leakage for Quantum Privacy Mechanisms [8.259231344795781]
We first show that an adversary's maximal expected $alpha$-gain is characterized by measured conditional R'enyi entropy. We prove that the $alpha$-leakage and maximal $alpha$-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured R'enyi capacity.
arXiv Detail & Related papers (2024-03-21T14:58:07Z)
High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize [55.0090961425708]
We propose a new, simplified high probability analysis of AdaGrad for smooth, non- probability problems. We present our analysis in a modular way and obtain a complementary $mathcal O (1 / TT)$ convergence rate in the deterministic setting. To the best of our knowledge, this is the first high probability for AdaGrad with a truly adaptive scheme, i.e., completely oblivious to the knowledge of smoothness.
arXiv Detail & Related papers (2022-04-06T13:50:33Z)
Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
arXiv Detail & Related papers (2021-09-24T14:48:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.