Optimal Universal Bounds for Quantum Divergences

• URL: http://arxiv.org/abs/2603.09885v1
• Date: Tue, 10 Mar 2026 16:42:50 GMT
• Title: Optimal Universal Bounds for Quantum Divergences
• Abstract summary: We identify a universal structural principle underlying the smoothing of classical divergences.<n>We derive optimal universal bounds for smoothed quantum divergences.<n>In particular, we obtain optimal universal bounds for the hypothesis testing divergence.
• Score: 2.7074235008521246
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum R'enyi divergences of arbitrary order and the hypothesis testing divergence. Our inequalities relate divergences of different orders through bounds of the form $D_β^{\varepsilon} \le D_α+ \mathrm{correction}$ and $D_β^{\varepsilon} \ge D_α+ \mathrm{correction}$, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis testing divergence.

Related papers

• Generalized nonparametric regression in reproducing kernel Hilbert spaces: Consistency and rates of convergence [0.0]
  We develop a comprehensive theory for M-lev in reproducing spaces.<n>For Sobo spaces we obtain new rates that connect to spaces of smoothness with mixed smoothness functions.
  arXiv  Detail & Related papers  (2026-06-22T08:12:31Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• The Procrustean Bed of Time Series: The Optimization Bias of Point-wise Loss [53.542743390809356]
  This paper aims to provide a first-principles analysis of the Expectation of Optimization Bias (EOB)<n>Our analysis reveals a fundamental paradigm paradox: the more deterministic and structured the time series, the more severe the bias by point-wise loss function.<n>We present a concrete solution that simultaneously achieves both principles via DFT or DWT.
  arXiv  Detail & Related papers  (2025-12-21T06:08:22Z)
• From Tail Universality to Bernstein-von Mises: A Unified Statistical Theory of Semi-Implicit Variational Inference [0.12183405753834557]
  Semi-implicit variational inference (SIVI) constructs approximate posteriors of the form $q() = int k(| z) r(dz)$<n>This paper develops a unified "approximation-optimization-statistics'' theory for such families.
  arXiv  Detail & Related papers  (2025-12-05T19:26:25Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• Entropic Mirror Descent for Linear Systems: Polyak's Stepsize and Implicit Bias [55.72269695392027]
  This paper focuses on applying entropic mirror descent to solve linear systems.<n>The main challenge for the convergence analysis stems from the unboundedness of the domain.<n>To overcome this without imposing restrictive assumptions, we introduce a variant of Polyak-type stepsizes.
  arXiv  Detail & Related papers  (2025-05-05T12:33:18Z)
• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning [1.5229257192293195]
  We investigate the uniform convergence of subdifferential mappings from empirical risk to population risk in nonsmooth, non-valued to deterministic optimization.<n>These guarantees offer new insight into the geometry of problems arising in robust statistics and related applications.
  arXiv  Detail & Related papers  (2024-05-16T17:49:46Z)
• Convergence of Adam Under Relaxed Assumptions [72.24779199744954]
  We show that Adam converges to $epsilon$-stationary points with $O(epsilon-4)$ gradient complexity under far more realistic conditions. We also propose a variance-reduced version of Adam with an accelerated gradient complexity of $O(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-04-27T06:27:37Z)
• High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance [59.211456992422136]
  We propose algorithms with high-probability convergence results under less restrictive assumptions. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes in optimization.
  arXiv  Detail & Related papers  (2023-02-02T10:37:23Z)
• Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization [11.017632675093628]
  Coordinate-type subgradient methods for addressing nonsmooth problems are relatively underexplored due to the set of properties of the Lipschitz-type assumption.
  arXiv  Detail & Related papers  (2022-06-30T02:17:11Z)
• Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry [6.85316573653194]
  We give curvature-dependant convergence rates for weakly convex functions defined on a manifold of 1-bounded geometry. We compute these bounds explicitly for some manifold commonly used in the optimization literature. We present self-contained proofs of fully general bounds on the norm of the differential of the exponential map.
  arXiv  Detail & Related papers  (2020-08-06T08:30:35Z)
• The semiring of dichotomies and asymptotic relative submajorization [0.0]
  We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum.
  arXiv  Detail & Related papers  (2020-04-22T14:13:26Z)

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

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.