Geometric Optimization over Quantum State Spaces: Tight Uncertainty Relations and Resource Certification

Abstract

Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable non-convex optimization problem. We introduce a generic support-function-based outer-approximation framework for solving concave-minimization (or convex-maximization) problems over the quantum state space. By mapping the problem onto a reduced $\mathcal{Z}$-space, we characterize the exact quantum boundary through supporting half-spaces derived from the largest eigenvalues of effective observables. This yields an effective method that produces tight bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. As an initial application, we recover the exact variance-based uncertainty relations of [PRL \textbf{119}, 170404 (2017)] and efficiently compute optimal entropic uncertainty relations (EURs). Our results reveal that standard analytical and majorization-based EUR bounds are fundamentally loose for generic measurements, and we show that the resulting exact bounds directly enhance quantum steering detection under asymmetric settings. We further apply the framework to determine the maximal athermality resource certifiable from a restricted measurement scenario. Our method thus provides a universal computational tool for exploring the boundaries of quantum state space and the limits of quantum resources.

13 pages, 4 figures, revised version with reorganized presentation and improved structure; main results and conclusions remain unchanged, with one new application added