Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable nonconvex 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 Schwonnek et al. [State-independent uncertainty relations and entanglement detection in noisy systems, Phys. Rev. Lett. 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 certified 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.

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• Quantum foundations
• Quantum information theory
• Uncertainty relation