Geometric Optimization for Tight Entropic Uncertainty Relations

Abstract

Entropic uncertainty relations play a fundamental role in quantum information theory. However, determining optimal (tight) entropic uncertainty relations for general observables remains a formidable challenge and has so far been achieved only in a few special cases. Motivated by Schwonnek \emph{et al.} [PRL \textbf{119}, 170404 (2017)], we recast this task as a geometric optimization problem over the quantum probability space. This procedure leads to an effective outer-approximation method that yields tight entropic uncertainty bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. We benchmark our approach against existing analytical and majorization-based bounds, and demonstrate its practical advantage through applications to quantum steering.

Figures (4)

• Figure 1: Visualization of the convergence of the outer-approximating polytope. The figure illustrates the optimization of Shannon entropy of Haar-random POVMs with $m=4$ outcomes and Hilbert space of dimension $d=100$ (generated via https://heyredhat.github.io/qbism/05random.html). The red square denotes the optimal vertex.
• Figure 2: Comparison of entropic uncertainty bounds for the two-measurement setting $\mathcal{M}_2$. The red line represents the optimal bound derived from the proposed outer-approximation algorithm, showing a clear advantage over analytical bounds.
• Figure 3: Comparison of entropic uncertainty bounds for the three-measurement setting $\mathcal{M}_3$. The proposed algorithm yields the tightest possible bound allowed by quantum mechanics.
• Figure 4: Steering detection thresholds for isotropic states using measurement setting $\mathcal{M}_2$. A lower threshold indicates stronger noise robustness in detecting steerability.

Theorems & Definitions (1)

• Definition 1: Support Function