From top quarks to enhanced quantum key distribution: A Framework for Optimal Predictability of Quantum Observables

Predicting the outcomes of quantum measurements is a cornerstone of quantum information theory and a key resource for quantum technologies. Here, we introduce a comprehensive framework for quantifying the predictability of measurements on a bipartite quantum system using error measures inherited from statistical learning theory: the Bayes risk and inference variance. We derive analytical expressions for the optimal measurement that minimizes the prediction error for any arbitrary observable and any two-qubit state. We establish a direct, quantitative link between the ability to surpass the fundamental limit of local unpredictability and the presence of Einstein-Podolsky-Rosen steering. Additionally, by optimizing measurement choices according to the minimal Bayes risk, we propose a modified entanglement-based quantum key distribution protocol achieving higher secure key rates than the standard BB84 protocol, demonstrating enhanced resilience to noise. We apply our framework in two scenarios: perfect Bell states affected by local amplitude-damping noises, and top-antitop quark pairs produced in high-energy colliders. Our work offers a novel perspective on quantum correlations, connecting statistical inference, fundamental quantum phenomena, and cryptographic applications.