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

TL;DR

This paper develops a framework for predicting quantum measurement outcomes using statistical learning tools, linking quantum correlations with cryptography and demonstrating improved quantum key distribution protocols.

Contribution

It introduces a novel framework for quantifying measurement predictability, deriving optimal measurements, and applying these insights to enhance quantum cryptography and analyze high-energy particle states.

Findings

Optimal measurements minimize prediction error for any observable.

Surpassing local unpredictability limits indicates EPR steering.

Modified quantum key distribution protocol achieves higher secure key rates.

Abstract

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.