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

TL;DR

The paper develops a statistical-learning framework for quantum predictability using Bayes risk $L^{\!*}$ and inference variance $\Delta$, deriving analytic optimal measurements for any two-qubit state and linking the ability to surpass local unpredictability to EPR steering in Bell-diagonal states. It then shows how Bayes-risk optimization can enhance entanglement-based QKD, yielding higher secure-key rates than BB84 under local amplitude-damping noise and imperfect sources, including top-antitop quark states. The authors connect averaging over all A-observables to steering criteria via Carlson elliptic integrals and steering ellipsoids, establishing a deep operational link between quantum correlations and cryptographic performance. They also explore the practical relevance of their framework to high-energy physics platforms, highlighting both current limitations and the potential for collider-based quantum cryptography as experimental capabilities advance.

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.

Figures (12)

• Figure 1: Diagram showing the predictability problem addressed in Sec. \ref{['sec:Predictability of quantum observables']}, supporting Eqs. \ref{['eq_bayes_risk_opt_problem']} and \ref{['eq_inference_variance_opt_problem']}: Bob is intended to predict Alice's local measurement $x$ outcome with $f^*(y)$. These local measurements, $\mathcal{O}_A$ and $\mathcal{O}_B$, are performed over a resource state $\rho_{AB}$. Bob's measurements are chosen to minimize the prediction error measures, the Bayes risk $\text{L}^{\!*}$\ref{['eq:BayesRisk']} and the inference variance $\Delta$\ref{['eq:InferenceVariance']}.
• Figure 2: Contour plot of Bayes risk $\overline{\text{L}^{\!*}}_{\!\!\min}$ [Eq. \ref{['eq:BayesRiskAverage']}] and inference variance $\overline{\Delta}_{\min}$ [Eq. \ref{['eq:QEntropyAverage']}], in the tetrahedron spanned by Bell-diagonal states. Separable states correspond to the octahedron, while the yellow contours denote the corresponding local unpredictability thresholds: $\overline{\text{L}^{\!*}}_{\!\!\min}=1/4$ and $\overline{\Delta}_{\min}=1/6$.
• Figure 3: Contour lines of the average Bayes risk $\overline{\text{L}^{\!*}}_{\!\!\min}$ (left), Eq. \ref{['eq:BayesRiskAverage']}, and inference variance $\overline{\Delta}_{\min}$ (right), Eq. \ref{['eq:QEntropyAverage']}, as functions of the local damping parameters $p_A$ and $p_B$. Their corresponding local predictability thresholds are shown as a continuous white lines and the steering inequalities $F_2^{\text{CJWR}}\leq1$ and $F_3^{\text{CJWR}}\leq1$ are also displayed as solid and dashed yellow lines, respectively.
• Figure 4: Contour plot of the average Bayes risk $\text{L}^{\!*}_{\min}$ for the statistical mixture of the $gg$ and $q\bar{q}$ processes, weighted by $w_{gg} \in [0,1]$, in the production of $t\bar{t}$ pairs. The local predictability threshold, $\overline{\text{L}^{\!*}}_{\!\!\min,l} = 1/4$, is shown as a continuous white line. Also shown are the three steering inequalities described in Appendix \ref{['app_steering']}: $F_2^{\text{CJWR}} \leq 1$ (solid yellow line), $F_3^{\text{CJWR}} \leq 1$ (dashed yellow line), and $F_{\text{Haar}} \leq 1$ which coincides with the local predictability threshold (white line). The region where the resource state of the $t\bar{t}$-pair exhibits entanglement is bounded by dashed red lines.
• Figure 5: Average inference variance, $\overline{\Delta}_{\min}$ for the statistical mixture of the $gg$ and $q\bar{q}$ processes, weighted by $w_{gg} \in [0,1]$, in the production of $t\bar{t}$ pairs. The local predictability threshold, $\Delta_{\min,l} = 1/6$, is shown as a continuous white line. Also shown are the three steering inequalities described in Appendix \ref{['app_steering']}: $F_2^{\text{CJWR}} \leq 1$ (solid yellow line), $F_3^{\text{CJWR}} \leq 1$ which coincides with the local predictability threshold (white line), and $F_{\text{Haar}} \leq 1$ (yellow dash-dotted line). The region where the resource state of the $t\bar{t}$-pair exhibits entanglement is bounded by dashed red lines.