Semidefinite Programming for Optimal Quantum Cloning: A Computational Framework

TL;DR

This paper introduces a computational framework using Semidefinite Programming to find explicit operator representations for quantum cloning, enabling practical implementation and security analysis.

Contribution

It provides a unified, numerical method to derive explicit Kraus operators for various quantum cloning scenarios, bridging the gap between theoretical limits and practical realization.

Findings

Successfully certifies global optimality of cloning operations

Automatically extracts operational Kraus operators from optimal solutions

Analyzes security of BB84 protocol under realistic noise conditions

Abstract

While algebraic derivations establish theoretical limits for quantum cloning, practical implementations require explicit operator representations that are often unavailable analytically. We present a computational framework that reformulates cloning optimization as a search over completely positive trace-preserving maps using the Choi-Jamiolkowski isomorphism and Semidefinite Programming. The framework (i) numerically certifies global optimality through primal-dual strong duality and (ii) automatically extracts operational Kraus operators from the optimal Choi matrix via spectral decomposition. We systematically treat universal, phase-covariant, asymmetric, and entanglement cloning scenarios, providing -for the first time - a unified computational catalogue of explicit, implementable Kraus representations across all major cloning families, including higher-order processes and arbitrary input state distributions. As an application, we analyse optimal cloning attacks on BB84 under depolarizing noise, demonstrating how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation enables community validation and extension.