Efficient Computational Information Geometric Analysis of Physically Allowed Quantum Cloning Attacks for Quantum Key Distribution Protocols

In secret quantum communications the best eavesdropping attacks on quantum cryptography are based on imperfect cloning machines. The incoherent attack, based on quantum cloning, is the most common eavesdropping strategy. Using a probe, the eavesdropper imperfectly clones the sender’s quantum state which keeps one copy and sends the other. The physically allowed transformations of Eve’s quantum cloner on Bob’s qubit can be described in terms of Completely Positive (CP), trace preserving maps. The map of the quantum cloner compresses the Bloch-ball, as an affine map. This affine map has to be a complete positive, trace preserving map, which shrinks the Bloch ball. The effects of a quantum cloner can be given in tetrahedron representation. In this paper we show a new, quantum information theoretical representation of eavesdropping detection, focused on the Four-state (BB84) and Six-state quantum cryptography protocols. We use a fundamentally new computational geometrical method to analyze the informational theoretical impacts of cloning activity on the quantum channel. The proposed algorithm uses Delaunay tessellation and convex hull calculation on the Bloch sphere, with respect to quantum relative entropy as distance measure. The improved core-set approach can be used to analyze efficiently the informational theoretical impacts of physically allowed quantum cloning attacks. Key-Words: Quantum Cryptography, Quantum Cloning, Quantum Informational Distance