Computational Geometric Analysis of Physically Allowed Quantum Cloning Transformations for Quantum Cryptography

The safety of quantum cryptography relies on the no-cloning theorem. In secret quantum communications, an eavesdropper cannot clone the sent qubits perfectly, however the best eavesdropping attacks for quantum cryptography are based on imperfect cloning machines. The eavesdropper’s physically allowed quantum evolutions on the sent qubit can be described in terms of the quantum state’s geometry. We use a fundamentally new computational geometrical method to analyze the informational theoretical impacts of cloning activity on the quantum channel. Our method uses Delaunay tessellation and convex hull calculation, with respect to quantum relative entropy as distance measure. The security analysis is focused on the four state (BB84) and Six state quantum cryptography protocols. The proposed geometrical method can be used to analyze efficiently the informational theoretical impacts of physically allowed quantum cloning transformations. Key-Words: Quantum Cryptography, Quantum Cloning, Quantum Informational Distance