Quantum Informational Geometry for Secret Quantum Communication

The problem of quantum cloning is closely connected to quantum cryptography. In quantum cryptography, an eavesdropper on the quantum channel can not copy perfectly the sent quantum states, however in many cases the cloning machine is known to be the most powerful eavesdropping strategy against quantum cryptographic protocols. The geometric interpretation of quantum states investigates distances between two different quantum states. In our method we use quantum relative entropy as an informational distance between quantum states. We show a geometrical approach to analyze the security of quantum cryptography, based on quantum relative entropy and Delaunay triangulation on the Bloch sphere. In our security analysis, we use an approximation algorithm from classical computational geometry to determine the smallest enclosing ball of balls using core-sets.