Optimal asymmetric cloning machine on a great circle using no-signalling condition

Abstract

Assuming the condition of no superluminal signalling, we got an upper bound on the quality of all asymmetric $1\to 2$ cloning machines, acting on qubits whose Bloch vectors lie on a great circle. Then we constructed an $1\to 2$ cloning machine, which asymmetrically clone all qubits corresponding to this great circle, and this machine matches with that upper bound, and hence this is optimal one.

6 pages, Latex