Optimal asymmetric cloning machine on a great circle using no-signalling   condition
  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. 
Samir Kunkri, Md. Manirul Ali, Geetu Narang, Debasis Sarkar