Editing Asymmetric Universal 1-2 Cloning (section)

==Notation==
**<math>|\psi\rangle_{in}:</math>The original input state
**<math>\rho_{\psi_{in}}:</math> The density matrix of the input pure state equal to <math>|\psi_{in}\rangle\langle\psi_{in}|</math>
**<math>s_0:</math> The scaling parameter of the first clone
**<math>s_12:</math> The scaling parameter of the second clone
**<math>\rho_{a}^{out}:</math> The output density matrix of the first clone (equal to <math>|\psi_{a}^{out}\rangle\langle\psi_{a}^{out}|</math> if the output state is pure)
**<math>\rho_{b}^{out}:</math> The output density matrix of the second clone (equal to <math>|\psi_{b}^{out}\rangle\langle\psi_{b}^{out}|</math> if the output state is pure)
**<math>\hat{I}:</math> The ``Identity" or completely mixed density matrix
**<math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle:</math> state of the ancillary qubits before preparation phase
**<math>|\psi\rangle_{m_1,n_1}:</math>  state of the ancillary qubits after the preparation
**<math>c_j:</math> amplitudes (or coefficients) of the prepared state $|\psi\rangle_{m_1,n_1}$. These coefficients are being used to control the flow of the information between the copies before starting the cloning process.
**<math>P_{k,l}:</math> The CNOT gate where the control qubit is <math>k</math> and the target qubit is <math>l</math> 
**<math>|\psi\rangle_{out}:</math> The total output of the asymmetric cloning circuit
**<math>|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle):</math> Bell state
**<math>|+\rangle =  \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle):</math> Plus state. The eigenvector of Pauli X