Editing Asymmetric Universal 1-2 Cloning (section)

===Special case with bell state:===
<u>'''Stage 1'''</u> Cloner state preparation
# Prepare the original qubit and two additional blank qubits <math>m</math> and <math>n</math> in pure states: <math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle</math>
# Prepare <math>|\psi\rangle_{m_1,n_1} = a|\Phi^{+}\rangle_{m_1,n_1} + b|0\rangle_{m_1} |+\rangle_{n_1}</math>, where <math>|\Phi^{+}\rangle</math> is a [[Bell state]] and <math>|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)</math>. In this case, the density matrix representation of the output states will be:</br>
<math>\rho_{a}^{out} = (1 - b^2) |\psi\rangle\langle\psi| + \frac{b^2}{2} \hat{I}</math></br>
<math>\rho_{b}^{out} = (1 - a^2) |\psi\rangle\langle\psi| + \frac{a^2}{2} \hat{I}</math></br>
<u>'''Stage 2'''</u> Cloning Circuit
* The cloning circuit is exactly the same as the general case. after the cloning circuit, the output state will be:</br>
<math>a|\psi\rangle_{in} |\Phi^+\rangle_{m,n} + b|\psi\rangle_m |\Phi^+\rangle_{in,n}</math>
The reduced density matrix of two clones A and B can be written in terms of their fidelities:</br>
<math>\rho_{a}^{out} = F_a |\psi\rangle\langle\psi| + (1 - F_a)|\psi^{\perp}\rangle\langle\psi^{\perp}|</math></br>
<math>\rho_{b}^{out} = F_b |\psi\rangle\langle\psi| + (1 - F_b)|\psi^{\perp}\rangle\langle\psi^{\perp}|</math></br>
<u>'''Stage 3'''</u> Discarding ancillary state
* The same as the general case.