Editing Probabilistic Cloning (section)

===Two qubit case===
'''General Informaton:''' The special case of the above probabilistic cloning machine is the following protocol for two nonorthogonal qubit states. The input states for this machine are presented in the following form:</br>
<math>|\psi_{\pm}\rangle = cos\eta|0\rangle \pm sin\eta|1\rangle, \quad \eta \in [0,\pi/4]</math></br>
where <math>|0\rangle</math> and <math>|1\rangle</math> are two orthogonal bases of a single qubit.</br></br>
'''<u>Stage 1</u>''' Ancilla preparation</br>
# Prepare two blank states $|0\rangle|0\rangle$. One of these states is the blank state that we will copy on it and the other one is the ancilla. </br></br>
'''<u>Stage 2</u>''' Unitary Evolution</br></br>
'''Input:''' <math>|\psi_{\pm}\rangle</math>, <math>|0\rangle|0\rangle</math></br>
'''Output:''' <math>U|\psi_{\pm}\rangle_x|0\rangle_y|0\rangle_z</math></br>
#  Perform the unitary transformation expressed as:</br>
<math>U|\psi_{\pm}\rangle_x|0\rangle_y|0\rangle_z = \sqrt{p}|\psi_{\pm}\rangle_x|\psi_{\pm}\rangle_y|0\rangle_z + \sqrt{1-p}|\Phi\rangle_{xy}|1\rangle_z</math></br>
here we labelled the three qubits by x, y and z</br></br>
'''<u>Stage 3</u>''' Measurements </br>
'''Input:''' <math>\sqrt{p}|\psi_{\pm}\rangle_x|\psi_{\pm}\rangle_y|0\rangle_z + \sqrt{1-p}|\Phi\rangle_{xy}|1\rangle_z</math></br>
'''Output:''' <math>|\psi_{\pm}\rangle_x|\psi_{\pm}\rangle_y</math> with probabiliy p
# Measure qubit z in the standard basis (<math>|0\rangle</math> and <math>|1\rangle</math> basis).
##'''If''' the output of the measurement is <math>|0\rangle</math>
##'''Then''' the protocol is successful and the final state of the machine are <math>|\psi_{\pm}\rangle_x|\psi_{\pm}\rangle_y</math>
##'''Else''' the protocol failed and the final state of the machine is <math>|\Phi\rangle_{xy}</math>