Editing Probabilistic Cloning (section)

==Protocol Description==
The probabilistic cloning machine can only perform effectively for a special set of input states. It was shown that states chosen from a set <math>S = \{|\psi_1\rangle, |\psi_2\rangle, ..., |\psi_n\rangle,\}</math> can be probabilistic-ally cloned if and only if the <math>|\psi_i\rangle</math> are linearly independent. This protocol in general consists of three stages as follows:
===General case===
'''<u>Stage 1</u>''' Ancilla preparation
# Prepare an orthonormal set of states such as <math>\{|A_k\rangle\}</math> where <math>\langle A_k|A_l\rangle = \delta_{kl}</math> for <math>k,l= 0,1,...,n</math>. These states act as ancillary states of the protocol. </br></br>
'''<u>Stage 2</u>''' Unitary Evolution</br></br>
'''Input:''' <math>|\psi_i\rangle</math>, <math>|0\rangle</math>, <math>|A\rangle</math></br>
'''Output:''' <math>U|\psi_i\rangle|0\rangle|A\rangle</math>
# Perform the unitary transformation of the following form:</br>
<math>U|\psi_i\rangle|0\rangle|A\rangle = \sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math></br></br>
'''<u>Stage 3</u>''' Measurements </br>
'''Input:''' <math>\sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math></br>
'''Output:''' <math>|\psi_i\rangle|\psi_i\rangle</math> with probability <math>p_i</math>
#  Measure the ancilla states on the basis <math>{|A_k\rangle}</math>.
##'''If''' The output of the measurement is the state <math>|A_0\rangle</math>
##'''Then''' the protocol is successful and the output is the desired clones.
##'''Else''' Abort

===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>

===The quantum circuit===
Finally, the quantum circuit which illustrates the above stages can be described to consist of following gates and parts which have been also shown in the figure in [[Probabilistic Cloning#Outline|Outline]]
* Reverse Controlled <math>U_1</math> gate: A controlled unitary gates with the <math>x</math> (original) qubit as the control qubit and the <math>z</math> qubit as the operational qubit. The unitary gate <math>U_1</math> acts only if the control qubit is <math>|0\rangle</math>. The unitary <math>U_1</math> is:</br>
<math>U_1 = sin\beta |0\rangle\langle 0| + cos\beta |1\rangle\langle 0| + cos\beta |1\rangle\langle 0| - sin\beta |1\rangle\langle 1|</math></br>
where the <math>\beta = arcsin\Bigg(\sqrt{\frac{1+tan^4\eta}{2}}\Bigg)</math>
* A normal [[CNOT]] gate: The control qubit is <math>y</math> and the operational qubit is <math>x</math> (The flip occurs if the control qubit is <math>|1\rangle</math>)
* Reverse Controlled $U_2$ gate: A controlled unitary gates with the <math>x</math> (original) qubit as the control qubit and the <math>y</math> qubit as the operational qubit. The unitary gate <math>U_2</math> acts only if the control qubit is <math>|0\rangle</math>. The unitary <math>U_2</math> is:</br>
<math>U_2 = sin\delta |0\rangle\langle 0| + cos\delta |1\rangle\langle 0| + cos\delta |1\rangle\langle 0| - sin\delta |1\rangle\langle 1|</math></br>
where the <math>\delta = arcsin[(\sqrt{\frac{2}{1+tan^4\eta}} + \sqrt{\frac{2}{1+tan^{-4}\eta}})/2]</math>
* [[Hadamard gate]]: A Hadamard gate on qubit <math>y</math>
* Measurement part: Measuring qubit <math>z</math> in the standard basis.