Editing Phase Co-variant Cloning (section)

==Protocol Description==
===General Information===
*The to-be-cloned states of the phase-covariant cloner are equatorial states of the form:</br>
<math>|\psi(\phi)\rangle = \frac{1}{\sqrt{2}} (|0\rangle + e^{i\phi}|1\rangle)</math></br>
*Equatorial states could be written in density matrix representation as:</br>
<math>|\psi(\phi)\rangle\langle\psi(\phi)| = \frac{1}{2}(\mathbb{I} + cos\phi\sigma_x + sin\phi\sigma_y)</math></br>
where <math>\sigma_x</math> and <math>\sigma_y</math> are [[Pauli Operators]].</br>
*The action of a general phase-covariant QCM is described by a map $T$ acting as follows:</br>
<math>T(|\psi(\phi)\rangle\langle\psi(\phi)|) = \eta|\psi(\phi)\rangle\langle\psi(\phi)| + (1-\eta)\frac{\mathbb{I}}{2}</math></br>
where <math>\eta</math> is the shrinking factor. This relation holds for all <math>\phi</math>, which guarantees that the cloning machine to act equally well on all the equatorial states. Now we investigate different types of the protocol:
===Phase-covariant cloning without ancilla===
'''Input:''' <math>|\psi(\phi)\rangle|0\rangle</math>
'''Output:''' <math>\frac{1}{\sqrt{2}}(|00\rangle + cos\eta e^{i\phi}|10\rangle + sin\eta e^{i\phi} |01\rangle)</math>
#Perform a unitary transformation described as follows:</br>
<math>U_{pc}|00\rangle = |00\rangle</math></br>
<math>U_{pc}|10\rangle = cos\eta|10\rangle + sin\eta|01\rangle</math></br>	
===Phase-covariant cloning with ancilla===
'''<u>Stage 1</u>''' Cloner state preparation
# Prepare a Bell state <math>|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math>
'''<u>Stage 2</u>''' Cloner transformation</br>
'''Input:''' <math>|\psi(\phi)\rangle_{A}|\Phi^{+}\rangle_{BM}</math></br>
'''Output:''' <math>U_{pca}|\psi(\phi)\rangle_{A}|\Phi^{+}\rangle_{BM}</math></br>
# Perform the unitary transformation described as follows:</br>
<math>U_{pca}|0\rangle|0\rangle|0\rangle = |0\rangle|0\rangle|0\rangle</math></br>
<math>U_{pca}|1\rangle|0\rangle|0\rangle = (cos\eta|1\rangle|0\rangle + sin\eta|0\rangle|1\rangle)|0\rangle</math></br>
<math>U_{pca}|0\rangle|1\rangle|1\rangle = (cos\eta|0\rangle|1\rangle + sin\eta|1\rangle|0\rangle)|1\rangle</math></br>
<math>U_{pca}|1\rangle|1\rangle|1\rangle = |1\rangle|1\rangle|1\rangle</math></br></br>
'''<u>Stage 3</u>''' Discarding ancillary state
# Discard the extra state. mathematically, trace out the ancilla