Editing Phase Co-variant Cloning (section)

===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: