Editing Phase Co-variant Cloning (section)

==Properties==
*Fidelity Claims
#'''Fidelity of the asymmetric case without ancilla case:'''</br>
<math>F_A = \frac{1}{2}(1 + cos\eta) \quad F_B = \frac{1}{2}(1 + sin\eta)</math>
#'''Fidelity of the symmetric case without ancilla case:''' The special case of the asymmetric phase-covariant cloning with <math>\eta = \pi/4</math>. This fidelity is larger than the fidelity of the [[Quantum Cloning|Universal QCM]]:</br>
<math>F_{A,B} = \frac{1}{2}(1 + \frac{1}{\sqrt{2}}) \approx 0.8535 > \frac{5}{6}</math></br>
#'''Fidelity of the general case with ancilla case:''' The same as the without ancilla case:</br>
<math>F_A = \frac{1}{2}(1 + cos\eta) \quad F_B = \frac{1}{2}(1 + sin\eta)</math>
*'''Case 1 (without ancilla):''' The state of each copy which is a subsystem of this two-qubit system is described with a [[density matrix]] which can be obtained as below</br>
	<math>\rho_A = Tr_B (|\psi(\phi)_F\rangle\langle\psi(\phi)_F|)</math></br>
	<math>\rho_B = Tr_A (|\psi(\phi)_F\rangle\langle\psi(\phi)_F|)</math></br></br>
	<math>\circ</math> The symmetric case occurs when <math>\eta = \pi/4</math></br>
*'''Case 2 (with ancilla case):''' The unitary transformation can also be described in terms of the fidelity of one of the copies and [[Pauli operators]]:</br>
	<math>U_{pca} = F_A \mathbb{I}_{ABM} + (1-F_A)\sigma_z \otimes \sigma_z \otimes \mathbb{I} + \sqrt{F_A(1-F_A)}(\sigma_x \otimes \sigma_x + \sigma_y \otimes \sigma_y)\otimes \mathbb{I}</math>