Editing State Dependent N-M Cloning (section)

==Properties==
* State-dependent cloning machine only transforms a set of two non-orthogonal input states, parametrized as follows:
<math>|a\rangle = cos\theta |0\rangle + sin\theta |1\rangle,</math></br>
<math>|b\rangle = sin\theta |0\rangle + cos\theta |1\rangle,</math></br>
where <math>\theta \in [0, \pi/4]</math> and their scalar product (or inner product) is specified as <math>S = \langle a|b \rangle = sin 2\theta</math>.
* The condition that transformation should be unitary adds following constraint on the scalar product of final states:</br>
<math>\langle\alpha_{NM}|\beta_{NM}\rangle = (\langle a|b \rangle)^N = S^N</math>
*Fidelity Claims
** '''Optimal global fidelity:''' The average global fidelity of the machine for both sets of input states in terms of the inner(scalar) product of the two sets will be:</br>
<math>F_g^{opt}(N,M) = \frac{1}{2}(1 + S^{N+M} + \sqrt{1 - S^{2N}} \sqrt{1 - S^{2M}})</math>
** '''Subsystem fidelity:''' The fidelity of the '''density matrix''' of the subsystem with the original state <math>a</math> in terms of the inner product (similar fidelity for the subsystem b):</br>
<math>F_{sd}(N,M) = \langle a|\rho_{\alpha}|a\rangle = A^2(1 + S^2 + 2S^M) + B^2(1 + S^2 - 2S^M) + 2AB(1 - S^2)</math></br>
where the reduced density matrix of the subsystem <math>a</math> (The quantum state of a subsystem in density matrix representation) is described as:</br>
<math>\rho_{\alpha} = (A+B)^2 |a\rangle\langle a| + (A-B)^2 |b\rangle\langle b| + (A^2 - B^2)S^{M-1}(|a\rangle\langle b| + |b\rangle\langle a|)</math></br>
A and B are presented in the [[State Dependent N-M Cloning#Pseudo Code|Pseudo Code]] section.