Editing Universal Superposition of Orthogonal States (section)

==Protocol Description==
'''<u>Stage 1</u>''' Preparation and operation</br></br>
*'''Input:''' The ancillary state <math>|a\rangle = \alpha|1\rangle + \beta |0\rangle</math>, <math>|\psi\rangle</math>, <math>|\psi^\perp\rangle</math>
*'''Output:''' <math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle  |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math>
*At this stage the following gates will be performed respectively:
#'''S gate (Controled Swap):''' Performing this gate will interchange the input qubits. The total states before and after performing this gate are as follows:
##<math>|\Psi_0\rangle = \alpha|1\rangle |\psi\rangle |\psi^\perp\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math>
##<math>|\Psi_1\rangle = \alpha|1\rangle |\psi^\perp\rangle  |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math>
#'''CNOT gate:''' The total state after this step is as follows:
##<math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle  |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> 

'''<u>Stage 2</u>''' Measurements</br></br>
*'''Input:''' <math>|\Psi_2\rangle</math>
*'''Output:''' The superposed state <math>|\Psi\rangle</math>
#Measure qubit 1 (the ancilary qubit) in X basis and the qubit 2 in Z basis.
##'''Pure Output Case:''' 
###''if:''' The output of the X measurement is 0 AND the output of Z measurement is 1
###'''Then:''' Accept the round
###'''Else:''' Reject.
### The successful output is in the form:
<math>|\Psi\rangle = C(\alpha |\psi\rangle + \beta e^{i\eta} |\psi^\perp\rangle)</math></br>
<math>e^{i\eta}</math> is a relative phase which is <math>\frac{\langle1|\psi\rangle}{\langle0|\psi^\perp\rangle}</math></br>
##'''Mixed Output Case:''' 
###Always accept. The protocol is perfect.
### The output will be:<math>|\Psi^{\mu,\nu} \rangle = C (\alpha |\psi\rangle + \beta e^{i\eta_{\mu,\nu}} |\psi^\perp\rangle) </math>, <math>e^{i\eta_{\mu,\nu}}</math> is a relative phase which depends on the outputs of the measurements but in all cases, the superposition has the desired form and weights.