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Probabilistic Cloning: Difference between revisions
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# Prepare an orthonormal set of states such as <math>\{|A_k\rangle\}</math> where <math>\langle A_k|A_l\rangle = \delta_{kl}</math> for <math>k,l= 0,1,...,n</math>. These states act as ancillary states of the protocol. </br></br> | # Prepare an orthonormal set of states such as <math>\{|A_k\rangle\}</math> where <math>\langle A_k|A_l\rangle = \delta_{kl}</math> for <math>k,l= 0,1,...,n</math>. These states act as ancillary states of the protocol. </br></br> | ||
'''<u>Stage 2</u>''' Unitary Evolution</br> | '''<u>Stage 2</u>''' Unitary Evolution</br> | ||
'''Input:''' <math>|\psi_i\rangle</math>, <math>|0\rangle</math>, <math>|A\rangle</math> | '''Input:''' <math>|\psi_i\rangle</math>, <math>|0\rangle</math>, <math>|A\rangle</math></br> | ||
'''Output:''' <math>U|\psi_i\rangle|0\rangle|A\rangle</math> | '''Output:''' <math>U|\psi_i\rangle|0\rangle|A\rangle</math> | ||
# Perform the unitary transformation of the following form:</br> | # Perform the unitary transformation of the following form:</br> | ||
<math>U|\psi_i\rangle|0\rangle|A\rangle = \sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math></br></br> | <math>U|\psi_i\rangle|0\rangle|A\rangle = \sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math></br></br> | ||
'''<u>Stage 3</u>''' Measurements </br> | '''<u>Stage 3</u>''' Measurements </br> | ||
'''Input:''' <math>\sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math> | '''Input:''' <math>\sqrt{p_i}|\psi_i\rangle|\psi_i\rangle|A_0\rangle + \sum_{j=1}^{n} c_{ij}|\Phi^{j}_{AB}\rangle|A_j\rangle</math></br> | ||
'''Output:''' <math>|\psi_i\rangle|\psi_i\rangle</math> with probability <math>p_i</math> | '''Output:''' <math>|\psi_i\rangle|\psi_i\rangle</math> with probability <math>p_i</math> | ||
# Measure the ancilla states on the basis <math>{|A_k\rangle}</math>. | # Measure the ancilla states on the basis <math>{|A_k\rangle}</math>. | ||
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##'''Then''' the protocol is successful and the output is the desired clones. | ##'''Then''' the protocol is successful and the output is the desired clones. | ||
##'''Else''' Abort | ##'''Else''' Abort | ||
===Two qubit case=== | ===Two qubit case=== | ||
'''General Informaton:''' The special case of the above probabilistic cloning machine is the following protocol for two nonorthogonal qubit states. The input states for this machine are presented in the following form:</br> | '''General Informaton:''' The special case of the above probabilistic cloning machine is the following protocol for two nonorthogonal qubit states. The input states for this machine are presented in the following form:</br> | ||