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Probabilistic Cloning
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===General case=== '''<u>Stage 1</u>''' Ancilla preparation # 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></br> '''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> # 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> '''<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></br> '''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>. ##'''If''' The output of the measurement is the state <math>|A_0\rangle</math> ##'''Then''' the protocol is successful and the output is the desired clones. ##'''Else''' Abort
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