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(Created page with " This example protocol achieves the functionality of Quantum Cloning Machine (QCM). Phase-covariant cloning describes is a special State Dependent N-M Cloning|state-...") |
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==Pseudo Code== | ==Pseudo Code== | ||
===General Information=== | |||
''' | *The to-be-cloned states of the phase-covariant cloner are equatorial states of the form:</br> | ||
#Perform a unitary | <math>|\psi(\phi)\rangle = \frac{1}{\sqrt{2}} (|0\rangle + e^{i\phi}|1\rangle)</math></br> | ||
<math>|\ | *Equatorial states could be written in density matrix representation as:</br> | ||
<math>|\ | <math>|\psi(\phi)\rangle\langle\psi(\phi)| = \frac{1}{2}(\mathbb{I} + cos\phi\sigma_x + sin\phi\sigma_y)</math></br> | ||
where <math>\sigma_x</math> and <math>\sigma_y</math> are [[Pauli Operators]].</br> | |||
<math> | *The action of a general phase-covariant QCM is described by a map $T$ acting as follows:</br> | ||
<math>T(|\psi(\phi)\rangle\langle\psi(\phi)|) = \eta|\psi(\phi)\rangle\langle\psi(\phi)| + (1-\eta)\frac{\mathbb{I}}{2}</math></br> | |||
where <math>\eta</math> is the shrinking factor. This relation holds for all <math>\phi</math>, which guarantees that the cloning machine to act equally well on all the equatorial states. Now we investigate different types of the protocol: | |||
===Phase-covariant cloning without ancilla=== | |||
'''Input:''' <math>|\psi(\phi)\rangle|0\rangle</math> | |||
'''Output:''' <math>\frac{1}{\sqrt{2}}(|00\rangle + cos\eta e^{i\phi}|10\rangle + sin\eta e^{i\phi} |01\rangle)</math> | |||
#Perform a unitary transformation described as follows:</br> | |||
<math>U_{pc}|00\rangle = |00\rangle</math></br> | |||
<math>U_{pc}|10\rangle = cos\eta|10\rangle + sin\eta|01\rangle</math></br> | |||
===Phase-covariant cloning with ancilla=== | |||
'''<u>Stage 1</u>''' Cloner state preparation | |||
# Prepare a Bell state <math>|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math> | |||
'''<u>stage 2</u>''' Cloner transformation</br> | |||
'''Input:''' <math>|\psi(\phi)\rangle_{A}|\Phi^{+}\rangle_{BM}</math></br> | |||
'''Output:''' <math>U_{pca}|\psi(\phi)\rangle_{A}|\Phi^{+}\rangle_{BM}</br> | |||
# Perform the unitary transformation described as follows:</br> | |||
<math>U_{pca}|0\rangle|0\rangle|0\rangle = |0\rangle|0\rangle|0\rangle</math></br> | |||
<math>U_{pca}|1\rangle|0\rangle|0\rangle = (cos\eta|1\rangle|0\rangle + sin\eta|0\rangle|1\rangle)|0\rangle</math></br> | |||
<math>U_{pca}|0\rangle|1\rangle|1\rangle = (cos\eta|0\rangle|1\rangle + sin\eta|1\rangle|0\rangle)|1\rangle</math></br> | |||
<math>U_{pca}|1\rangle|1\rangle|1\rangle = |1\rangle|1\rangle|1\rangle</math></br> | |||
'''<u>Stage 3</u>''' Discarding ancillary state | |||
# Discard the extra state. mathematically, trace out the ancilla | |||
==Further Information== | ==Further Information== | ||
The phase-covariant QCM has a remarkable application in quantum cryptography since it is used for some of the eavesdropping strategies on the [[BB84 Quantum Key Distribution|BB84 QKD]]. This is due to the fact that states which are being used as BB84 states, are equatorial states and these are the only states that the eavesdropper is interested in. Both protocols mentioned in [[Phase Variant Cloning#Pseudo Code|Pseudo Code]] can be used for this analysis. | The phase-covariant QCM has a remarkable application in quantum cryptography since it is used for some of the eavesdropping strategies on the [[BB84 Quantum Key Distribution|BB84 QKD]]. This is due to the fact that states which are being used as BB84 states, are equatorial states and these are the only states that the eavesdropper is interested in. Both protocols mentioned in [[Phase Variant Cloning#Pseudo Code|Pseudo Code]] can be used for this analysis. | ||