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Asymmetric Universal 1-2 Cloning: Difference between revisions
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'''Tags:'''Building blocks, [[Quantum Cloning]], Universal Cloning, asymmetric cloning, symmetric cloning, copying quantum states, quantum functionality, symmetric or [[Optimal Universal N-M Cloning|Optimal Cloning]], [[Probabilistic Cloning|Probabilistic Cloning]] | '''Tags:'''Building blocks, [[Quantum Cloning]], Universal Cloning, asymmetric cloning, symmetric cloning, copying quantum states, quantum functionality, symmetric or [[Optimal Universal N-M Cloning|Optimal Cloning]], [[Probabilistic Cloning|Probabilistic Cloning]] | ||
==Assumptions== | |||
* We assume that the original input qubit is unknown and the protocol is independent of the original input state (universality). | |||
* The output copies are not identical and we are able to control the likelihood (fidelity) of the output copies to the original state by pre-preparing the ancillary states with special coefficients. | |||
==Outline== | |||
In this protocol, there are three main stages. The first stage is a preparation stage. we prepare two ancillary states with special coefficients which will lead to the desired flow of the information between copied states. The original state will not be engaged at this stage. At the second stage the cloner circuit will act on all the three states (the original and two other states) and at the second stage the two copied state will appear at two of the outputs and one of the outputs should be discarded. The Procedure will be as following: | In this protocol, there are three main stages. The first stage is a preparation stage. we prepare two ancillary states with special coefficients which will lead to the desired flow of the information between copied states. The original state will not be engaged at this stage. At the second stage the cloner circuit will act on all the three states (the original and two other states) and at the second stage the two copied state will appear at two of the outputs and one of the outputs should be discarded. The Procedure will be as following: | ||
* Prepare two blank states and then perform a transformation taking these two states to a new state with pre-selected coefficients in such a way that the information distribution between two final states will be the desired.[[File:Asymmetriccloner.jpg|right|thumb|1000px| Graphical representation of the network for the asymmetric cloner. The CNOT gates are shown in the cloner section with control qubit (denoted as <math>\bullet</math> ) and a target qubit (denoted as <math>\circ</math> ). We separate the preparation of the quantum copier from the cloning process itself.]] | * Prepare two blank states and then perform a transformation taking these two states to a new state with pre-selected coefficients in such a way that the information distribution between two final states will be the desired.[[File:Asymmetriccloner.jpg|right|thumb|1000px| Graphical representation of the network for the asymmetric cloner. The CNOT gates are shown in the cloner section with control qubit (denoted as <math>\bullet</math> ) and a target qubit (denoted as <math>\circ</math> ). We separate the preparation of the quantum copier from the cloning process itself.]] | ||
*Perform the cloner circuit. the cloner circuit consists of four CNOT gate acting in all the three input qubits. For every CNOT gate, we have a control qubit (which indicates whether or not the CNOT should act) and a target qubit (which is the qubit that CNOT gate acts on it and flips it). | *Perform the cloner circuit. the cloner circuit consists of four CNOT gate acting in all the three input qubits. For every CNOT gate, we have a control qubit (which indicates whether or not the CNOT should act) and a target qubit (which is the qubit that CNOT gate acts on it and flips it). | ||
* The two asymmetric clones will appear at two of the outputs (depending on the preparation stage) and the other output should be discarded. | * The two asymmetric clones will appear at two of the outputs (depending on the preparation stage) and the other output should be discarded. | ||
==Notations Used== | |||
== | |||
**<math>|\psi\rangle_{in}:</math>The original input state | **<math>|\psi\rangle_{in}:</math>The original input state | ||
**<math>\rho_{\psi_{in}}:</math> The density matrix of the input pure state equal to <math>|\psi_{in}\rangle\langle\psi_{in}|</math> | **<math>\rho_{\psi_{in}}:</math> The density matrix of the input pure state equal to <math>|\psi_{in}\rangle\langle\psi_{in}|</math> | ||
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**<math>|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle):</math> Plus state. The eigenvector of Pauli X | **<math>|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle):</math> Plus state. The eigenvector of Pauli X | ||
==Properties== | |||
*The protocol assumes that the original input qubit is unknown and the protocol is independent of the original input state (universality). | *The protocol assumes that the original input qubit is unknown and the protocol is independent of the original input state (universality). | ||
*The output copies are not identical and we are able to control the likelihood (fidelity) of the output copies to the original state by pre-preparing the ancillary states with special coefficients. | *The output copies are not identical and we are able to control the likelihood (fidelity) of the output copies to the original state by pre-preparing the ancillary states with special coefficients. | ||
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<math>F_a = 1 - \frac{b^2}{2}, F_b = 1 - \frac{a^2}{2}</math> | <math>F_a = 1 - \frac{b^2}{2}, F_b = 1 - \frac{a^2}{2}</math> | ||
==Pseudo Code== | |||
'''General Case''' | '''General Case''' | ||
For more generality, we use the [[density matrix]] representation of the states which includes [[mixed states]] as well as [[pure states]]. For a simple pure state <math>|\psi\rangle</math> the density matrix representation will be <math>\rho_{\psi} = |\psi\rangle\langle\psi|</math>. Let us assume the initial qubit to be in an unknown state <math>\rho_{\psi}</math>. Our task is to clone this qubit universally, i.e. input-state independently, in such a way, that we can control the scaling of the original and the clone at the output. In other words, we look for output which can be represented as below:</br> | For more generality, we use the [[density matrix]] representation of the states which includes [[mixed states]] as well as [[pure states]]. For a simple pure state <math>|\psi\rangle</math> the density matrix representation will be <math>\rho_{\psi} = |\psi\rangle\langle\psi|</math>. Let us assume the initial qubit to be in an unknown state <math>\rho_{\psi}</math>. Our task is to clone this qubit universally, i.e. input-state independently, in such a way, that we can control the scaling of the original and the clone at the output. In other words, we look for output which can be represented as below:</br> | ||