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Asymmetric Universal 1-2 Cloning: Difference between revisions
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<math>\rho_{a}^{out} = s_0 \rho_{\psi} + \frac{1 - s_0}{2} \hat{I}</math></br> | <math>\rho_{a}^{out} = s_0 \rho_{\psi} + \frac{1 - s_0}{2} \hat{I}</math></br> | ||
<math>\rho_{b}^{out} = s_1 \rho_{\psi} + \frac{1 - s_1}{2} \hat{I}</math></br> | <math>\rho_{b}^{out} = s_1 \rho_{\psi} + \frac{1 - s_1}{2} \hat{I}</math></br> | ||
Here we assume that the original qubit after the cloning is “scaled” by the factor $s_0$, while the copy is scaled by the factor <math>s_1</math>. These two scaling parameters are not independent and they are related by a specific inequality.</br> | Here we assume that the original qubit after the cloning is “scaled” by the factor $s_0$, while the copy is scaled by the factor <math>s_1</math>. These two scaling parameters are not independent and they are related by a specific inequality.</br></br> | ||
<u>'''Stage 1'''</u> Cloner State Preparation | <u>'''Stage 1'''</u> Cloner State Preparation | ||
# Prepare the original qubit and two additional blank qubits <math>m</math> and <math>n</math> in pure states: <math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle</math> | # Prepare the original qubit and two additional blank qubits <math>m</math> and <math>n</math> in pure states: <math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle</math> | ||
#Prepare <math>|\psi\rangle_{m_1,n_1} = c_1|00\rangle + c_2|01\rangle + c_3|10\rangle + c_4|11\rangle</math>, where the complex <math>c_i</math> coefficients will be specified so that the flow of information between the clones will be as desired. </br>At this stage the original qubit is not involved, but this preparation stage will affect the fidelity of the clones at the end of the process. | |||
#To prepare the <math>|\psi\rangle_{m_1,n_1}</math> state, a Unitary gate must be performed so that: | |||
<math>U|00\rangle = |\psi\rangle_{m_1,n_1}</math> | <math>U|00\rangle = |\psi\rangle_{m_1,n_1}</math> | ||
*Use following relations to specify $c_j$ in terms of <math>a</math> and <math>b</math>:</br> | *Use following relations to specify $c_j$ in terms of <math>a</math> and <math>b</math>:</br> | ||
<math>c_1 = \sqrt{\frac{s_0 + s_1}{2}}</math>, <math>c_2 = \sqrt{\frac{1 - s_0}{2}}</math>, <math>c_3 = 0</math>, <math>c_4 = \sqrt{\frac{1 - s_1}{2}}</math></br> | <math>c_1 = \sqrt{\frac{s_0 + s_1}{2}}</math>, <math>c_2 = \sqrt{\frac{1 - s_0}{2}}</math>, <math>c_3 = 0</math>, <math>c_4 = \sqrt{\frac{1 - s_1}{2}}</math></br> | ||
these $c_j$ satisfy the scaling equations and also the normalization condition of the state <math>|\psi\rangle_{m_1,n_1}</math>. They are being used to control the flow of information between the clones</br> | these $c_j$ satisfy the scaling equations and also the normalization condition of the state <math>|\psi\rangle_{m_1,n_1}</math>. They are being used to control the flow of information between the clones</br></br> | ||
<u>'''Stage 2'''</u> Cloning Circuit | <u>'''Stage 2'''</u> Cloning Circuit | ||
*The cloning circuit consists of four CNOT gates acting on original and pre-prepared qubits from stage 2. We call the original qubit <math>|\psi\rangle_{in}</math>, ``first qubit", the first ancillary qubit of $|\psi\rangle_{m_1,n_1}$, ``second qubit" and the second one, ``third qubit". The CNOT gates will act as follows: | *The cloning circuit consists of four CNOT gates acting on original and pre-prepared qubits from stage 2. We call the original qubit <math>|\psi\rangle_{in}</math>, ``first qubit", the first ancillary qubit of $|\psi\rangle_{m_1,n_1}$, ``second qubit" and the second one, ``third qubit". The CNOT gates will act as follows: | ||
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# Third CNOT acts on first and second qubit while the second qubit is control and the first qubit is the target. | # Third CNOT acts on first and second qubit while the second qubit is control and the first qubit is the target. | ||
# Forth CNOT acts on first and third qubit while the third qubit is control and the first qubit is the target. | # Forth CNOT acts on first and third qubit while the third qubit is control and the first qubit is the target. | ||
*Mathematically the cloning part of the protocol can be shown as: | *Mathematically the cloning part of the protocol can be shown as:</br> | ||
<math>|\psi\rangle_{out} = P_{3,1} P_{2,1} P_{1,3} P_{1,2} |\psi\rangle_{in} |\psi\rangle_{m_1,n_1}</math> | <math>|\psi\rangle_{out} = P_{3,1} P_{2,1} P_{1,3} P_{1,2} |\psi\rangle_{in} |\psi\rangle_{m_1,n_1}</math></br></br> | ||
<u>'''Stage 3''' Discarding ancillary state | <u>'''Stage 3'''</u> Discarding ancillary state | ||
*Discard one of the extra states. The output states will be the first and second (or third) output. | *Discard one of the extra states. The output states will be the first and second (or third) output.</br></br> | ||
'''Special case with bell state''' | '''Special case with bell state:'''</br> | ||
<u>'''Stage 1'''</u> Cloner state preparation | <u>'''Stage 1'''</u> Cloner state preparation | ||
# Prepare the original qubit and two additional blank qubits <math>m</math> and <math>n</math> in pure states: <math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle</math> | # Prepare the original qubit and two additional blank qubits <math>m</math> and <math>n</math> in pure states: <math>|0\rangle_{m_0} \otimes |0\rangle_{n_0} \equiv |00\rangle</math> | ||
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<math>\rho_{b}^{out} = F_b |\psi\rangle\langle\psi| + (1 - F_b)|\psi^{\perp}\rangle\langle\psi^{\perp}|</math></br> | <math>\rho_{b}^{out} = F_b |\psi\rangle\langle\psi| + (1 - F_b)|\psi^{\perp}\rangle\langle\psi^{\perp}|</math></br> | ||
<u>'''Stage 3'''</u> Discarding ancillary state | <u>'''Stage 3'''</u> Discarding ancillary state | ||
The same as the general case. | * The same as the general case. | ||
==Relevant Papers== | ==Relevant Papers== | ||