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Asymmetric Universal 1-2 Cloning: Difference between revisions
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==Pseudo Code== | ==Pseudo Code== | ||
===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> | ||
<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 | Here we assume that the original qubit after the cloning is “scaled” by the factor <math>s_0</math>, 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> | ||
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#To prepare the <math>|\psi\rangle_{m_1,n_1}</math> state, a Unitary gate must be performed so that: | #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 | *Use following relations to specify <math>c_j</math> 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 | these <math>c_j</math> 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 | *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 <math>|\psi\rangle_{m_1,n_1}</math>, ``second qubit" and the second one, ``third qubit". The CNOT gates will act as follows: | ||
#First CNOT acts on first and second qubit while the first qubit is control and the second qubit is the target. | #First CNOT acts on first and second qubit while the first qubit is control and the second qubit is the target. | ||
#Second CNOT acts on first and third qubit while the first qubit is control and the third qubit is the target. | #Second CNOT acts on first and third qubit while the first qubit is control and the third qubit is the target. | ||
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<u>'''Stage 3'''</u> 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.</br></br> | *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:=== | |||
<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> | ||