State Dependent N-M Cloning: Difference between revisions

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State Dependent N-M Cloning (edit)

Revision as of 18:43, 16 March 2019

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==Pseudo Code==
==Pseudo Code==
'''General Case'''
'''Input:''' <math>|a\rangle^{\otimes N} \otimes |0\rangle^{\otimes M-N}</math> or <math>|b\rangle^{\otimes N} \otimes |0\rangle^{\otimes M-N}</math></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>
'''Output:''' <math>|\alpha_{NM}\rangle = (A + B)|a^M\rangle + (A - B)|b^M\rangle</math> or <math>|\beta_{NM}\rangle = (A - B)|a^M\rangle + (A + B)|b^M\rangle</math>
<math>\rho_{a}^{out} = s_0 \rho_{\psi} + \frac{1 - s_0}{2} \hat{I}</math></br>
#Perform a unitary on N input states of the form <math>|x\rangle^{\otimes N}</math>, with <math>x = a, b</math> and also on <math>(M-N)</math> blank states, which its action is discribed by:</br>
<math>\rho_{b}^{out} = s_1 \rho_{\psi} + \frac{1 - s_1}{2} \hat{I}</math></br>
<math>|\alpha_{NM}\rangle = U_{NM} (|a\rangle^{\otimes N} \otimes |0\rangle^{\otimes M-N}) = (A + B)|a^M\rangle + (A - B)|b^M\rangle</math>,</br>
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>
<math>|\beta_{NM}\rangle = U_{NM} (|b\rangle^{\otimes N} \otimes |0\rangle^{\otimes M-N}) = (A - B)|a^M\rangle + (A + B)|b^M\rangle</math>, </br>
<u>'''Stage 1'''</u> Cloner State Preparation
where</br>
# 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>
<math>A = \frac{1}{2}\sqrt{\frac{1 + S^N}{1 + S^M}}, \quad B = \frac{1}{2}\sqrt{\frac{1 - S^N}{1 - S^M}}</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>
*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>
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
*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.
#Second CNOT acts on first and third qubit while the first qubit is control and the third 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.
*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></br></br>
<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>
'''Special case with bell state:'''</br>
<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 <math>|\psi\rangle_{m_1,n_1} = a|\Phi^{+}\rangle_{m_1,n_1} + b|0\rangle_{m_1} |+\rangle_{n_1}</math>, where <math>|\Phi^{+}\rangle</math> is a [[Bell state]] and <math>|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)</math>. In this case, the density matrix representation of the output states will be:</br>
<math>\rho_{a}^{out} = (1 - b^2) |\psi\rangle\langle\psi| + \frac{b^2}{2} \hat{I}</math></br>
<math>\rho_{b}^{out} = (1 - a^2) |\psi\rangle\langle\psi| + \frac{a^2}{2} \hat{I}</math></br>
<u>'''Stage 2'''</u> Cloning Circuit
* The cloning circuit is exactly the same as the general case. after the cloning circuit, the output state will be:</br>
<math>a|\psi\rangle_{in} |\Phi^+\rangle_{m,n} + b|\psi\rangle_m |\Phi^+\rangle_{in,n}</math>
The reduced density matrix of two clones A and B can be written in terms of their fidelities:</br>
<math>\rho_{a}^{out} = F_a |\psi\rangle\langle\psi| + (1 - F_a)|\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
* The same as the general case.


==Further Information==
==Further Information==
One of the most important applications of quantum cloning is to analyze the security of [[Quantum Key Distribution]] (QKD) protocols. Usually, an eavesdropper is supposed in QKD protocols os assumed to be able to perform any attacks, including copying of states used by parties involved in the protocol. Special states which are being used in different QKD protocols will lead to different state-dependent cloners for analyzing cloning attacks.
One of the most important applications of quantum cloning is to analyze the security of [[Quantum Key Distribution]] (QKD) protocols. Usually, an eavesdropper is supposed in QKD protocols os assumed to be able to perform any attacks, including copying of states used by parties involved in the protocol. Special states which are being used in different QKD protocols will lead to different state-dependent cloners for analyzing cloning attacks.
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