Editing State Dependent N-M Cloning
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==Assumptions== | ==Assumptions== | ||
* We assume that no [[ancillary states]] (test qubits) are not needed for this state-dependent protocol. | * We assume that no [[ancillary states]] (test qubits) are not needed for this state-dependent protocol. | ||
* We assume that this state-dependent QCM is symmetric. | * We assume that this state-dependent QCM is [[symmetric]]. | ||
* We assume that the transformation is [[unitary]], acting on the Hilbert space of <math>M</math> qubits and thus, the following relation holds between the inner product of input and output states:</br> | * We assume that the transformation is [[unitary]], acting on the Hilbert space of <math>M</math> qubits and thus, the following relation holds between the inner product of input and output states:</br> | ||
<math>\langle\alpha_{NM}|\beta_{NM}\rangle = (\langle a|b \rangle)^N = S^N</math> | <math>\langle\alpha_{NM}|\beta_{NM}\rangle = (\langle a|b \rangle)^N = S^N</math> | ||
* We assume that protocol is optimal on the input states meaning that it will give the maximum possible [[fidelity]] value on average for a given set of states. | * We assume that protocol is optimal on the input states meaning that it will give the maximum possible [[fidelity]] value on average for a given set of states. | ||
==Outline== | ==Outline== | ||
This state-dependent QCM can only act effectively on two sets of states. These states are non-orthogonal (orthogonal case is trivial). For a state-dependent cloner, the only thing that we need is to perform a transformation which takes N identical input states and <math>M-N</math> blank states, to M copies such that it yields the best copies for the given special input states. It is important to note that this transformation is useful when we have a prior information about the input states. | This state-dependent QCM can only act effectively on two sets of states. These states are non-orthogonal ([[orthogonal case is trivial]]). For a state-dependent cloner, the only thing that we need is to perform a transformation which takes N identical input states and <math>M-N</math> blank states, to M copies such that it yields the best copies for the given special input states. It is important to note that this transformation is useful when we have a prior information about the input states. | ||
== | ==Notations Used== | ||
*<math>|a\rangle, |b\rangle:</math> Two nonorthogonal input states | *<math>|a\rangle, |b\rangle:</math> Two nonorthogonal input states | ||
*<math>S:</math> The inner (scalar) product of two input states <math>|a\rangle</math> and <math>|b\rangle</math> | *<math>S:</math> The inner (scalar) product of two input states <math>|a\rangle</math> and <math>|b\rangle</math> | ||
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A and B are presented in the [[State Dependent N-M Cloning#Pseudo Code|Pseudo Code]] section. | A and B are presented in the [[State Dependent N-M Cloning#Pseudo Code|Pseudo Code]] section. | ||
== | ==Pseudo Code== | ||
'''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> | '''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> | ||
'''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> | '''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> |