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State Dependent N-M Cloning: Difference between revisions
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'''Tags:''' [[Quantum Cloning#Protocols|Non-Universal Cloning]], State-Dependent Cloning, [[Category: Building Blocks]] [[:Category: Building Blocks|Building Blocks]], [[Quantum Cloning]], Universal Cloning, asymmetric cloning, copying quantum states, [[:Category: Quantum Functionality|Quantum Functionality]][[Category: Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]],symmetric or [[Optimal Universal N-M Cloning|Optimal or Symmetric Cloning]], [[Probabilistic Cloning]] | '''Tags:''' [[Quantum Cloning#Protocols|Non-Universal Cloning]], State-Dependent Cloning, [[Category: Building Blocks]] [[:Category: Building Blocks|Building Blocks]], [[Quantum Cloning]], Universal Cloning, asymmetric cloning, copying quantum states, [[:Category: Quantum Functionality|Quantum Functionality]][[Category: Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]],symmetric or [[Optimal Universal N-M Cloning|Optimal or Symmetric Cloning]], [[Probabilistic Cloning]] | ||
==Assumptions== | ==Assumptions== | ||
* We assume that no ancillary ( | * 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 | * 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 | 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== | ==Notations Used== | ||
*<math>|a\rangle, |b\rangle:</math> Two nonorthogonal input states | *<math>|a\rangle, |b\rangle:</math> Two nonorthogonal input states | ||
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==Properties== | ==Properties== | ||
* State-dependent cloning machine only transforms a set of two | * State-dependent cloning machine only transforms a set of two non-orthogonal input states, parametrized as follows: | ||
<math>|a\rangle = cos\theta |0\rangle + sin\theta |1\rangle,</math></br> | <math>|a\rangle = cos\theta |0\rangle + sin\theta |1\rangle,</math></br> | ||
|b\rangle = sin\theta |0\rangle + cos\theta |1\rangle,</math></br> | |b\rangle = sin\theta |0\rangle + cos\theta |1\rangle,</math></br> | ||
where <math>\theta \in [0, \pi/4]</math> and their scalar product (or inner product) is specified as <math>S = \langle a|b \rangle = sin 2\theta</math>. | where <math>\theta \in [0, \pi/4]</math> and their scalar product (or inner product) is specified as <math>S = \langle a|b \rangle = sin 2\theta</math>. | ||
* | * The condition that transformation should be unitary adds following constraint on the scalar product of final 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> | ||
*Fidelity Claims | *Fidelity Claims | ||