Editing Optimal Universal N-M Cloning

This protocol achieves the functionality of [[Quantum Cloning]]. Symmetric or Optimal Universal <math>N \rightarrow M</math> quantum cloning machine (QCM) transformation, acting on N original qubits, <math>(M - N)</math> blank qubits and an extra system which usually contains the states of the QCM itself, and produces M identical copies. Mathematically, this transformation is a [[Unitary transformation]] (It preserves the inner product). This machine is universal meaning it is not depending on the input states and works for all the possible inputs. Also, the machine is symmetric so all the output clones are the same and have the same [[fidelity]] with the original states. We investigate the optimal protocol which has the maximum possible average fidelity over all the states (produces best clones in average for all of the states).

'''Tags:''' Universal Cloning, optimal cloning, N to M cloning, symmetric cloning,  [[Asymmetric Universal 1-2 Cloning|Asymmetric Cloning]], [[Category: Building Blocks]][[:Category: Building Blocks|Building Blocks]], [[Quantum Cloning]], copying quantum states, [[:Category: Quantum Functionality|Quantum Functionality]][[Category: Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], [[Probabilistic Cloning]] 

==Assumptions==
* We assume that all of the original states are identical and also all of the output copies will be identical at the end of the protocol (In other words, the final output state belongs to the symmetric subspace of M qubits).
* We assume that the protocol is an approximate deterministic cloning protocol, meaning that in every round it produces approximate copies of the original states.
==Outline==
The steps of the protocol will be as follow:
* Prepare your N original states and <math>(M - N)</math> blank states. This machine acts on these states and also on internal states of the QCM.
* Perform the operation of the QCM which is a transformation taking these states to M identical states as close as possible to the original states. This unitary operation can be implemented by quantum gates (or other equivalent quantum computing models)
*Discard the extra machine states that have been used in the previous step. Mathematically this means that you should <math>trace out</math> the states of the machine. The final output states will be the M approximately similar copies.

==Notation==
**<math>|\psi\rangle^{\otimes N}:</math> N initial states
**<math>|\psi^{\perp}\rangle:</math> The state orthogonal to <math>|\psi\rangle</math>
**<math>R_{j}:</math> ancillary and internal states of the QCM
**<math>U_{N,M}:</math> Unitary operation describing the QCM
**<math>F_{N \rightarrow M}:</math> Fidelity of the USQCM showing how close the M output copies are to the N original states

==Properties==
*'''The protocol-'''
**assumes that all of the original states are identical and also all of the output copies will be identical at the end of the protocol (In other words, the final output state belongs to the symmetric subspace of M qubits).
**assumes that the protocol is an approximate deterministic cloning protocol, meaning that in every round it produces approximate copies of the original states.
*'''Fidelity of the <math>N \rightarrow M</math> UQCM:''' <math>F_{N \rightarrow M} = \frac{MN + M + N}{M(N + 2)}</math>
*'''Special case of 1 qubit to 2 qubits:''' <math>F_{1 \rightarrow 2} = \frac{5}{6}</math>

==Protocol Description==
'''Input:''' j qubits where <math>R_{j}</math> are ancillary and internal states of the QCM.

<u>'''Stage 1'''</u> State preparation
#Prepare N initial states: <math>|\psi\rangle^{\otimes N}</math> and <math>M - N</math> blank states: <math>|0\rangle^{\otimes M - N}</math>
<u>'''Stage 2'''</u> Unitary transformation
# Perform the following unitary transformation on input state <math>|N\psi\rangle = |\psi\rangle^{\otimes N} |0\rangle^{\otimes M - N} |R\rangle</math>
<math>U_{N,M} |N\psi\rangle = \sum_{j=0}^{M-N} \alpha_{j} |(M - j)\psi, j\psi^{\perp}\rangle \otimes R_{j}(\psi)</math></br>
where <math>\alpha_{j} = \sqrt{\frac{N + 1}{M + 1}} \sqrt{\frac{(M - N)!(M - j)!}{(M - N - j)! M!}}</math></br>
<u>'''Stage 3:'''</u> Trace out the QCM state
#Trace out the state of the QCM in <math>R_{j}</math> states.

==Further Information==
<div style='text-align: right;'>''*contributed by Mina Doosti''</div>