Editing Optimal Universal N-M Cloning
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
== Functionality Description== | |||
Universal <math>N \rightarrow M</math> quantum cloning machine (QCM), transformation acting on N original qubits, $(M - N)$ 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:'''Building blocks, [[Quantum Cloning]], Universal Cloning, optimal cloning, N to M cloning, symmetric cloning, copying quantum states, quantum functionality, [[Asymmetric Universal 1-2 Cloning|Asymmetric Cloning]] | ||
== Requirements == | |||
==Outline== | |||
==Example:== | |||
===Outline=== | |||
---- | |||
The steps of the protocol will be as follow: | 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. | * 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) | * 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. | *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. | ||
===Properties=== | |||
== | ---- | ||
*'''Notations Used:''' | |||
**<math>|\psi\rangle^{\otimes N}:</math> N initial states | **<math>|\psi\rangle^{\otimes N}:</math> N initial states | ||
**<math>|\psi^{\perp}\rangle:</math> The state orthogonal to <math>|\psi\rangle</math> | **<math>|\psi^{\perp}\rangle:</math> The state orthogonal to <math>|\psi\rangle</math> | ||
Line 19: | Line 23: | ||
**<math>F_{N \rightarrow M}:</math> Fidelity of the USQCM showing how close the M output copies are to the N original states | **<math>F_{N \rightarrow M}:</math> Fidelity of the USQCM showing how close the M output copies are to the N original states | ||
*'''The protocol-''' | *'''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 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. | **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 | *'''Fidelity of the $N \rightarrow M$ 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> | *'''Special case of 1 qubit to 2 qubits:''' <math>F_{1 \rightarrow 2} = \frac{5}{6}</math> | ||
== | ===Pseudo Code=== | ||
---- | |||
'''Input:''' j qubits where <math>R_{j}</math> are ancillary and internal states of the QCM. | '''Input:''' j qubits where <math>R_{j}</math> are ancillary and internal states of the QCM. | ||
Line 38: | Line 42: | ||
#Trace out the state of the QCM in <math>R_{j}</math> states. | #Trace out the state of the QCM in <math>R_{j}</math> states. | ||
== | ==Relevant Papers== | ||