Universal Superposition of Orthogonal States: Difference between revisions
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*'''Output:''' <math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | *'''Output:''' <math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ||
*At this stage the following gates will be performed respectively: | *At this stage the following gates will be performed respectively: | ||
#'''S gate (Controled Swap):''' Performing this gate will interchange the input qubits. The total states before and after performing this gate are as follows: | |||
#<math>|\Psi_0\rangle = \alpha|1\rangle |\psi\rangle |\psi^\perp\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ##<math>|\Psi_0\rangle = \alpha|1\rangle |\psi\rangle |\psi^\perp\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ||
#<math>|\Psi_1\rangle = \alpha|1\rangle |\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ##<math>|\Psi_1\rangle = \alpha|1\rangle |\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ||
#'''CNOT gate:''' The total state after this step is as follows: | |||
#<math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ##<math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ||
'''<u>Stage 2</u>''' Measurements</br></br> | '''<u>Stage 2</u>''' Measurements</br></br> | ||
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*'''Output:''' The superposed state <math>|\Psi\rangle</math> | *'''Output:''' The superposed state <math>|\Psi\rangle</math> | ||
#Measure qubit 1 (the ancilary qubit) in X basis and the qubit 2 in Z basis. | #Measure qubit 1 (the ancilary qubit) in X basis and the qubit 2 in Z basis. | ||
##'''Pure Output Case:''' | |||
##''if:''' The output of the X measurement is 0 AND the output of Z measurement is 1 | ###''if:''' The output of the X measurement is 0 AND the output of Z measurement is 1 | ||
##'''Then:''' Accept the round | ###'''Then:''' Accept the round | ||
##'''Else:''' Reject. | ###'''Else:''' Reject. | ||
* The successful output is in the form: </br> | * The successful output is in the form: </br> | ||
<math>|\Psi\rangle = C(\alpha |\psi\rangle + \beta e^{i\eta} |\psi^\perp\rangle)</math></br> | <math>|\Psi\rangle = C(\alpha |\psi\rangle + \beta e^{i\eta} |\psi^\perp\rangle)</math></br> | ||
<math>e^{i\eta}</math> is a relative phase which is <math>\frac{\langle1|\psi\rangle}{\langle0|\psi^\perp\rangle}</math></br> | <math>e^{i\eta}</math> is a relative phase which is <math>\frac{\langle1|\psi\rangle}{\langle0|\psi^\perp\rangle}</math></br> | ||
#'''Mixed Output Case:''' Always accept. The protocol is perfect. | |||
* The output will be: | * The output will be:<math>|\Psi^{\mu,\nu} \rangle = C (\alpha |\psi\rangle + \beta e^{i\eta_{\mu,\nu}} |\psi^\perp\rangle) </math> | ||
<math>|\Psi^{\mu,\nu} \rangle = C (\alpha |\psi\rangle + \beta e^{i\eta_{\mu,\nu}} |\psi^\perp\rangle) </math | *<math>e^{i\eta_{\mu,\nu}}</math> is a relative phase which depends on the outputs of the measurements but in all cases, the superposition has the desired form and weights. | ||
<math>e^{i\eta_{\mu,\nu}}</math> is a relative phase which depends on the outputs of the measurements but in all cases, the superposition has the desired form and weights. | |||
==Further Information== | ==Further Information== | ||
# [https://arxiv.org/abs/1708.04360 DKK(2017)] The above protocol | # [https://arxiv.org/abs/1708.04360 DKK(2017)] The above protocol | ||
# [https://arxiv.org/abs/1505.04955 OGHW(2016)] The first paper that talks about and proves the no-superposition theorem. Also in this paper, they present a probabilistic protocol for superposing two arbitrary (but not completely unknown) states where we know the overlaps of them with a fixed reference state. this protocol, is also restricted to a set of input states. | # [https://arxiv.org/abs/1505.04955 OGHW(2016)] The first paper that talks about and proves the no-superposition theorem. Also in this paper, they present a probabilistic protocol for superposing two arbitrary (but not completely unknown) states where we know the overlaps of them with a fixed reference state. this protocol, is also restricted to a set of input states. | ||