Universal Superposition of Orthogonal States: Difference between revisions

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###'''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.
##'''Mixed Output Case:'''  
* The output will be:<math>|\Psi^{\mu,\nu} \rangle = C (\alpha |\psi\rangle + \beta e^{i\eta_{\mu,\nu}} |\psi^\perp\rangle) </math>
###Always accept. The protocol is perfect.
*<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.
### The output will be:<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.


==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.
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