- #1
koroshii
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- Homework Statement
- Recall that in the Quantum Teleportation protocol, Alice and Bob start withthe state [itex](\alpha\ket{0}+\beta\ket{1})\otimes\left(\frac{\ket{00}+\ket{11}}{\sqrt{2}}\right)[/itex], and end in one of the four following states probabilistically, after Alice’s measurement: [tex]\ket{00}\otimes(\alpha\ket{0}+\beta\ket{1})[/tex] [tex]\ket{10}\otimes(\alpha\ket{0}-\beta\ket{1})[/tex] [tex]\ket{01}\otimes(\beta\ket{0}+\alpha\ket{1}) [/tex] [tex]|11〉⊗(-\beta\ket{0}+\alpha\ket{1}) [/tex]
Give the 2×2 matrix representation of the unitary operator Bob must apply to turn his state into the desired state [itex]\alpha\ket{0}+\beta\ket{1}[/itex] for each of the four possibilities when Alice communicates the result of her measurement (00, 01, 10, or 11).
- Relevant Equations
- This appears to be the last step of the Quantum Teleportation protocol in my notes
"Alice sends a 2-bit classical message to Bob telling him her measurement result. He applies a local unitary; i.e., [itex]I_{A_1, A_2}\otimes U_B[/itex], to transform his state to [itex]\ket{\phi} = \alpha\ket{0}+\beta\ket{1}[/itex]"
My trouble might be from how I interpret the problem. Alice and Bob are entangled. After Alice makes the measurement both of their states should collapse to one of these states with a certain probability. (Unless my understand of how entanglement is wrong.) The way I am understand the question is to find an operator [itex]U_B [/itex] (using the first one as an example) such that
[tex]I_{A_1, A_2}\otimes U_B\left(\ket{00}\otimes(\alpha\ket{0}+\beta\ket{1})\right) = \alpha\ket{0}+\beta\ket{1}[/tex]
[tex] = I_{A_1, A_2}\ket{00}\otimes U_B(\alpha\ket{0}+\beta\ket{1}) = \alpha\ket{0}+\beta\ket{1}[/tex]
My professor doesn't explicitly say it but I'm assuming [itex] I_{A_1, A_2} [/itex] is an identity, so the first term is unaffected. That already seems to make this equation impossible. If the first term can somehow be ignored, [itex]U_B = I[/itex] and I could pretty easily find the other unitary operators.
This whole section on entanglement and non-locality really hit me by surprise and I am still trying to wrap my head around it. Could someone point me in the right direction?
Thank you.
[tex]I_{A_1, A_2}\otimes U_B\left(\ket{00}\otimes(\alpha\ket{0}+\beta\ket{1})\right) = \alpha\ket{0}+\beta\ket{1}[/tex]
[tex] = I_{A_1, A_2}\ket{00}\otimes U_B(\alpha\ket{0}+\beta\ket{1}) = \alpha\ket{0}+\beta\ket{1}[/tex]
My professor doesn't explicitly say it but I'm assuming [itex] I_{A_1, A_2} [/itex] is an identity, so the first term is unaffected. That already seems to make this equation impossible. If the first term can somehow be ignored, [itex]U_B = I[/itex] and I could pretty easily find the other unitary operators.
This whole section on entanglement and non-locality really hit me by surprise and I am still trying to wrap my head around it. Could someone point me in the right direction?
Thank you.