How does this work? You can rewrite \(\psi_{XAB}\) as:
\[\psi_{XAB} = (e_{00} (-\alpha u_B - \beta d_B) +
e_{01} (-\alpha u_B + \beta d_B) +
e_{10} ( \alpha d_B + \beta u_B) +
e_{11} ( \alpha d_B - \beta u_B)) / 2\]
The four states for particle B on the RHS can all be transformed by an
appropriate operation into:
\[\alpha u_B + \beta d_B\]
up to an overall phase, either by doing nothing or by measuring a spin,
i.e. \(U_{nn}\) applied to the term following \(e_{nn}\) will always yield \(\alpha u_B +
\beta d_B\) if you choose:
\[U_{00} = - I\]
\[U_{01} = - \sigma_z\]
\[U_{10} = \sigma_x\]
\[U_{11} = i \sigma_y\]
So after Bob receives a classical two-bit message from Alice telling him which of the \(e_{nn}\) she measured particles X and A to be in, he can then perform the corresponding unitary operation \(U_{nn}\) on particle B ... and the result will be that particle B will end up in the same state as particle X was in initially!
But Bob can't do anything until he receives the message from Alice telling him the result of her measurement. Nothing is "teleported" until he performs the correct, matching operation himself, and he can't know what that is until he has heard from Alice.
Greg Egan
npub12c…gd38p
2025-01-24 09:45:31 UTC
in reply to nevent1q…m78r
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npub12cuzzqzv8e8y7na77a9zv47mx2v6pec60qs5h7takzmv5p0mw0fsmgd38pPublished at
2025-01-24 09:45:31 UTCEvent JSON
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"content": "How does this work? You can rewrite \\(\\psi_{XAB}\\) as:\n\n\\[\\psi_{XAB} = (e_{00} (-\\alpha u_B - \\beta d_B) +\n e_{01} (-\\alpha u_B + \\beta d_B) +\n e_{10} ( \\alpha d_B + \\beta u_B) +\n e_{11} ( \\alpha d_B - \\beta u_B)) / 2\\]\n\nThe four states for particle B on the RHS can all be transformed by an\nappropriate operation into:\n\n \\[\\alpha u_B + \\beta d_B\\]\n\nup to an overall phase, either by doing nothing or by measuring a spin,\ni.e. \\(U_{nn}\\) applied to the term following \\(e_{nn}\\) will always yield \\(\\alpha u_B +\n\\beta d_B\\) if you choose:\n\n\\[U_{00} = - I\\]\n\n\\[U_{01} = - \\sigma_z\\]\n\n\\[U_{10} = \\sigma_x\\]\n\n\\[U_{11} = i \\sigma_y\\]\n\nSo after Bob receives a classical two-bit message from Alice telling him which of the \\(e_{nn}\\) she measured particles X and A to be in, he can then perform the corresponding unitary operation \\(U_{nn}\\) on particle B ... and the result will be that particle B will end up in the same state as particle X was in initially!\n\nBut Bob can't do anything until he receives the message from Alice telling him the result of her measurement. Nothing is \"teleported\" until he performs the correct, matching operation himself, and he can't know what that is until he has heard from Alice.",
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