It might be fun to work through an example of quantum "teleportation", to see why you need a classical message as well as an entangled pair. (The following is adapted from "Quantum Theory: Concepts and Methods" by Asher Peres.)
Let A and B be two spin-(1/2) particles in the entangled state:
\[ \psi_{AB} = (u_A d_B - d_A u_B) / \sqrt{2}\]
where \(\{u_P, d_P\}\) are the spin-up and spin-down states for particle P.
Let X be a spin-(1/2) particle in an unknown state:
\[$\psi_X = \alpha u_X + \beta d_X\]
giving a total state for the three particles:
\[\psi_{XAB} = (\alpha u_X + \beta d_X) (u_A d_B - d_A u_B) / \sqrt{2}\]
or, expanding this out:
\[\psi_{XAB} = (\alpha u_X u_A d_B - \alpha u_X d_A u_B +
\beta d_X u_A d_B - \beta d_X d_A u_B) / \sqrt{2}\]
Suppose Alice has particle A of the entangled pair, along with particle X that she wants to "teleport" to Bob, while Bob has particle B.
Alice performs a measurement on X and A that resolves
their state into one of the following four orthonormal states:
\[e_{00} = (u_X d_A - d_X u_A) / \sqrt{2}\]
\[$e_{01} = (u_X d_A + d_X u_A) / \sqrt{2}\]
\[e_{10} = (u_X u_A - d_X d_A) / \sqrt{2}\]
\[e_{11} = (u_X u_A + d_X d_A) / \sqrt{2}\]
Given the result of that measurement, if Bob performs the correct operation
on particle B for each possible outcome, that will guarantee that after he's done so, any subsequent measurements on B's spin degree of freedom will be identical to those you'd get for the original state of particle X.
Greg Egan
npub12c…gd38p
2025-01-24 09:42:17 UTC
Author Public Key
npub12cuzzqzv8e8y7na77a9zv47mx2v6pec60qs5h7takzmv5p0mw0fsmgd38pPublished at
2025-01-24 09:42:17 UTCEvent JSON
{
"id": "ccd113a85bea79c0df203c966b4fa4225d36eac99daa1270e20ffa7637a68481",
"pubkey": "563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3",
"created_at": 1737711737,
"kind": 1,
"tags": [
[
"proxy",
"https://mathstodon.xyz/@gregeganSF/113882676454425414",
"web"
],
[
"proxy",
"https://mathstodon.xyz/users/gregeganSF/statuses/113882676454425414",
"activitypub"
],
[
"L",
"pink.momostr"
],
[
"l",
"pink.momostr.activitypub:https://mathstodon.xyz/users/gregeganSF/statuses/113882676454425414",
"pink.momostr"
],
[
"-"
]
],
"content": "It might be fun to work through an example of quantum \"teleportation\", to see why you need a classical message as well as an entangled pair. (The following is adapted from \"Quantum Theory: Concepts and Methods\" by Asher Peres.)\n\nLet A and B be two spin-(1/2) particles in the entangled state:\n\n\\[ \\psi_{AB} = (u_A d_B - d_A u_B) / \\sqrt{2}\\] \n\nwhere \\(\\{u_P, d_P\\}\\) are the spin-up and spin-down states for particle P.\n\nLet X be a spin-(1/2) particle in an unknown state:\n\n\\[$\\psi_X = \\alpha u_X + \\beta d_X\\]\n\ngiving a total state for the three particles:\n\n\\[\\psi_{XAB} = (\\alpha u_X + \\beta d_X) (u_A d_B - d_A u_B) / \\sqrt{2}\\]\n\nor, expanding this out:\n\n\\[\\psi_{XAB} = (\\alpha u_X u_A d_B - \\alpha u_X d_A u_B +\n \\beta d_X u_A d_B - \\beta d_X d_A u_B) / \\sqrt{2}\\]\n\nSuppose Alice has particle A of the entangled pair, along with particle X that she wants to \"teleport\" to Bob, while Bob has particle B.\n\nAlice performs a measurement on X and A that resolves\ntheir state into one of the following four orthonormal states:\n\n \\[e_{00} = (u_X d_A - d_X u_A) / \\sqrt{2}\\]\n\n\\[$e_{01} = (u_X d_A + d_X u_A) / \\sqrt{2}\\]\n\n \\[e_{10} = (u_X u_A - d_X d_A) / \\sqrt{2}\\]\n\n \\[e_{11} = (u_X u_A + d_X d_A) / \\sqrt{2}\\]\n\nGiven the result of that measurement, if Bob performs the correct operation\non particle B for each possible outcome, that will guarantee that after he's done so, any subsequent measurements on B's spin degree of freedom will be identical to those you'd get for the original state of particle X.",
"sig": "ba1a94723763ae52c11af32774327ce2ca316fd301b20f946abd96ccdaa659b5bef643becbc34bdc3b78507861a20e2d5fc98084f2ad50cf1fa57ec584d8cab4"
}