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What does it mean to teleport a state through another state? According to most of the online sources, we use a bell state for teleportation. What would be the outcome if we use any other possible state $|\phi\rangle$ to teleport $|\psi\rangle$? Would that even make sense to do?

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  • $\begingroup$ You have to be more specific. What sort of state do you have in mind? If it's maximally entangled then it doesn't really matter, you'll just need to change the protocol a bit. If it is not maximally entangled then your teleportation won't be perfect, but what exactly the problem is depends on what exactly the state is. $\endgroup$ Apr 19 at 14:54
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As you say, it just means to replace the Bell state with the state $|\phi\rangle$. Of course you can do that in the protocol. Why would you want to? There are a couple of options:

  • Maybe you don't have a Bell state (for example, $|\phi\rangle$ is your best attempt at producing a Bell state, but it's not perfect).
  • If you incorporate some additional unitaries into the state $|\phi\rangle$, such as $|\phi\rangle=(|0+\rangle+|1-\rangle)/\sqrt{2}$, you can actually use the teleportation to also apply a unitary to your state. In this case, a Hadamard gate.
  • Teleportation need not be a real operation. There's the whole study of CP-maps and the Choi-Jamiolkowski isomorphism which is kind of the above two items combined, but is not generally considered a real operation, but more of a mathematical tool.
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  • $\begingroup$ Thank you very much, that confirmed my intuition about the topic! I'm right now investigating CP-maps and how work with mixed states in a circuit model. I guess it should then also be possible to teleport a mixed state through a pure state? (I should probably open another question). Can you recommend any good resources to get started on that? $\endgroup$
    – Thommy257
    Apr 19 at 15:09
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    $\begingroup$ To be honest, it's really not any different to using pure states. You just have to do everything in the density matrix formalism for consistency. Of, you take a purification of your density matrix. $\endgroup$
    – DaftWullie
    Apr 19 at 15:16
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Answering the question in the title, the general idea is that if $|\phi\rangle$ is some bipartite (hopefully) entangled state, shared by two parties, call them Alice and Bob, that means that Alice and Bob's measurement results will be correlated, for many (possibly all) choice of measurements they can perform.

One can exploit such correlations to "teleport" some (local) state $|\psi\rangle$ from (say) Alice to Bob. To do this, Alice performs measurements after entangling her share of $|\phi\rangle$ with $|\psi\rangle$. Her measurement results thus contain some information about $|\psi\rangle$. But these are also correlated with Bob's measurement results. It is therefore possible, under suitable conditions, to perform some local measurement on Bob's side of the state that result in his state being identical to $|\psi\rangle$.

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