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I'm not an expert on this, but I've tried to read this article and this article from Nature. I've also seen this answer which claims that you can use quantum teleportation combined with a classical channel to send an EPR pair that can be used later to perform quantum teleportation again.

Can you create a setup a system in which a finite number of particles can be used for quantum teleportation both ways between two people? It was ambiguous to me whether a continuous classical connection was needed to support the quantum teleportation system or whether only an initial setup was needed.

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I will try to answer your question. It is not so natural to think about the number of particles needed since any experimental setup you use for this will require quite (a lot) of particles. Usually one rather thinks about how many qubits you need to be able to store at a specific time. That is, how good quantum memory you need. For quantum teleportation you only need to store the qubit being teleported for as long as it takes to send classical communication between the parties (speed of light).

Regarding the classical communication, you need to send two classical bits per round of teleportation. That is every time you teleport one qubit from A to B you need to send two bits from A to B over some classical channel.

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