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When we talk about superdense coding, we say that Alice and Bob share an entangled qubit and then after this, Alice transmits her qubit over to Bob after performing the necessary manipulation.

What does this mean experimentally? How do you transmit a single qubit of an entangled qubit over?

And, if I understand correctly, only when Bob has both the Alice's and his qubits of the entangled pair can he recover the Bell state that the entangled pair is in.

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It means moving the qubit from one place to another. The fact that the qubit happens to be entangled with some other qubit elsewhere is irrelevant. The technical details of transmission depend on the physical implementation of the qubit. It could be transmission of a photon through free space or fiber or physical transportation of a device with the qubit from one place to another.

In principle, it could even be quantum state teleportation, although this is silly in the context of superdense coding because qubit teleportation itself requires transmission of two classical bits.

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Your understanding is almost correct. As said in the previous answer it is irrelevant that they are entangled. However, it is crucial for bob to have both to be aware of the actual state. This is a somewhat complicated argument to make but instant communication is impossible because Alice is not aware of what she is communicating and Bob isn’t aware of what he is receiving from Alice. If Bob has both he can ideally measure.

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