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Suppose there are 3 parties, of which 2 pairs share an EPR pair and can communicate classically. What is a protocol that results in the third pair sharing an EPR pair?

That is the problem I'm given, I'm lost on how to do it, let alone what it means to "share an EPR pair" in the end.

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For two parties to share an EPR pair means that each party has one qubit, and these two qubits together are in state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ (or one of the other Bell pairs).

The entanglement swapping protocol is described, for example, in Wikipedia. It is the same as traditional (I almost wrote "classical" but caught myself in time) quantum teleportation, but applied to qubit in mixed state instead of pure state.

Let's say

  • Alice has qubit Q1,
  • Bob has qubits Q2 (entangled with Q1) and Q3,
  • and Carol has qubit Q4 (entangled with Q3).

If Bob teleports the state of qubit Q2 to Carol (using up the entanglement of qubits Q3 and Q4 in the process), Carol's qubit Q4 will end up entangled with Alice's qubit Q1.

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