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It is well-known that in https://arxiv.org/abs/quant-ph/0312190 that error correction can be performed by teleportation on the logical level, so prepare one logical Bell pair and (logical?) Bell measurement.

I'm wondering, is there a way, that we start with noisy physical Bell pairs, and can still perform such teleported-error-correction? e.g. in the Steane code case, if we use the standard Knill scheme, the ancilla state would be $|\overline{00}\rangle + |\overline{11}\rangle/\sqrt{2}$, $|\overline{0}\rangle$ is encoded with 7 physical qubits. Is there any way I can do this error-correction with $\left(|00\rangle+|11\rangle/\sqrt{2} \right)^{\otimes 7}$ where each of these physical Bell pair is noisy?

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Yes. You just prepare, on one side, a logical Bell pair and teleport one half of it through the noisy bell pairs. When you run error correction on it, it effectively corrects the errors that were in the noisy Bell pairs. You may want to check out sections 5.3 and 5.4 of https://arxiv.org/abs/quant-ph/9604024.

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  • $\begingroup$ Thanks, but I don't think I understand fully. Is it possible to draw a diagram? I'm not sure why you would need a logical Bell pair to start with. Basically what I want is, I have some state in the logical space, and I hope to use physical noisy Bell pairs to perform this teleported-error-correction. $\endgroup$
    – AndyLiuin
    Commented Apr 25 at 18:35
  • $\begingroup$ OK, sure you can do it that way around as well. It's all equivalent. $\endgroup$
    – DaftWullie
    Commented Apr 26 at 6:31

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