michael_nielsen

Of course, we could have just done the algebra for the whole 3-qubit system up front. It's really not very complicated, & that's how I've always done it in the past. But I noticed this approach a few hours ago, & rather like it. It's in some sense longer, but more elegant.

No claim that this is new! But it was new & pleasing to me, a slightly new way of seeing an old friend.

I really like the first part of the proof: it's just a quantum one-time pad. I don't quite know how to think about the second part of the proof. Why does measuring in the |+>, |-> basis work? It's easy to prove it does, but I don't have a really good explanation for why.

That's a measurement based uncomputation. If a qubit Q is a computational basis function of other qubits R, you can get rid of Q by measuring it in the X basis and using the result to control phasing operations on R. Useful tool. e.g. see Appendix C of https://arxiv.org/abs/1902.02134 

That appendix doesn't seem to address my question. Note that I'm extremely familiar with the facts you mention - I've written papers based on those observations (and a review: https://arxiv.org/pdf/quant-ph/0504097.pdf … ). But none of that gives a conceptual explanation.