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I know the common proof of the no-cloning theorem assuming the cloning apparatus acts unitarily on the system and leading this to a contradiction.

But what about a non-unitarian apparatus that involves measurements? I'm pretty sure cloning is still impossible, but I don't know how to prove it in full generality, since you could do very elaborate measurement procedures that could involve entanglement swapping or unitarily evolving parts.

Somebody knows a short proof?