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So we know we cannot fully copy a quantum state. But doesn't stimulated emission does just that? Say, a photon in a particular qubit state $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$ passes through an excited atom, which is then stimulated to produce another photon of the exact same state $|\psi\rangle$.

So how does this work? Clearly no (projective) measurement has been made on the photon by the atom, so it's a unitary evolution of some sort? Or perhaps that's where I misunderstand; that the evolution is some CP map instead because by focusing on the photon (or more accurately the quantum electromagnetic field) alone we have 'traced out' the mechanisms inside the atom?

If that's the case, then what is the map?

As a beginning student, this is somewhat puzzling.

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In short, you cannot have stimulated emission without spontaneous emission (hermiticity requirement) . The spontaneous part will destroy the fidelity of your copying

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