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I'd like to understand exactly what people mean when they speak of quantum channels. As I understand it, we can represent a channel by a set of Kraus operators, $M_i$, which satisfy $\sum_{i}M^{\dagger}_i M_i = 1$.

Any unitary operator (or even probabilistic combination of unitaries?) satisfies this definition. Any generalized measurement also satisfies this.

So is a quantum channel essentially either a unitary evolution or a measurement? Or is there something deeper that I am missing?

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    $\begingroup$ You're not missing much. It's those two options - with a continuous slider between the two. $\endgroup$ – Emilio Pisanty Dec 14 '18 at 16:51
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A quantum channel is a generalized measurement (POVM measurement) where you forget the outcome. (Note that this in particular includes unitaries: A "measurement" with only one outcome.)

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  • $\begingroup$ I assume that in the case of the unitary operator, the forgetting doesn't apply. Is that right? $\endgroup$ – user1936752 Dec 14 '18 at 16:10
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    $\begingroup$ @user1936752 That's the 'only one outcome' case, where you can't 'forget' anything as there's no information there to forget (as the measurement outcome is just 'the system exists'). $\endgroup$ – Emilio Pisanty Dec 14 '18 at 16:53

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