Why do channels arise from "failing to record measurement outcomes"? In Preskill's notes, the need for quantum channels began with the following situation: System A starts out in a pure state and interacts with system B, therefore forming a joint state of system AB. We then imagine measuring system B (the pointer) but fail to record the measurement outcomes, therefore, forcing us to take the final state being the sum of all post-measurement states weighed by their probability outcome.
My question is, what do we 'mean' by failing to record measurement outcomes? Is this a human error wherein we forget to record a measurement outcome, or is it something entirely different?
(Preskill's Lecture notes 3, page 11)
http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf
 A: The way I'd put it is that using channels is what you get when you want to describe the dynamics of part of a quantum state. In other words, there's a large quantum state evolving in the standard way via Schroedinger, but you only care about/have access to part of the state. This is therefore an "effective description" of the evolution.
Not having access to part of the system turns out to be mathematically equivalent to saying that that part of the system (which I'll refer to as the "environment" in the following) was somehow measured but we don't know the measurement outcomes.
This doesn't mean that there's necessarily actually someone measuring the environment. It means that the reduced dynamic on the system isn't affected by whether the environment was measured or not. By definition, we don't have access to the state of the environment, and thus won't know whether it was measured. As it turns out, it is often convenient to think of this situation as a measurement having been performed on the environment and us not knowing the outcome.
Another way to think about this is that we describe the reduced dynamics on the system in terms of what we'd get conditionally to someone measuring the environment and finding a specific outcome. Except we don't know the outcome and thus have to consider convex mixtures of these possibilities.
A: Quantum mechanical systems interact with the environment. If they do so, we can think of the environment as effectively measuring the system -- formally, this is completely equivalent. However, the information is "hidden" in the environment, and we don't have access to it (and it is typically very hard to access it, as the information is "smeared out" over the $\sim 10^{23}$ degrees of freedom of the environment). Thus, we can think of this as "measuring and ignoring the outcome", i.e., averaging over all possible outcomes (with the respective probabilities).
