Finding out about quantum systems

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One of the cool, frustrating things about quantum theory is how the once-innocuous concept of "measurement" gets really complicated. I'd like to understand how we find out about the universe around us, and how to reconcile (a) everyday experience, (b) experiments on quantum systems, and (c) our theory of quantum measurements. In this talk, I'll try to braid three [apparently] separate research projects into the beginnings of an answer. I'll begin from the premise that you make a measurement to find something out, then attack some specific questions: "How do we find out about quantum systems?", "What can we find out about quantum systems?", and finally "What do we actually know, afterward?" I'll give precise statements of these questions, then present [partial] answers.