02 June 1999, “Please Do”

You know what I'm really trying to get at here: I want the structure of quantum mechanics to be extremal for some kind of more abstract (i.e., QM-independent) information-disturbance problem. But the first thing that needs to be done is to pin down what quantum mechanics itself has to say.

Another thing that keeps coming to mind is that the assumptions behind Gleason's theorem might have something to do with the allegory I presented you with at Charlie and Theo's house. God said, “Ok I will make information gathering invasive, but not so invasive that you will be forever science-less.” The connection is this. I think one can split the assumption behind Gleason up into at least two parts: (1) The questions that can be asked of a quantum system only correspond to orthonormal bases over some vector space, with there being no good notion of measuring two distinct questions simultaneously (there being no good notion of an AND operation). And (2) It is the task of physical theory to give probabilities for the outcomes of those questions, but we can say at least this much about the probabilities. They are context-independent in the sense that, for a given outcome, it does not matter which physical question (which basis) we've associated it with. I.e., the probabilities can be assumed to be of the form of a frame function.