Yeah, this also bothered me. The notion of “probability distribution over quantum states” is not a good notion: the matrix I is both (|0\rangle \langle 0|+|1\rangle \langle 1|) and (|a\rangle \langle a|+|b\rangle \langle b|) for any other orthogonal basis. The fact that these should be treated equivalently seems totally arbitrary. The point is that density matrix mechanics is the notion of probability for quantum states, and can be formalized as such (dynamics of informational lower bounds given observations). I was sort of getting at this with the long “explaining probability to an alien” footnote, but I don’t think it landed (and I also don’t have the right background to make it precise)
Yeah, this also bothered me. The notion of “probability distribution over quantum states” is not a good notion: the matrix I is both (|0\rangle \langle 0|+|1\rangle \langle 1|) and (|a\rangle \langle a|+|b\rangle \langle b|) for any other orthogonal basis. The fact that these should be treated equivalently seems totally arbitrary. The point is that density matrix mechanics is the notion of probability for quantum states, and can be formalized as such (dynamics of informational lower bounds given observations). I was sort of getting at this with the long “explaining probability to an alien” footnote, but I don’t think it landed (and I also don’t have the right background to make it precise)