That’s a good point; ⟨A|B⟩=0 is a strong precise notation of “mutually exclusive” in quantum mechanics. (...)
I’d be remiss at this point not to mention Gleason’s theorem: once you accept that notion of mutually exclusive events, the Born rule comes (almost) automatically. There’s a relatively large camp that accepts Gleason’s theorem as a good proof of why the Born rule must be the correct rule, but there’s of course another camp that’s looking for more solid proofs. Just on a personal note, I really like this paper, but I haven’t seen much discussion about it anywhere.
But that’s kind of vague, and my whole introduction was sloppy. I added it after the fact; maybe should have stuck with just the “three experiments”.
The general idea of adding terms without interference effects when you average over phases is solid. I will have to think about it more in the context of alternative probability rules; I’ve never thought about any relation before.
From the wiki page, it sounds like a density matrix is a way of describing a probability distribution over wavefunctions. Which is what I’ve spent some time thinking about (though in this post I only wrote about probability distributions over a single amplitude). Except it isn’t so simple: many distributions are indistinguishable, so the density matrix can be vastly smaller than a probability distribution over all relevant wavefunctions.
And some distributions (“ensembles”) that sound different but are indistinguishable:
(...)
This is really interesting. It’s satisfying to see things I was confusedly wondering about answered formally by von-Neumann almost 100 years ago.
Yeah, some of these sorts of things that are really important for getting a good grasp of the general situation don’t often get any attention in undergraduate classes. Intro quantum classes often tend to be crunched for time between teaching the required linear algebra, solving the simple, analytically tractable problems, and getting to the stuff that has utility in physics. I happened to get exposed to density matrices relatively early as an undergraduate, but I think there’s probably a good number of students who didn’t see it until graduate school.
Roughly speaking, there’s two big uses for density matrices. One, as you say, is the idea of probability distributions over wavefunctions (or ‘pure states’) in the minimal way. But the other, arguably more important one, is simply being able to describe subsystems. Only in extraordinary cases (non-entangled systems) is a subsystem of some larger system going to be in a pure state. Important things like the no-communication theorem are naturally expressed in terms of density matrices.
Von Neumann invented/discovered such a huge portion of the relevant linear algebra behind quantum mechanics that it’s kind of ridiculous.
I’d be remiss at this point not to mention Gleason’s theorem: once you accept that notion of mutually exclusive events, the Born rule comes (almost) automatically. There’s a relatively large camp that accepts Gleason’s theorem as a good proof of why the Born rule must be the correct rule, but there’s of course another camp that’s looking for more solid proofs. Just on a personal note, I really like this paper, but I haven’t seen much discussion about it anywhere.
The general idea of adding terms without interference effects when you average over phases is solid. I will have to think about it more in the context of alternative probability rules; I’ve never thought about any relation before.
Yeah, some of these sorts of things that are really important for getting a good grasp of the general situation don’t often get any attention in undergraduate classes. Intro quantum classes often tend to be crunched for time between teaching the required linear algebra, solving the simple, analytically tractable problems, and getting to the stuff that has utility in physics. I happened to get exposed to density matrices relatively early as an undergraduate, but I think there’s probably a good number of students who didn’t see it until graduate school.
Roughly speaking, there’s two big uses for density matrices. One, as you say, is the idea of probability distributions over wavefunctions (or ‘pure states’) in the minimal way. But the other, arguably more important one, is simply being able to describe subsystems. Only in extraordinary cases (non-entangled systems) is a subsystem of some larger system going to be in a pure state. Important things like the no-communication theorem are naturally expressed in terms of density matrices.
Von Neumann invented/discovered such a huge portion of the relevant linear algebra behind quantum mechanics that it’s kind of ridiculous.