I haven’t closely read the details on the hypothetical experiments yet, but I want to comment on the technical details of the quantum mechanics at the beginning.
In quantum mechanics, probabilities of mutually exclusive events still add: P(A∨B)=P(A)+P(B). However, things like “particle goes through slit 1 then hits spot x on screen” and “particle goes through slit 2 then hits spot x on screen” aren’t such mutually exclusive events.
This may seem like I’m nit-picking, but I’d like to make the point by example. Let’s say we have a state |ψ⟩=1√2|A⟩−1√2|B⟩ where ⟨A|B⟩=0. If we simply add the complex amplitudes to try to calculate P(A∨B), we get 0; in actuality, we should get P(A∨B)=12+12=1 as we expect from classical logic.
Here’s where I bad-mouth the common way of writing the Born rule in intro quantum material asP(A)=|⟨ψ|A⟩|2 and the way I’d been using it. By writing the state as |ψ⟩ and the event as |A⟩ we’ve made it look like they’re both naturally represented as vectors in a Hilbert space. But the natural form of a state is as a density matrix, and the natural form of an event is as an orthogonal projection; I want to focus on events and projections. For mutually exclusive events A and B with projections ΠA and ΠB, the event A∨B has the corresponding projection ΠA∨B=ΠA+ΠB.
So where’s the adding of amplitudes? Let’s pretend I didn’t just say states are naturally density matrices and let’s take the same state |ψ⟩ from above and an arbitrary projection ΠX corresponding to some event. The Born rule takes the following form:
This is notably not just an A contribution plus a B contribution; the other terms are the interference terms. Skipping over what a density matrix is, let’s say we have a density matrix ρ=12ρ1+12ρ2. The Born rule for density matrices is
P(X)=tr(ρΠX)=12tr(ρ1ΠX)+12tr(ρ2ΠX)
Now this one is just a sum of two contributions, with no interference.
This ended up longer and more rambling than I’d originally intended. But I think there’s a lot to the finer details of how probabilities and amplitudes behave that are worth emphasizing.
Thanks for taking the time to write this response up! This made some things click together for me.
In quantum mechanics, probabilities of mutually exclusive events still add: P(A∨B)=P(A)+P(B). However, things like “particle goes through slit 1 then hits spot x on screen” and “particle goes through slit 2 then hits spot x on screen” aren’t such mutually exclusive events.
That’s a good point; ⟨A|B⟩=0 is a strong precise notation of “mutually exclusive” in quantum mechanics. I meant to say that “events whose amplitudes you add” would often naturally be considered mutually exclusive under classical reasoning. (“Slit 1 then spot x” and “slit 2 then spot x” sure sound exclusive). And that if the phases are unknown then the classical reasoning actually works.
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 Born rule takes the following form:
Ah! So the first Born rule you give is the only one I saw in my QM class way back when.
The second one I hadn’t seen. 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:
The wiki page: Therefore, unpolarized light cannot be described by any pure state, but can be described as a statistical ensemble of pure states in at least two ways (the ensemble of half left and half right circularly polarized, or the ensemble of half vertically and half horizontally linearly polarized). These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state.
This is really interesting. It’s satisfying to see things I was confusedly wondering about answered formally by von-Neumann almost 100 years ago.
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 haven’t closely read the details on the hypothetical experiments yet, but I want to comment on the technical details of the quantum mechanics at the beginning.
In quantum mechanics, probabilities of mutually exclusive events still add: P(A∨B)=P(A)+P(B). However, things like “particle goes through slit 1 then hits spot x on screen” and “particle goes through slit 2 then hits spot x on screen” aren’t such mutually exclusive events.
This may seem like I’m nit-picking, but I’d like to make the point by example. Let’s say we have a state |ψ⟩=1√2|A⟩−1√2|B⟩ where ⟨A|B⟩=0. If we simply add the complex amplitudes to try to calculate P(A∨B), we get 0; in actuality, we should get P(A∨B)=12+12=1 as we expect from classical logic.
Here’s where I bad-mouth the common way of writing the Born rule in intro quantum material asP(A)=|⟨ψ|A⟩|2 and the way I’d been using it. By writing the state as |ψ⟩ and the event as |A⟩ we’ve made it look like they’re both naturally represented as vectors in a Hilbert space. But the natural form of a state is as a density matrix, and the natural form of an event is as an orthogonal projection; I want to focus on events and projections. For mutually exclusive events A and B with projections ΠA and ΠB, the event A∨B has the corresponding projection ΠA∨B=ΠA+ΠB.
So where’s the adding of amplitudes? Let’s pretend I didn’t just say states are naturally density matrices and let’s take the same state |ψ⟩ from above and an arbitrary projection ΠX corresponding to some event. The Born rule takes the following form:
This is notably not just an A contribution plus a B contribution; the other terms are the interference terms. Skipping over what a density matrix is, let’s say we have a density matrix ρ=12ρ1+12ρ2. The Born rule for density matrices is
Now this one is just a sum of two contributions, with no interference.
This ended up longer and more rambling than I’d originally intended. But I think there’s a lot to the finer details of how probabilities and amplitudes behave that are worth emphasizing.
Thanks for taking the time to write this response up! This made some things click together for me.
That’s a good point; ⟨A|B⟩=0 is a strong precise notation of “mutually exclusive” in quantum mechanics. I meant to say that “events whose amplitudes you add” would often naturally be considered mutually exclusive under classical reasoning. (“Slit 1 then spot x” and “slit 2 then spot x” sure sound exclusive). And that if the phases are unknown then the classical reasoning actually works.
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”.
Ah! So the first Born rule you give is the only one I saw in my QM class way back when.
The second one I hadn’t seen. 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.
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.