This is a great thought experiment, thank you. I'm not totally clear how the machines could work without actually taking a measurement, though. It sounds like you're saying the 2nd machine (P = 0.5) takes measurements (and therefore "paints" the balls), but the other two don't?
I've heard of the apocryphal "half-silvered mirror", but I don't get why reflection isn't an observation/interaction there either.
> I don't get why reflection isn't an observation/interaction there either.
I know this comment is going to get lost in the noise, but that is a really excellent point, one of the best that has been raised here so far. This is a point that is often glossed over, but it is actually really important, and quite challenging to explain without getting deep into the weeds. The answer is that passing through a half-silvered mirror is an observation/interaction, but it is special because it can be practically reversed by using additional mirrors so that you can get back to a state where you can no longer tell what the outcome of the "measurement" was. All measurements are reversible in principle, but some are irreversible in practice because the number of things you'd have to reverse is just too large. And in particular, by the time a measurement has affected the state of any macroscopic system (like a ball) it is absolutely impossible to reverse in practice, though not in principle. This process of becoming irreversible-in-practice is called "decoherence".
I agree with this. But it's worth pointing out that lots of good physicists don't. It's a statement of the Everett/relative state/many worlds interpretation, which is simply too weird for many people to accept. That's why there are about a dozen other interpretations of quantum measurement theory, which are all weird in other ways that I can't accept.
The weird part is precisely this: "you can get back to a state where you can no longer tell what the outcome of the 'measurement' was." In other words, at lunchtime you believed that a horizontally polarised photon hit your nose at 10am in morning, and you were right. Now it's dinner time, you don't believe that, and you would be wrong if you did. If the Everett interpretation doesn't pose a massive challenge to your ideas about reality and human identity, you haven't understood it. There are physics professors who picture photons choosing which way to go at a beam splitter, then transmitting the news backwards in time, because that seems more plausible to them.
Of course, interpretations are not science. Everyone agrees how an experiment would go: any attempt to reverse the interaction of the photon with your nose and brain would fail, because thermodynamics. From a purely scientific viewpoint, it simply doesn't matter how many other yous are superposed in parallel universes, because their existence or lack of it has no consequences that (any of?) you can observe. But scientists are as fascinated by this as everyone else is.
> It's a statement of the Everett/relative state/many worlds interpretation
No, it isn't. I've said nothing about many-worlds, only reversibility. And on that point everyone agrees.
> Now it's dinner time, you don't believe that
You really need to read the link above. It goes into all that in great detail. But the TL;DR here is that if it's dinner time, you haven't actually reversed the measurement, notwithstanding your current mental state with respect to the photon.
I don't like claims to authority, but maybe you should read the papers I've published about Bell inequalities too? :-p
It sounds like you're saying that, in principle, physical processes are all reversible. (Although that is often thermodynamically impossible in practice.) You're also saying that it's impossible in principle for someone to learn the result of a measurement in the morning, then unlearn it when the measurement is reversed during the afternoon. I don't see how there could be a self-consistent interpretation of quantum measurement where both those things are true.
How am I supposed to do that? You haven't provided and references and your profile is empty.
> in principle, physical processes are all reversible
Correct. This is a straightforward mathematical property of the Schroedinger equation.
> it's impossible in principle for someone to learn the result of a measurement in the morning, then unlearn it when the measurement is reversed during the afternoon
That's right. But that's not because it's impossible to reverse the measurement. It's because when you reverse a measurement you don't just "unlearn" the result.
That's right. That's why when this experiment is actually done, the mirror is typically rigidly mounted to an optical bench, which is sitting on the surface of a planet. If the mirror were freely floating in zero G, the outcome would be different. It is a worthwhile exercise to calculate how small the mass of the mirror would have to be before you would actually notice a difference in the results.
> I'm not totally clear how the machines could work without actually taking a measurement, though.
Here you're hitting on the heart of the Measurement Problem. In QM as it is understood today, unlike classical mechanics, there are two fundamentally different kinds of interactions between objects: quantum interactions and measurement. Quantum interactions are linear changes to the wave function, while measurements perform a non-linear update to the wave function (it becomes one for the measured value and 0 everywhere else).
Unfortunately, we do not have any theory so far that explains what is the difference between a quantum interaction and a measurement. The experiment I described works with 'machines' that interact quantically with the 'balls', but does not reproduce if the machines measure the state of the balls.
I will note that in the Many Worlds Interpretation, the measurement problem is somewhat different - it states that the state of the universe is always described by a wave function, but that parts of the wave which are sufficiently separated can no longer perceive each other somehow, usually called branching. Precisely when, why or how this happens are just as unknown, though decoherence seems to play a role
Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
However, the wave function at different places interacts with the environment and start to shift in phase, eventually becoming unable to interfere with itself - this is called decoherence, and is a valid explanation about why and how we can't observe wave-like behaviors at large scales or in hot systems.
On the other hand, we can only postulate, based on observations, that when a particle interacts with a measurement device, the measurement device will show a single value with a probability determined by the amplitude of the particle's wave function at that point. We can postulate that the wave function collapses, or we can postulate that the device branches out into different devices in different worlds (enough such devices&worlds to achieve the probability distribution through observer selection somehow), or many other ways of formulating the Born rule. But whichever way you put it, this rule must be added to your system to predict experimental results, it does not derive from the Schrodinger equation.
>Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
I suppose it's destructive interference. It's qualitatively interesting, but its observation is complicated by orthogonal states: when you multiply orthogonal states you get zero. If you can thoroughly dismantle the state to observe it, you still can do it only on microscale, then you'll have a problem lifting it to macroscale evading destructive interference while orthogonal states are all over the place. Anyway, Schrodinger equation describes behavior of quantum states with mathematical precision and the math is quite conclusive that a linear equation behaves in a linear way. When you feel intuition doesn't get you much, you can resort to math, that's why math is seen as an indispensable part of science, because intuition isn't guaranteed to work, which is exactly your case.
>it does not derive from the Schrodinger equation
MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
> MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
This posits the notion of an observer that only observes one outcome, whereas the SE predicts that an observer will observe several different outcomes with different amplitudes. The MWI is postulating that we should only look at each outcome separately.
Furthermore, it is not possible to derive the actual probability value from the wave function amplitude without some additional postulate equivalent to the Born rule, for example that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome.
The result of calculation of the state of observer is linear evolution: the state of observer splits and entangles with the observed state and each part observes the respective outcome. Ironically Copenhagen gave the same result for Schrodinger's cat experiment: even before measurement it's known what states are in superposition and those states are "dead" and "alive", and it's still known without measurement too.
>that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome
If you mean the number, the norm of each state of observer that observes the respective outcome can be calculated. The statistics over the outcomes can be calculated too.
I've heard of the apocryphal "half-silvered mirror", but I don't get why reflection isn't an observation/interaction there either.