That's a really good question. I'd love to hear the answer from a physicist in the know.

Assume you could overcome the uncertainty principle or observer effect and you could fully determine the initial conditions of the entire system. You knew the position and momentum of every atom. You knew the spin of every electron. Etc. Etc. Etc.

Doesn't the inherent nature of quantum mechanics say that it's still impossible to predict the state of that system at some future time? Interactions only occur probabilistically, and there is no way to predict them a priori.

Physicist here.

In terms of predicting the "future state", it depends on what you want to call the "state".

If I know the complete wave function of the system, then knowing the wave function at a future point in time is trivial. Just apply the time evolution operator.

However, knowing the wave function at a given point in time doesn't tell me the position or momentum - it just tells me the probability with which I'll measure a given position or momentum. So knowing the "state" still means that my measurements will have random components.

On a slightly different note, when you talk about knowing the exact position and momentum of every particle, you're not talking about overcoming a physical limitation, but a mathematical one. To put it differently, if I know that the momentum is exactly zero, I do know that the position is. The problem is that the position is NaN. If the position isn't NaN, then I know longer know the momentum isn't precisely defined.

> If I know the complete wave function of the system, then knowing the wave function at a future point in time is trivial. Just apply the time evolution operator.

But wouldn't the regular measurement cause a collapse onto a randomly-chosen eigenstate of the measured operator? That is, if we have a PRNG based on a regularly measuring the lava lamp, then to predict the state after N steps, we not only have the issue of the randomly chosen N-th measurement but also have to take into account the random results of the N-1 previous measurements, which can potentially evolve into entirely new directions.

Overall, it's difficult to place a lava lamp over human time scales, as both are far from the usual quantum/classical limits: We know that even in thousands of years’ time, Earth will still revolve mostly deterministically (in the classical sense) around the sun. Similarly, electrons will hardly ever behave deterministically. Lava lamps and a couple of years are oddly in between.

You're a couple of abstraction levels lower than where I aimed my post :-)

You're absolutely right that everything breaks down after a measurement. However, I'd begun my hypothetical by assuming that we had some magical technique for getting the complete wave function. If our measurements again give us the complete wave function, then we just use the time evolution operator on that again.

I guess what I'm trying to say is that we should be okay after N-1 measurements, as long as we're allowed to see the result of that final measurement. You're right, though, that we rapidly lose the ability to make any predictions if some jerk keeps measuring the system. I think that there's also an Everettian argument that, if you haven't given me the complete wave function for the jerk making the measurements, then you didn't really give me the the complete wave function of the system. However, that's pushing outside my area of expertise.

>If I know the complete wave function of the system, then knowing the wave function at a future point in time is trivial. Just apply the time evolution operator.

I was under the impression that the Bell experiments indicated that there was randomness not accounted for by our inability to measure with 100% accuracy (because of the uncertainty principle). Doesn't the falsity of hidden variable theory mean that actual randomness is present in quantum events, and we can't predict the future state perfectly even if we had the exact wave function of the system?

I could be totally off base here; please set me on the right track!

That's a good question and you're actually pretty close to the right track. The catch is the difference between the wave function and the actual measurement. Quantum mechanics and Schrodinger's equation gives us the wave function. If you know the current wave function, there's a well established way to calculate the wave function at a future point in time.

The catch is that the wave function doesn't tell us values - only probabilities. So knowing the probability distribution at any given point in time doesn't make anything less random because it's all still probability and not actual measurements.

As an analogy, imagine a casino where the roulette wheel has an LCD label for each number. Each round, they change the layout of the wheel. Sometime, they make all the labels black. Other times, it's 2/3rds red and all the numbers are primes. They also have a big book in the corner that tells you what the layout of the roulette wheel will be each round. As a result, if I put down a bet, you can tell me the odds of my bet coming up each round. However, you still can't actually tell me what will WIN the round.

The layout of the labels is like the wave function, the pages of the book are the time evolution operator, and the roulette ball is the fundamental randomness of quantum mechanics. The results are still random, just as Bell said that they must be, but we are at least allowed to know the odds.