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I think what you're describing is the infamous Measurement Problem:

1. Hamiltonian evolution is linear

2. Measurements produce nonlinear projections onto eigenvectors in accordance with the Born rule

Note that this projection is a function of the initial state - but a probabilistic function, so therefore not reversible.

There isn't really a satisfying answer to this question yet. Decoherence explains how the system evolves to a mixture of eigenstates but not the collapse into a single one, nor the Born rule.

Note that this projection is a function of the initial state - but a probabilistic function, so therefore not reversible.

I understand what you want to say, but I think it is not correct to call it a function. The term »probabilistic function« does appear in Google results but it seems not to be a common one. I also spent a few minutes looking for a mathematical formalization of a random number generator as a function with no inputs returning a random number adhering to a given probability distribution but could not find anything. Mathematicians seem to be pretty good at avoiding problematic definitions and according to Wikipedia some do this deliberately. The best formalization I could find would be a function from the initial state to a random variable but I was unable to figure out in a couple of minutes how I would mathematically formalize the step from a random variable to a concrete result of a single trail.

Decoherence explains how the system evolves to a mixture of eigenstates but not the collapse into a single one [...]

I never managed to really wrap my head around decoherence, I always only had a quite fuzzy picture of it. But now it is starting to click, it's about the transition from superposition states to probabilistic mixtures, the transition from uncertainty due to superposition to uncertainty due to ignorance.

> The only other option seems to be that the final state is still a superposition and we have to reach for something like the multiverse interpretation to reconcile the fact that we indeed observe a definite final state.

What's wrong with that?

I just finished teaching a Jr/Sr undergrad Quantum Mechanics course last semester, so I'll take a stab at answering this.

There are two approaches to answering this question. One of them invokes the concept of "the collapse of the wavefunction" at the moment of measurement: when an observer actually measures the outcome of such an experiment, the universe probabilistically selects one of the possible outcomes as "true" and proceeds from there. This "collapse" process is explicitly non-unitary. You may ask, "What entities qualify as 'observers' to trigger collapse?", and the answer is basically "It's not well-defined." As you might gather from that, it's not clear to me that you can include an "observer" in this sense as part of the overall wavefunction of the universe at all, though perhaps people who have studied the foundations of quantum mechanics in more detail could give a better answer.

The other option is pretty much exactly what you've described in your final paragraph: you don't invoke "collapse of the wavefunction" at all, and the final quantum state of the universe after measurement is indeed a superposition of "|0> and observer who saw '0'" and "|1> and observer who saw '1'". Because anything we'd call an "observer" will be a large, complex system, there will very rapidly be a vast number of tiny, "pseudo-random" differences between the two branches of the wavefunction, ensuring that there won't be opportunities for the two branches to experience significant constructive or destructive interference in the future: they'll evolve pretty much completely independently from then on. (This concept is called "decoherence", which you've made reference to, and I'm definitely not an expert on it, but I'm told it's become quite important in research on quantum computing.) There's still a probabilistic question of which branch of the wavefunction a given observer might experience (and understanding why that happens is an enormously subtle question; I've seen some work by Sean Carroll among others trying to make progress on it), but once a given observer has measured a particular outcome to the experiment, the math of quantum mechanics guarantees that they'll continue to measure consistent results after that. This is more or less the Everett "many worlds" interpretation of quantum mechanics.

As you might gather, I'm partial to the latter interpretation, in part because it's where the math seems to most naturally lead (the first time I felt like I really understood quantum mechanics was after a very math-heavy graduate course taught by Robert Geroch), and in part because I've never been able to make sense of what wavefunction collapse might mean (and never been happy about its violation of unitarity).

As you might gather from that, it's not clear to me that you can include an "observer" in this sense as part of the overall wavefunction of the universe at all, though perhaps people who have studied the foundations of quantum mechanics in more detail could give a better answer.

I did some more reading and it seems that at least some try to do what I also considered, try to attribute the randomness in the outcome to the randomness and uncertainty in the initial state of the measuring device, observer, and environment. This then raises the question why all kinds of measuring devices produce the same distribution of measurement results if the concrete outcome essentially only depends on the initial state of the measuring device. And this also has the smell of a hidden variable theory where the hidden variables are in the measurement device, would this be compatible with EPR?

I also took away some things to follow up on from your comments on Everett but unlike you I don't really like multiverses and so this will have to wait for some time. I however agree that Everett makes the most sense with regard to completeness and consistency.

There's a third option, which historically arose out of the issues you describe with your first approach. This is to deny that quantum theory is a description of an objective, observer-independent reality. Rather it is a kind of "personal rule book" which informs an agent's beliefs about future interactions with the world.

https://en.wikipedia.org/wiki/Quantum_Bayesianism

https://arxiv.org/pdf/1003.5209.pdf

It's been a long time since my QM course, but, assuming by unitary you mean deterministic, then the answer is that time evolution of a particle's state is NOT deterministic. It is probabilistic. For large collections of particles, the probabilities come to overwhelmingly favor a single average state, but the state of any single particle remains non-deterministic.

This is in contrast to, for example, classical statistical mechanics where individual particles are considered to behave deterministically, but we simply lack the ability to measure their behavior. In (the Copenhagen interpretation of) QM there is no underlying determinism at all.

You might not like this idea (it's certainly non-intuitive), but decades of experiments have shown that the alternatives are all, most likely, wrong.

You are not too far off, as a first approximation you can say unitary means deterministic. But you are wrong with the time evolution not being deterministic, time evolution in quantum mechanics is deterministic. What seems not to be deterministic in the general case are measurement outcomes, that is where we take the Born rule, square the probability amplitudes, and get the probabilities for the different measurement outcomes.

But here lies the incompatibility I am asking about, time evolution is unitary and is in a sense incompatible with the Born rule stating that different measurement outcomes are possible. You can not have unitary time evolution and different possible measurement outcomes starting from the same initial state, at least not in a really obvious way.

> decades of experiments have shown that the alternatives are all, most likely, wrong

How?

The Bell Theorem shows that any local hidden variable theory wouldn't match the experimental data we already have.

There's other options for getting around the weirdness of the Copenhagen interpretation that aren't ruled out by the data we have, but most of them are even weirder.

What makes you think the Copenhagen interpretation is the least weird choice, given all its difficulties?

It's the simplest interpretation of the data we have.

The many-worlds interpretation is arguably simpler than the Copenhagen interpretation because it gets rid of the collapse postulate. And depending on how literally you take the Copenhagen interpretation, it may not even be self-consistent. At best it provides no explanation how the apparent collapse of the wave function happens, at worst it suggests that the world evolves according to two fundamentally incompatible laws, unitary time evolution and probabilistic projection onto eigenstates during measurement.

I don't want to argue for the many-worlds interpretation, I don't really like - and I am aware that this is a somewhat stupid thing to say about a physical theory. But I also have to admit that it makes sense from a point of simplicity and consistency and compares favorable to other interpretations.

How is an interpretation which violates unitary evolution and doesn’t even explain measurement “the simplest”?