Measurement problem and precise mathematical calculation

Question

The infamous measurement problem is a problem in the foundations of quantum mechanics: different people have different views how to understand this problem: some people even deny that there is any problem after all: from the same reason there will be a variety of opinions what would count as a solution of this problem. I don't want to dig deep into the philosophy but rather focus on mathematics. I feel very uncomfortable when I see formulas like $|e^+ \rangle |pointer \ shows \ up \rangle +|e^- \rangle|pointer \ shows \ down \rangle$ where $|e^+ \rangle, |e^- \rangle $ are eigenstates of spin-those I understand as vectors lying in $\mathbb{C}^2$ but I completely don't understand where vector like $|pointer \ shows \ up \rangle $ live. You could answer ,,in the Hilbert space of the apparatus'' but what is the dimension of this space and what is the Hamiltonian governing the dynamics? It looks like a black box and seeing it I simply don't know what is happening and I hesitate to admit that there is a measurement problem at all. To be precise: I'm not claiming that there is no problem, rather that I don't know whether there is a problem.
Suppose that somehow somebody managed to describe every microscopic interaction from the spin of the electron through a measuring apparatus (every interaction, every molecule in the apparatus, every air molecule and so on-practically impossible of course but let us assume this for the purpose of the thought experiment) up to my brain while I'm performing an experiment and doing observation-and suppose that this person was able to calculate the dynamics precisely (assuming just Schrodinger evolution, nothing else):

what would be a result of this calculation? Would it solve the measurement problem?

Somewhat similar issues were discussed in this discussion where this paper (titled ,,Understanding quantum measurement from the solution of dynamical models'') was mentioned. Is it true that the authors of this paper managed to achieve something similar to what I had just described?

Answer 1

A number of different issues all contribute to what physicists generally call the "measurement problem". On top of that, there are other problems which intersect with these issues, in particular what Travis Norsen (in his quantum foundations textbook) calls the "ontology problem". This other problem seems closely related to your question about where these mathematical structures "live".

Following Norsen's summary of the measurement problem (section 3.5 in his textbook), he notes three interrelated issues: (quotes from Norsen)

(1) "It is not clear exactly which set of physical interactions of processes in nature should count as 'measurements'. " (as opposed to mere interactions)

(2) Assuming that problem (1) has a resolution, allowing a "sharp boundary" between measurements and non-measurements (mere interactions), then another problem is that "there is something unbelievable about the idea that the world is fundamentally split into two distinct 'realms'. Surely a proper fundamental theory should describe the entire universe in a coherent, unified way."

(3) "Finally, 'the measurement problem' also sometimes denotes the theory's apparent inability to provide sensible results when it is modified in the obvious way in response to" problem (1) and problem (2).

And, for completeness, since this ties into your question, there's also what Norsen calls the 'ontology problem':

(4) "What kind of physical thing – what ontology, exactly – could the quantum wave function possibly represent, and how would that representation work?"

As you point out, we ideally want something in the mathematics to represent the physical pointer state of a physical measurement device, but is it plausible that a vector in Hilbert space could indeed represent such a thing?

Let's look at your main Supposition from the context of the ontology problem. Your starting point is a reasonable classical ontology where there really exist things like the spin of an actual single electron, "every molecule in the apparatus, every air molecule", etc. Then you map this onto the quantum formalism, where you allow superposition states of this classical ontology. Then, of course, you find that you always tend to end up with entangled superposition states in the mathematics. Now, to what aspects of your original classical ontology does this entangled state correspond? Can you back out of it and identify your original ontology, the state of each air molecule, etc? No; there's no clear-cut way to do that, without simply referring back to your original classical ontology and assuming that there must be an answer in that regime. So something has gone wrong in this analysis, and pretty much every straightforward resolution to it encounters what John Bell called the "shifty split" between the microscopic and macroscopic worlds.

The paper you cite is not immune to the Shifty Split complaint, and the idea that you can squint at a nearly-diagonal density matrix and say that it's exactly diagonal is just sweeping the above problems under the rug.

I'll leave the last word (for now) with a relevant quote from Bell, also taken from section 3.5 of Norsen's book. I think it would apply perfectly to the paper you cite.

"There can be no question then of identifying the quantum system S with the whole world W. There can be no question – without changing the axioms – of getting rid of the shifty split. Sometimes some authors of 'quantum measurement' theories seem to be trying to do just that. It is like a snake trying to swallow itself by the tail. It can be done – up to a point. But it becomes embarassing for the spectators even before it becomes uncomfortable for the snake." J.S. Bell, "Against Measurement", 1989.

Answer 2

The measurement problem is about an apparent conflict between what we see when we do a measurement and what a superficial reading of the post measurement state claims we should see.

Suppose that you have a system $S$ in a state $|a\rangle_S$ and measurement device $M$ in a ready state $|0\rangle_M$ and a Hamiltonian coupling them so that $$|a\rangle_S|0\rangle_M\to |a\rangle_S|a\rangle_M.$$ This seems straighforward: the measurement device indicates the value of the state of $S$.

Because the Schrodinger equation and other quantum mechanical equations of motion are linear, if the system $S$ is in a state $\sum_a\alpha_a|a\rangle_S$, then after the measurement the state under the same coupling is $$\sum_a\alpha_a|a\rangle_S|a\rangle_M.$$ This state has multiple values for the measurement device, which raises some problems. What does this state predict about the measurement result? What does it even mean for the measurement to have multiple outcomes?

Many solutions have been proposed for this problem. Many of those solutions, such as dynamical collapse theories, involve replacing quantum theory with a completely different theory. But if you keep quantum theory and work out what happens in realistic situations where observers and measuring instruments are treated as large systems that interact with the environment, then the spread of information about measurement outcomes prevents interference between different outcomes: decoherence. In that case, there would multiple versions of the measurement outcome that wouldn't interact with one another to a very good approximation.

For a summary of the controversy and proposed solutions, see:

https://arxiv.org/abs/0712.0149

For explanations of decoherence see

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/0707.2832