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I'm reading a Paper by Maximilian Schlosshauer on Decoherence, the measurement problem, and interpretations of quantum mechanics.

Here, One considers a system and apparatus. The whole system and apparatus starts with $$|\Psi(0)\rangle =\psi(0)\otimes |a(0)\rangle =\sum_n c_n|s_n\rangle \otimes |a_r\rangle $$ This state after the interaction evolve to $$|\Psi(t)\rangle =\sum_n d_n(t)|s_n\rangle \otimes |a_n\rangle $$ The right hand side is a superposition of system-apparatus states.

The paper says,

The expansion of the final composite state is in general not unique, and therefore the measured observable is not uniquely defined either. This is the problem of the preferred basis.

I don't understand quite well. The composite system has basis $\{|s_n\rangle \otimes |a_n\rangle \}$, and in this basis the final state is expanded uniquely. If we choose say different apparatus basis, we will have different expansion but isn't the correspondence between the system states and dial measurement would also be changed. Therefore, we will have different apparatus state corresponding to same system state. In any case, the system case would be unique. How does the any preference arise here?

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I think EE18's answer is generally correct but at times I'm not 100% clear on it, so I'll write a bit more in case it's helpful.

First I need to give some context. The preferred basis problem arises in the context of trying to derive as many of the macroscopic measurement postulates as you can, starting from the microscopic (Schrödinger evolution). There are multiple things that would need to be derived, and one thing that must be determined is "Which basis should the particle to collapse in?". Because if you do a measurement of spin-z, it collapses to either $|z\rangle$ or $|-z\rangle$, whereas an x measurement collapses to different states $|x\rangle, |-x\rangle$, where those states are the eigenvectors of $\sigma_z$ and $\sigma_x$. Obviously those are different outcomes, and the question is, can we derive which basis the collapse happens in, just from microscopic evolution?

Just looking at the post-measurement state that would come from Schrödinger evolution, it is not possible in general to identify a unique basis that is "the collapsed basis". That is because the same state may be written in a similar format in a different basis:

$$|\Psi(t)\rangle =\sum_n d_n(t)|s_n\rangle \otimes |a_n\rangle =\sum_m d'_m(t)|s'_m\rangle \otimes |a'_m\rangle $$

...and so just the final result of Schrödinger evolution is not only insufficient to derive the exact collapsed post-measurement state, it also is insufficient to even determine which basis the collapse would have been in. This is the preferred basis problem (not really a solvable problem, it's just pointing out an issue that exists).

Note that this is not just trivial, because the form given is a special form. The general basis expansion of a tensor product basis is

$$|\psi \rangle = \sum_n \sum_m c_{mn} |s_m\rangle \otimes |b_n\rangle $$

Note also that the $|a_n\rangle, |a'_m\rangle$ in the equation for $|\Psi(t)\rangle$ above are not a basis for the environment, which would have way more degrees of freedom than just the $n$ possible measurement outcomes of the system. But the $|s_n\rangle$ etc are a basis for the system since they're the eigenvectors of a hermitian operator.

Finally, note that there are attempts to solve this problem, the one I've read about is Einselection, which gives criteria for states to "survive" the measurement process. I have read that Einselection may help, but is still not enough in general to pick out a "special basis" for any measurement scenario, so it may not be a complete solution to the problem.

Finally finally, note that real measurements actually don't even heed the QM measurement postulates to begin with, i.e. we for example may measure momentum through a series of position measurements... so of course the post-measurement state is not a momentum eigenstate. I think that gets lost in all of this sometimes. The whole measurement formalism in the usual postulates is just artificial anyway, so I don't see it as necessarily necessary to derive. But feel free to have your own take on it.

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  • $\begingroup$ This is an awesome, thank you so much. The context given at the start of the second paragraph was particularly insightful for me. If this was my question I would accept the answer :) thanks again! $\endgroup$
    – EE18
    Commented Mar 12, 2023 at 4:59
  • $\begingroup$ I should add that Schlosshauer seems to suggest (I am not there yet) that decoherence DOES solve the preferred basis problem and the problem of nonobservability of quantum interference (in macroscopic systems), but NOT the problem of outcomes. It seems like you disagree with that assessment though? $\endgroup$
    – EE18
    Commented Mar 12, 2023 at 5:02
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    $\begingroup$ I have a memory of reading that Einselection may sometimes determine a unique basis, but does not always. I don't remember the source right now, so I think that would be best to leave that as a loose end and I don't want to guarantee it. It's good you brought it up; I'll edit the question accordingly. I think also the "finally finally" comment is important for context. $\endgroup$ Commented Mar 12, 2023 at 5:13
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I am reading Schlosshauer's book right now and figured I'd give a stab at this question.

The crux, I believe, of the preferred basis problem is that there is no a priori clear correspondence rule between the mathematical formalism of von-Neumann measurement (which you gave) and the physical measurement which we are seeking to represent. This notion of correspondence rule is important; as discussed at the start of Chapter 2 of Ballentine [1], all of physical theory involves ultimately using a correspondence rule between the real world and our mathematical representation thereof.

I think the point of the preferred basis problem is to observe that there is no clear correspondence rule to use. When you constructed the mathematical representation of measurement, you wanted to show how an observable $S$ (with eigenstates $|s_n\rangle$) was being measured on the system. Thus you might suppose that you could say that your apparatus had "measured" various outcomes for the system by mapping back a given term (say, $|s_n\rangle|a_n\rangle$) from the superposition you gave to a corresponding physical outcome (apparatus pointing to $a_n$ so system measured to be in state with value $s_n$). But within the formalism of quantum mechanics you can do a change of basis (in general) and so obtain some other sum of terms to write this same state. If you apply the same correspondence rule then you have (in contradiction to what we see in the world) concluded that you have in fact measured something else (corresponding to the observable which has as eigenkets the system states involved in the new sum). Therein lies the "problem".

It should be noted that this problem applies much more general than just to measurement of laboratory systems. Depending on your interpretation of quantum mechanics it is ostensibly something we need to solve to explain why we only observe certain sorts of states in the world around us (for example, position states rather than superposition of position states -- which form an equally valid basis for some observables).

[1] L. E. Ballentine, “Quantum Mechanics: A Modern Development,” World Scientific, New York, 1998.

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