Once and for all: Is the preferred basis problem in the Everettian Interpretation of QM considered solved by decoherence or not? THere are a few people who claim that it's not, but it seems the vast majority of the literature says it has been solved by Zurek, Joos, Zeh, Saunders and Wallace.
So which is true and why?
Unfortunately, physicists and philosophers disagree on what exactly the preferred basis problem is, and what would constitute a solution. Wojciech Zurek was my PhD advisor, and even he and I don't agree. I wish I could give you an objective answer, but the best I can do is state the problem as I see it.
In my opinion, the most general version of the problem was best articulated by Adrian Kent and Fey Dowker near the end their 1996 article "On the Consistent Histories Approach to Quantum Mechanics" in the Journal of Statistical Physics. Unfortunately, this article is long so I will try to quickly summarize the idea.
Kent and Dowker analyzed the question of whether the consistent histories formalism provided a satisfactory and complete account of quantum mechanics (QM). Contrary to what is often said, consistent histories and many-worlds need not be opposing interpretations of quantum mechanics [1]. Instead, consistent histories is a good mathematical framework for rigorously identifying the branch structure of the wavefunction of the universe [2]. Most many-world'ers would agree that unambiguously describing this branch structure would be very nice (although they might disagree on whether this is "necessary" for QM to be a complete theory).
In my opinion, the situation is almost exactly analogous to the question of whether an abstract formulation of classical mechanics (e.g. Lagrangian mechanics) is satisfactory in the absence of a clear link between the mathematical formalism and our experiences. I could write down the math of Lagrangian mechanics very compactly, but it would not feel like a satisfactory theory until I told you how to link it up with your experiences (e.g. this abstract real scalar x = the position coordinate of a baseball) and you could then use it to make predictions. Similarly, a unitarily evolving wavefunction of the universe is not useful for making predictions unless I give you the branch structure which identifies where you are in wavefunction as well as the possible future, measurement-dependent versions of you. I would claim that the Copenhagen cook book for making predictions that is presented in introductory QM books is a correct but incomplete link between the mathematical formalism of QM and our experiences; it only functions correctly when (1) the initial state of our branch and (2) the measurement basis are assumed (rather than derived).
Anyways, Dowker and Kent argue that consistent histories might be capable of giving a satisfactory account of QM if only one could unambiguously identify the set of consistent histories describing the branch structure of our universe [3]. They point out that the method sketched by other consistent historians is often circular: the correct "quasi-classical" branch structure is said to be the one seen by some observer (e.g. the "IGUSes" of Murray Gell-Mann and Jim Hartle), but then the definition of the observer generally assumes a preferred set of quasi-classical variables. They argue that either we need some other principle for selecting quasi-classical variables, or we need some way to define what an observer is without appealing to such variables. Therefore, the problem of identifying the branch structure has not been solved and is still open.
I like to call this "Kent's set-selection problem". I consider it the outstanding question in the foundations of quantum mechanics, and I think of the preferred basis problem as a sort of special case.
The reason I say special case is that the preferred basis problem answers the question: how does the wave function branch when there is a preferred decomposition into system and environment (or into system and measuring apparatus). However, the boundaries of what we intuitively identify as systems (like a baseball) are not always well-defined. (What happens as atoms are removed from the baseball one by one? When does the baseball cease to be a useful system?) In this sense, I say that the decoherence program as led by Zeh, Zurek, and others is an improvement but not a complete solution.
Sorry that's not as clear as you would like, but that's the state of the field as I see it. There's more work to be done!
[1] Of course, some consistent histories make ontological claims about how the histories are "real", where as the many-worlders might say that the wavefunction is more "real". In this sense they are contradictory. Personally, I think this is purely a matter of taste.
[2] Note that although many-worlders may not consider the consistent histories formalism the only way possible to mathematically identify branch structure, I believe most would agree that if, in the future, some branch structure was identified using a completely different formalism, it could be described at least approximately by the consistent histories formalism.
[3] They argue that this set should be exact, rather than approximate, but I think this is actually too demanding and most Everettians would agree. David Wallace articulates this view well.
First some background. The preferred basis problem arises when you consider multiple worlds to be actual different worlds. This is not the same as Everett interpretation of quantum mechanics (coming up soon), it is deWitt's reading of the said interpretation.
The way it works is that an observer picks a number of measurement operators $M_k$ which satisfy $\sum_k M_k^\dagger M_k = \mathbf{1}$. Now the measurement process is no longer a collapse but is instead viewed as an interaction between the observer and the measured system. We imagine that the observer has memory of what was observed, and we shall describe the memory as a quantum state with basis $|0\rangle, |n\rangle$, plus a pre-measurement state $|NA\rangle$. The system starts in some state $|\psi\rangle$ (let's keep it pure for simplicity, but all this works for mixed states too).
Now to the interaction. The joint state of the observer and the system is $|\psi\rangle \otimes |NA\rangle$ which goes to $\sum_k M_k |\psi\rangle \otimes |k\rangle$. This basically means that whenever $M_k$ result has been obtained the observer's memory has recorded the result correctly to have been $k$. This operation is a unitary operation and therefore has an associated Hamiltonian. The job of the observer is to create a device that generates such a Hamiltonian.
So the worlds in this interpretation are entirely observer dependent. The only preferred basis you get is the Schmidt basis that arises from the measurement and the resulting observer being in a mixed state. If the observer and the measured system were well isolated inside some other measurement device then another third observer would actually be able to observe the entanglement between the observer and the measured system and only really see a single world, although for the observer inside there are many worlds (it would be neat to violate Bell inequalities with a physicist and a photon). These worlds arise from the observer's memory.
Finally, you might say that the memory basis of the observer is essentially arbitrary. This is correct. It's arbitrary to the extent that in principle it could be anything. But in practice picking states that are not stable in the world would not produce an evolutionary advantage as they could not be used to faithfully store information. I guess you could say this is where evolution and the interpretation of quantum mechanics meet.
This question is from 2013, so maybe this is too late as being a relevant answer, but I am going to write it anyway. The goal in this answer is to address how some proponents of the Everettian (many-worlds) interpretation can claim the preferred basis problem as being solved. In the past, I have been very skeptical of this claim, but I am now starting to really understand the claim. I would have written it as a research paper if I could find a way to do that but it seems useless to write this as a research paper, hence this answer for the community benefit.
First, on a fairly general criticism of any theory of the preferred basis. Suppose you have the universe consisting of two subsystem $S$ and $A$, with Hilbert space factorization $\mathcal{H} \equiv \mathcal{H}_S \otimes \mathcal{H}_A$ and $|\psi\rangle \in \mathcal{H}$. It is well-known that the Schmidt (which is largely SVD) decomposition of $|\psi\rangle$ may not be unique even though singular values are unique. You can find one example in this paper by Meir Hemmo and Orly Shenker - see also this YouTube video, where Meir Hemmo explains their critique in the paper.
So the Schmidt decomposition is not enough, and indeed the einselection (environment-assisted superselection) theory of Wojciech Zurek does not actually rely on the Schmidt decomposition. As Zurek notes in the linked YouTube video, it is already known that quantum states alone are insufficient to determine the preferred basis. We need a physical theory to determine the preferred basis, which breaks what Meir Hemmo calls as the global basis symmetry and what may alternatively refered to as basis invariance or basis irrelevance in physics.
The einselection theory exactly does that, which relies on (to simplify heavily) the three-subsystem Hilbert space factorization: $\mathcal{H} = \mathcal{H}_S \otimes \mathcal{H}_A \otimes \mathcal{H}_E$. Environment $E$ is necessary to force one particular pointer basis via interaction Hamiltonian $H_{AE}$, which is between $A$ and $E$. In particular, up to a reasonable approximation, we could say that out of all possible Schmidt decompositions, $E$ picks out which decomposition determines the preferred basis via $[H_{AE},O] = 0$, where $O$ is the pointer observable for $A$. This preferred basis choice preserve probability information for each outcome of $SA$ even when interacting with $E$ over time.
Again, the upshot is that this is a theory of the preferred basis that relies on the interaction Hamiltonian, instead of individual quantum states. Furthermore, decoherence is just one aspect of the einselection theory. So according to the Zurek-style Everettian interpretation, the preferred basis problem is solved. Combined with a theory of collapse in the Everettian interpretation that considers it as arising out of entanglement and practically irreversible decoherence, the only problem that remains in the Everettian interpretation is how the Born rule is justified in the interpretation, which is an entirely different question at least for the Everettian interpretation proponents.
Now the einselection theory of preferred basis is not without a potentially valid critique. For some reason I cannot find papers stating this, but the point is that in the einselection theory, decoherence is never completely achieved, so it is wrong to ignore small off-diagonal entries of density matrices in the preferred basis. This comes from the issue that the einselection theory is a dynamic theory of preferred basis, not a theory that depends on quantum states. The pointer basis may not equal to a Schmidt decomposition basis for each quantum state at each time. In some cases, these off-diagonal entries eventually have significant influences on how the universe evolves, but this is not what our universe seems to be like. Whether this argument is convincing depends on who you ask.
You could also question why probability (coefficient) information being preserved can settle the pointer basis. Couldn't it be possible that some preferred measurement basis allows $E$ to change outcome probability over time? Why should this be impossible? There is this thing about whether this question is even valid, and another thing about how this is addressed, assuming its validity. Zurek has tried to answer this question with his vision of quantum Darwinism, which relies on information redundancy across subsystems. For this point, I have not been fully convinced, but you might be.
There is also the naturalness problem as well. For any interpretation of quantum mechanics, the einselection theory is a physical theory in sense that for some interaction Hamiltonian, it chooses what basis is to be preferred. But this physical theory is very different from other physical theories. This problem becomes worse in the Everettian interpretation. We are supposed to believe in the Everettian interpretation because it is most straightforward interpretation out of quantum mechanics mathematics (canonical or path integral formalism). Yet we now face this extra theory that chooses why some basis is to be preferred. Again, Zurek answers this with quantum Darwinism - the privileged basis is not physical but more of the one that is most information-wise redundant. But while I have become more convinced of the approach, there are still loads of issues. For example, what privileges a particular Hilbert space factorization? Can information advantage ever be a reason for the privileged basis or the privileged Hilbet space factorization?
Or one can ask: is locality really justified as a reason for the privileged basis? If so, could not this be question-begging to the past? Is spacetime more fundamental than we think in the past? Why do we seem to observe the universe in a more localized basis than what quantum mechanics seems to suggest? Can quantum field theory address this answer? Does an algebraic quantum field theory of local observables address this question? Effective Hamiltonians that pick out the emergent privileged basis are different from the fundamental Hamiltonian, and in such a case, does the einselection theory really work? Is the privileged basis relative to an observer? In such a case, what does an observer even mean in the Everettian interpretation? All these questions suggest that the original vision of the Everettian interpretation that takes mathematics of quantum mechanics as it is has largely been lost in the modern variants of the Everettian interpretation. If so, what would be the reason to prefer it over others?
For the question of why the Everettian interpretation may still be preferred despite loss of the original vision, I think it largely comes down to providing a theory of collapse in terms of entanglement and decoherence without violating unitary evolution. Additionally, the Everettian interpretation brings the measurement process directly into mathematics of quantum mechanics, which squares well with the vision that quantum mechanics should capture everything about the universe.
I also think the einselection theory surprisingly seems to work well with the Copenhagen interpretation, where we do not have to address the question of why measurements have to be in a particular basis, such as why probability information has to be preserved in the preferred basis when entangled with environment. For the old-style Copenhagen interpretation, collapse occurs instantaneously, so no issue of probability information being preserved over time. The einselection theory simply picks out why some measurement apparatus has a privileged basis, and collapse occurs in that basis. Thus, it seems surprising that the einselection theory is not much mentioned with regards to the Copenhagen interpretation.
There are other criticisms of the einselection theory as well, but the point is that a criticism of some preferred basis theory based solely on quantum states (basis invariance, basis symmetry) is very likely a bad critique, and so many people, including myself in the past, have fallen into such traps.
