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I've seen the preferred basis problem referred to in many places, but have not seen a clear explanation of what the problem is. For example, this question asks whether the problem has been solved, but I wasn't able to glean anything from the discussion on what the problem was. Zurek 2001 gives the following:

the original MWI does not address the "preferred basis question" posed by Einstein

with a footnote quoting Einstein as saying this:

When the system is a macrosystem and when ψ1 and ψ2 are 'narrow' with respect to the macrocoordinates, then in by far the greater number of cases this is no longer true for ψ = ψ1 + ψ2. Narrowness with respect to macrocoordinates is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.

The first sentence seems obvious to me. I'm baffled by the second sentence. Some states are narrow and some are not. What's the problem, and what does this have to do with a basis or a preferred basis?

Zurek, "Decoherence, einselection, and the quantum origins of the classical,"

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up vote 3 down vote accepted

Narrowness is precisely the essence of the preferred basis problem.

Consider: some states are narrow, some are not. Given that some are narrow and some are not, why should 'narrowness' come about as a meaningful concept at all? Why should this quality be an interesting one?

Consider the position of a pointer. We don't interpret non-narrowly pointed states of the pointer as physical, basically by the very definition of what we mean by "a pointer" (after all, we take for granted that some systems can only occupy narrow states, and these are called pointer states in common terminology). In quantum mechanics, the non-narrow states of the pointer are perfectly valid. Then how does the pointer come in practice to inhabit the narrow states?

The answer is that the pointer has a preferred basis (or something which is almost but not quite an orthonormal basis, in the case of literal pointers having different positions): a basis in which its environment tends preferentially to interact, so that the information about the state of the pointer which is encoded in that basis gets copied in other systems, and is therefore strongly correlated. (This is the notion of quantum Darwinism: the information best suited to be reproduced elsewhere, comes to spread faster than it could be stopped, giving rise in practise to decoherence in the basis in which that information is represented.) The question then arises: how does one determine that basis, and why should this basis be priviledged in our experience of the world? For instance, whatever the superposition to which we supposedly belong, according to the MWI, we percieve a strong tendancy for objects to be spatially localised. Why? How does one explain the way in which the myriad potential microscopic worlds merge into distinguishable macroscopic worlds? Why should a superposition state seem, from the inside, like a decomposition with respect to any particular basis?

That is the preferred basis problem, in a nutshell.

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