I know there are various definitions of the "preferred basis problem." I'm trying to understand what Wikipedia is saying here.

... a quantum state can equally well be described (e.g.) as having a well-defined position or as being a superposition of two delocalised states...

Do they mean something like this?

$$|\psi\rangle = |x_1\rangle = \frac{1}{2}(|x_1\rangle + |x_2\rangle) + \frac{1}{2}(|x_1\rangle - |x_2\rangle)$$

(with $|x_n\rangle$ being position eigenstates)

Or something else? And then in this answer:

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.

Are they just talking about this kind of interaction?

$$|p_?\rangle \otimes \frac{1}{2}(|x_1\rangle + |x_2\rangle) \mapsto \frac{1}{2}(|p_1\rangle |x_1\rangle + |p_2\rangle|x_2\rangle)$$

The idea being that this entanglement in this basis is somehow more natural than others? This is of course a very standard kind of interaction in QM 101. Is the idea of the preferred basis problem that we shouldn't take the form of this interaction for granted, and that it needs to be explained somehow?



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