In decoherence theory, we try to remove the measurement postulate from Q.M to replace it by unitary evolutions.
Consider a two level system $S$, an apparatus $A$ and the environment $E$.
A first model of measurement is the following :
If $S$ is in $|\psi_S \rangle = \alpha |0\rangle + \beta |1\rangle$, I will have an interaction with the apparatus such that :
$$|\psi_S \rangle |A_i \rangle \rightarrow \alpha |0\rangle |A_0 \rangle + \beta |1 \rangle |A_1\rangle $$
However, this model is not good enough to describe a measurement because, if $\alpha=\beta=\frac{1}{\sqrt{2}}$, I have :
$$\frac{1}{\sqrt{2}} \left( |0\rangle |A_0 \rangle + |1 \rangle |A_1\rangle \right) = \frac{1}{\sqrt{2}}\left( |-\rangle |A_- \rangle + |+ \rangle |A_+\rangle \right) $$
Where I defined $|\pm\rangle = \frac{1}{\sqrt{2}} \left(|0\rangle \pm |1\rangle \right) $ (same idea for the apparatus).
$|0\rangle$ and $|1\rangle$ are eigenstates of $S_z$ spin, and $|+\rangle$ and $|-\rangle$ are eigenstates of $S_x$ spin.
The exact reason why this model is not good are still a little obscure for me (see What was exactly the preferred basis problem in decoherence ).
But what I understood from it is that it is not a good model because with this way of rewriting the vector, one could say "we measured $S_z$", but another one could say "we measured $S_x$".
Then, to solve the problem, we say that actually the apparatus+system also entangle with the environment. And this interaction remove the ambiguity of the measurement basis.
In practice it means that we also have :
$$\frac{1}{\sqrt{2}} \left( |0\rangle |A_0 \rangle + |1 \rangle |A_1\rangle \right)|E_i\rangle \rightarrow \frac{1}{\sqrt{2}} \left( |0\rangle |A_0 \rangle |E_0\rangle + |1 \rangle |A_1\rangle |E_1\rangle \right)$$
And then there is apparently no more the ambiguity basis when we take the partial density matrix of $S+A$, it is written in Chap IV, part A of
Decoherence, einselection, and the quantum origins of the classical
From Zurek.
We have :
$$\rho_{S,A}=\frac{1}{2} \left( |0 A_0\rangle \langle 0 A_0 | + |1 A_1\rangle \langle 1 A_1 | \right) $$
But for me there is still a basis ambiguity.
Indeed, the matrix is the identity in $span(|0 A_0\rangle, |1 A_1 \rangle)$. Then if I write for example : $|\phi_{\pm}\rangle = \frac{1}{\sqrt{2}} \left(|0 A_0 \rangle \pm |1 A_1 \rangle \right) $ for example, I also have :
$$\rho_{S,A}=\frac{1}{2} \left( |\phi_+\rangle \langle \phi_+ | + |\phi_-\rangle \langle \phi_- | \right) $$
Thus, we have a lot of basis in which $\rho_{S,A}$ is diagonal : there is somehow still a basis ambiguity.
However, maybe we are only looking for basis in which the states of the mixture are separable ? And for this case there is only the basis : $(|0A_0\rangle, |1,A_1\rangle)$.
But I am not sure of this.
In summary : how does the interaction with the environment solve the basis ambiguity problem ?