What was exactly the preferred basis problem in decoherence? I am reading some papers of Zurek, for example this one :

Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?

He starts to introduce a way to define a measurement in Q.M.
Consider we have a system $S$ and an apparatus $A$.
We assume our system $S$ is initially in the state :
$$ |\psi_S \rangle = \sum c_s |s \rangle$$
We want to measure the observable $S=\sum_s s |s\rangle \langle s |$.
The process of measurement can be understood as an interaction that will entangle the apparatus and the system, for example :
$$ |\psi_S \rangle \otimes |A_{ini} \rangle \rightarrow |\psi_{f} \rangle = \sum c_s |s \rangle |A_s \rangle $$
The state of the apparatus is now 1:1 correlated to the state of the system. Knowing the state of our apparatus we are then able to know in which state our system is.
But further, in the paper, he gives the objection that actually we have a choice of basis problem. I illustrate it here.
Let's consider for example $S$ is a qubit, and $|\psi_S \rangle = \frac{|0\rangle + |1\rangle} {\sqrt{2}}$
After the interaction we have :
$$ |\psi_f \rangle = \frac{|0\rangle |A_0\rangle + |1 \rangle |A_1\rangle}{2}$$
We can do a simple change of basis : $|\pm\rangle = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}$, $|A_\pm\rangle = \frac{|A_0\rangle \pm |A_1\rangle}{\sqrt{2}}$
And we find (this is a common property on bell states actually) : 
$$ |\psi_f \rangle = \frac{|-\rangle |A_-\rangle + |+ \rangle |A_+\rangle}{2}$$
Zurek says that there is then a basis ambiguity, so that we cannot understand the measurement such as the proposed unitary evolution.

My question is : why is there a problem here ?
What I think but I'm really not sure about that is.
We have a problem because when we do a measurement we want to measure a given observable, for example the value of the spin $S_z$. And in practice you will read the value that will display on your apparatus, either it is $+1$ and your system is in $|1\rangle$, either it is $0$ and your system is in $|0\rangle$.
Because of the basis ambiguity, when we look at our apparatus, we will have a number, let's say "1". Does that mean that our apparatus is in $|A_+\rangle$ and then our system is in $|+\rangle$, or does that mean our apparatus is in $|A_1\rangle$ and our system is in $|1\rangle$. As those two basis play the same role mathematically we cannot conclude.
Is it the exact reason why we do have a problem with this protocole ? I want to be sure.
 A: The problem that Zurek points out is that the roles of the control and the target seem to swap after the basis change. The direction of the flow of information becomes uncertain. 
Zurek often gives this example of modeling quantum measurements as CNOT operations. If the control state is $\alpha | 0 \rangle + \beta | 1 \rangle$ and the target (the measurement device) state is in $|A_0 \rangle$, after interaction, the total state becomes $\alpha | 0 \rangle |A_0\rangle + \beta | 1 \rangle |A_1\rangle$. Similarly, if the target state is $|A_1\rangle$, then the total state becomes $\alpha | 0 \rangle |A_1\rangle + \beta | 1 \rangle |A_0\rangle$. To summarize, if the control bit is 0, it does nothing, and if it is 1, it flips the target. So far so good. 
Now if you write that in the superposition basis, the truth table looks a bit different. You simply build the new truth table using the transformation rules from the old truth table and law of superposition (try this yourself). You will find out that the new truth table looks like the following:
$|\pm\rangle|A_{+}\rangle \to |\pm \rangle |A_+\rangle$
$|\pm\rangle|A_{-}\rangle \to |\mp \rangle |A_-\rangle$
Now the "target" seems to play the role of control in the CNOT operation! 
Reference: "ENVIRONMENT–INDUCED DECOHERENCE AND THE
TRANSITION FROM QUANTUM TO CLASSICAL", J.P. Paz and W.H. Zurek, 1999 Les Houches Summer School lecture notes (preprint)
