De-coherence and Wigners friend?

Question

So in the thought experiment Wigners friend the paradox is ultimately due to a difference of descriptions of density matrices.

If the physical variable that is measured of the spin system is denoted by $z$, where $z$ takes the possible outcome values $0$ or 1, the above Wigner's friend situation is modelled in the RQM context as follows: $F$ models the situation as the before-after-transition:

$$ \alpha | 0 \rangle_S + \beta | 1 \rangle_S \to |1 \rangle _S $$

of the state of $S$ relative to him (here it was assumed that $F$ received the outcome $z = 1$ in his measurement of $S$). In RQM language, the fact $z = 1$ for the spin of $S$ actualized itself relative to $EF$ during the interaction of the two systems.

A different way to model the same situation is again an outside (Wigner's) perspective. From that viewpoint, a measurement by one system ($F$) of another ($S$) results in a correlation of the two systems. The state displaying such a correlation is equally valid for modelling the measurement process. However, the system with respect to which this correlated state is valid changes. Assuming that Wigner ($W$) has the information that the physical variable $z$ of $S$ is being measured by $F$, but not knowing what $F$ received as result, $W$ must model the situation as:

$$ (\alpha |0 \rangle_S + \beta |1 \rangle_S) | \perp \rangle_F \to \alpha ( | 0 \rangle_S \otimes | 0 \rangle_F ) + \beta (|1\rangle_S \otimes |1 \rangle_F) $$

where ${F}$ is considered the state of $F$ before the measurement, and ${\displaystyle |1\rangle _{F}}{\displaystyle |1\rangle _{F}}$ and ${\displaystyle |0\rangle _{F}}{\displaystyle |0\rangle _{F}}$ are the states corresponding to $F$'s state when he has measured $1$ or $0$ respectively. This model is depicting the situation as relative to $W$, so the assigned states are relative states with respect to the Wigner system. In contrast, there is no value for the $z$ outcome that actualizes with respect to $W$, as he is not involved in the measurement.

The assumptions boil down to those assumptions are

(Q): Quantum theory is correct. (C): Agent's predictions are information-theoretically consistent. (S): A measurement yields only one single outcome.

Now, while decoherence does not solve the measurement problem it everyone seems to agree that it is a step in the right direction. Does de-coherence add value to the discussion when people question axiom $(S)$?

To make more explicit the time evolution can be reformulated in terms of density matrices to remove axiom C. Let's say my wavefunction is in superposition of

$$|\psi \rangle_S = \frac{1}{\sqrt{2}}(\alpha | 0 \rangle_S + \beta | 1 \rangle_S)$$

The corresponding density matrix is:

$$ \rho_S = \begin{bmatrix} |\alpha|^2 & \alpha^* \beta \\ \beta^* \alpha & |\beta|^2 \end{bmatrix} $$

Now, the time average of the density matrix is decoherence of the density matrix which happens due to decoherence is given by:

$$ \langle \rho_S \rangle_t = \begin{bmatrix} |\alpha|^2 & 0 \\ 0 & |\beta|^2 \end{bmatrix} $$

And proceeds to use the density matrix $\rho'$

$$\rho' = \langle \rho_S \rangle_t \tag{D}$$

Via this maneuver I reformulate the problem and still get the contradiction $\rho' \neq \rho$ (replacing axiom (S) with (D)). To me its very obvious one of them is using approximations statement $(D)$. More explicitly:

One of them is using:

$$i \hbar \frac{\partial \rho }{\partial t} = [H,\rho]$$

and the other:

$$i \hbar \frac{\partial \langle \rho \rangle_t}{\partial t} = [H,\langle \rho \rangle_t]$$

The real problem I encounter when I try to explain myself is $(D)$ is a valid replacement of $(S)$. What happens if I put both Wigner and his friend in box (the entire setup).

Edit: In response to Mateus Araújo's answer:

"if the description of the measurement becomes the same if the Friend's decoherence reaches Wigner, i.e., if the Friend is not inside a decoherence-proof box."

Yes, this does happen. Once, decoherence reaches the Wigner. Then Wigner is also using the equation:

$$i \hbar \frac{\partial \langle \rho \rangle_t}{\partial t} = [H,\langle \rho \rangle_t]$$

As a result neither of them see any discrepancy now for both of them their descriptions of the density matrices match.

Answer 1

Ok, now I understand your question. You were saying that the problem is that the description of the Friend's measurement was different from the Friend's and Wigner's points of view, and were asking if the description of the measurement becomes the same if the Friend's decoherence reaches Wigner, i.e., if the Friend is not inside a decoherence-proof box.

The answer is yes, it does. But that's not an interesting answer, as you're simply denying the premise of the argument, that the Friend is inside a decoherence-proof box.

A more interesting answer is to assert that it is in principle impossible to do the experiment, that the Friend's decoherence will always engulf Wigner. That would require changing the laws of quantum mechanics, though, as it predicts no such inescapable decoherence. When people go this route they usually postulate some objective collapse model (which, by the way, has never been made compatible with relativity).