# Wigners Friend where is the paradox?

ManyNames
So i try and try to understand why physics should suggest a quantum entanglement of wigner friend and the observed system, because instantaneous measurement makes a collapse in the system, there should be no question when the wave function is determined.

Suppose we consider two equations:

$\zeta_{\delta^{\alpha=i}_{\beta=i}} |\psi (t)>=\sum_n |\psi_n (\alpha_{\delta}, \beta_{\delta*} (t))|^2$

and

$\zeta_{\delta^{\alpha=j}_{\beta=j}} |\psi (t)> \ne \sum_n |\psi_n (\alpha_{\delta}, \beta_{\delta*} (t))|^2$

It's destinctive immediately that eq. 1 has a squared modulus value solution, whereas the equation with the subscripts containing $j$ in eq. 2 shown here as $\delta^{\alpha=j}_{\beta=j}$ do not lead to a real positive value of 1. The Dirac Delta is used trivially to express when both equations can be valid. In these equations, $\alpha$ is the observer and $\beta$ is the observed. Reference of the states of $\alpha$ and $\beta$ is given by the association of the collapse when both $\alpha$ and $\beta$ converge in a collapse.

If the observer and the observed have not collapsed, then the remain in a superpositioning in a joint state with the observed system:

$|\psi \alpha_{\delta_{t_1}} ... ... \psi \alpha_{\delta_(t_2)}> + |\psi \beta_{\delta_{t_1}}... ... \psi \beta_{\delta_{t_2}}>$

The joint state can be seen as evolving linearly, and being in a superpositining state then by:

$\zeta_{\delta^{\alpha=i}_{\beta=j}} |\psi (t)>=|\psi \alpha(t)> + |\psi \beta(t)>$

Remember that iff $\alpha$ and $\beta$ have values of $i$ can they collapse -interaction between the scientist and the state happens - no question of when the wave function truely has collapsed as the scientist holds it as an absolute fact of the exeriment.

Let's take $\alpha*$ as the second observer. If $|\psi \alpha_{\delta_{t_1}} ... ... \psi \alpha_{\delta_(t_2)}> + |\psi \beta_{\delta_{t_1}}... ... \psi \beta_{\delta_{t_2}}>$ converges into a collapse (the collapsed states as the observer and the observed) in an earlier time than which the second observer measures the experiment, then $\alpha*(t)<\alpha (t)$. Information on the state of $\beta$ is then assertained in an early period. The description of:

\zeta_{\delta^{\alpha=i}_{\beta=i}} |\psi (t_1)>=\sum_n |\psi (\alpha, \beta(t_1)|^2

states that the wave function of the observed system has collapsed, and time has been expressed in a past time sense $t_1$. Since one of the questions of Wigners Friend is if the first observer and the system are themselves in a state of quantum superpositioning?

How could it be though, if not vanishingly small, since we can state that the observers wave function and the observed wavefunction coheres, and the observed system has observable attributes; it seems valid to state the wave function collapsed when the meaurement was first taken, not whether or not a second observer is required.

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Gold Member
The supposed paradox is that the friend's lab does not seem to settle down into a collapsed state. If you actually model the evolution they remain superposed, despite that the friend should see an outcome. So it seems Wigner's description of his friend contradicts the definite observation of an outcome. The convergence to collapsed state that you describe does not occur.

However it isn't really a paradox for two reasons.

Firstly the friends observation of an outcome is related to the state of his measuring device. If Wigner traced over the environment within the lab, he would see even his superposed state describes decoherence of the device pointer observable and thus it is fine to take an ignorance based view of its statistics (i.e. there is an outcome but I don't know it).

An even deeper reason it isn't a paradox though is that one can arrange the paradox exactly as it is in Classical theories with an epistemic limit like Spekkens toy model and you can see Wigner's superposed state is just a result of him tracking the environment. His maximum information state isn't as "tight" as that of the friend who only tracks the device.

There's nothing Quantum about Wigner's friend as these Classical theories have it.