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I haven't seen this line of argument used. So I'm curious on it's validity and if it already exists in the literature.

In the thought experiment Wigner's friend the disparity of descriptions of the density matrices only happens not because Wigner's friend evolves in a method which is forbidden by the laws of quantum mechanics. The experiment is as follows:

In this thought experiment, Wigner posits that his friend is in a laboratory, and Wigner lets the friend perform a quantum measurement on a physical system (this could be a spin system). This system is assumed to be in a superposition of two distinct states, say, state 0 and state 1. When Wigner's friend measures the system in the 0/1-basis, according to quantum mechanics, they will get one of the two possible outcomes (0 or 1) and the system will collapse into the corresponding state.

Now Wigner himself models the scenario from outside the laboratory, knowing that inside, his friend will at some point perform the 0/1-measurement on the physical system. According to the linearity of the quantum mechanical equations, Wigner will assign a superposition state to the whole laboratory (i.e. the joint system of the physical system together with the friend): The superposition state of the lab is then a linear combination of "system is in state 0/ friend has measured 0" and "system is in state 1/ friend has measured 1".

Let Wigner now ask his friend for the result of the measurement. Whichever answer the friend gives (0 or 1), Wigner would then assign the state "system is in state 0/ friend has measured 0" or "system is in state 1/ friend has measured 1" to the laboratory. Therefore, it is only at the time when he learns about his friend's result that the superposition state of the laboratory collapses.

However, unless Wigner is considered in a "privileged position as ultimate observer", the friend's point of view must be regarded as equally valid, and this is where an apparent paradox comes into play: From the point of view of the friend, the measurement result was determined long before Wigner had asked about it, and the state of the physical system has already collapsed. When exactly did the collapse occur? Was it when the friend had finished their measurement, or when the information of its result entered Wigner's consciousness? As Wigner says, he could ask his friend, "What did you feel about the [measurement result] before I asked you?" The question of what result the friend has seen is surely "already decided in his mind", Wigner writes, which implies that the friend–system joint state must already be one of the collapsed options, not a superposition of them. Wigner concludes that the linear time evolution of quantum states according to the Schrödinger equation cannot apply when the physical entity involved is a conscious being.

But the true inconsistency comes when he's able to describe a density matrix which is different from Wigner. I think "comprehension" (I'm going to use this word in the Wittgenstein sense) is some kind of operation (or compuation) which cannot be described by quantum mechanics (or quantum information theory).

Has this been argued? (I wouldn't be surprised if Penrose used this line of argument)

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  • Please describe the thought experiment if you are going to reference it.
    – David Gudeman
    Aug 11, 2022 at 12:04
  • @DavidGudeman done. I thought providing a link was sufficient.
    – More Anonymous
    Aug 11, 2022 at 12:17
  • "There is a paradox only if we suppose that a density matrix (i.e. a probability distribution) is something 'physically real' and 'absolute'. But now the dilemma disappears when we recognize [that the density matrix] represents, not a physical situation, but only a certain state of knowledge about a range of possible physical situations", Jaynes. Already von Neumann showed that the timing of the collapse has no effect on anything observable. The "comprehension" is indeed not described by quantum mechanics, it is described by statistics and is called Bayesian conditioning on new data.
    – Conifold
    Aug 11, 2022 at 12:41
  • @Conifold what is the new data here? There is nothing stopping the outside observer from saying well I am the "less privileged observer" I will use the density matrix of Wigner's friend. Which density matrix will provide a more accurate description?
    – More Anonymous
    Aug 11, 2022 at 12:52
  • The outside observer does not know the friend's density matrix, that's what stops him from using it. Once he finds it out he gets new data and "comprehends" it by conditioning his probabilities on it. Neither of them is privileged, their available data is just different until they communicate. The only "privileged observer" is the device that actually interacts with the spin system, and different observers condition their probabilities at different times, as the interaction data becomes available to them.
    – Conifold
    Aug 11, 2022 at 13:04

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The theory of quantum computation implies that any physical system, including a person, can be simulated by a universal quantum computer:

http://www.daviddeutsch.org.uk/wp-content/ItFromQubit.pdf

So your solution requires throwing out quantum theory and the theory of computation, which is a disadvantage.

The Wigner's friend paradox comes from the idea of collapse not from quantum mechanics. If there is no collapse then there is no need to pick an observer favoured by collapse. The fact that we only see one value when we do a measurement is explained by decoherence, which prevents interactions between different measurement results:

https://arxiv.org/abs/1111.2189

There are multiple versions of the measurement result and the observer after the measurement, but they can't interact so you only see one result.

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The error in the argument as quoted in the question is this:

the friend's point of view must be regarded as equally valid

No, it needn't be. Not all points of view are equally valid.

When the friend in the lab does the measurement, either a wave function collapse occurs or it does not. If it does occur, then Wigner's linear model of the friend+lab system is wrong. If it doesn't occur, then his friend's nonlinear model is wrong. The models are different and at most one can be correct, even if we can't experimentally distinguish them right now, just as only one theory of quantum gravity is correct even if we can't tell which one.

In the implausible, but not philosophically impossible, event that Wigner and his friend can perform measurements with such precision that they can determine whether the other, almost-orthogonal branch of the wave function still exists or whether it has been pruned, they should use that ability to determine which model is correct, and then both use the correct model. If they can't do any distinguishing experiment, then it doesn't matter which model either of them uses. Wigner does not have to use a linear model, and his friend does not have to use a nonlinear model.

The author of this thought experiment would have you believe that even after Wigner and his friend meet and compare notes, they are obliged to reach incompatible conclusions—Wigner that the wave function collapsed later, his friend that it collapsed earlier—despite having access to exactly the same data, solely because of who they are. That sort of subjectivism is antithetical to the scientific worldview and it has never been a feature of any mainstream physical theory, despite strangely widespread beliefs to the contrary.

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