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Consider a sub-system in a pure state, expressed on the basis of the observable $O$, whose only eigenstates are $|a\rangle$ & $|b\rangle$. The state is $\frac{1}{\sqrt{2}}(|a\rangle+|b\rangle)$.

There is also a measuring device that measures $O$, initially separated from the sub-system. It is in state $|o\rangle$. The device might or might not include a human being, it doesn't matter.

Eventually they both interact. We can see this interaction in (at least) two ways:

  1. According to only the "deterministic-wave-dynamic" (Schrödinger's equation, or whatever equivalent): We consider the global system that contains both sub-system and measurer, which after the interaction becomes always only $\frac{1}{\sqrt{2}}(|a\rangle|Measured_A\rangle+|b\rangle|Measured_B\rangle)$. Repeating this experiment many times will then yield set of states that are all exactly the same.

  2. According to the "deterministic-wave-dynamic", and also to another additional dynamic, "probabilistic-collapse-dynamic": We consider the global system that contains both sub-system and measurer, which after the interaction becomes either $|a\rangle|Measured_A\rangle$ or $|b\rangle|Measured_B\rangle)$, with such probability given by the wave-dynamic. In this case, $50:50$. Repeating this experiment many times will then yield a statistical mixture of both different states.

Experiment shows that the final observed result is the one given by $2.$, not $1.$. If I'm not mistaken, this procedure is called "the Copenhagen interpretation". However, it is still possible to claim that 1. is correct by using "the Many Worlds interpretation". It states that the measurer is capable of interacting and thus observing only the result with which he is correlated. The $50:50$ probability observed is attributed to the idea, "if there are many worlds, what are the chances that I am in this one?"

Now, both of these are called "interpretations". This means that, despite their apparent differences, they should always predict the exact same results for all possible experiments.

But that is not true here. I can think of an experiment that differentiates them.

The global system is made to interact with another external measurer, which measures observable $X$, whose only eigenstates are:

$$|+\rangle = \frac{1}{\sqrt{2}} \left(|a\rangle|Measured_A\rangle + |b\rangle|Measured_B\rangle\right) \\ |-\rangle = \frac{1}{\sqrt{2}} \left(|a\rangle|Measured_A\rangle - |b\rangle|Measured_B\rangle \right) $$

A measurement of this observable yields |−⟩ 50% of the time for the Copenhagen interpretation, but never yields |−⟩ for the Many Worlds interpretation. Of course this experiment is unfeasible because the enormous complexity of the first measuring device. But it is possible in theory, which is enough for me to say:

They are not interpretations. They are entirely different theories. Is this correct?

Juan Perez
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    You are talking about WIgner's friend paradox: https://en.wikipedia.org/wiki/Wigner%27s_friend Note however that in the experiment that you propose the second observer doesn't know what the first observer measured... in fact, they haven't measured anything until their wave function collapses. – Roger V. May 26 '21 at 14:11
  • Does the collapse of the wave function happen immediately everywhere? In my answer, I discuss Copenhagen vs Many Worlds. – mmesser314 May 26 '21 at 14:20
  • @RogerVadim It's not Wigner's Friend since it's not about an apparent "contradiction" between both observers. What the first observer sees doesn't even matter. Only what the second observer measures, which is not even comparable with the first observation. The second observer will get a set of results from repeating the experiment, and that's all. They will either be collapse-compatible or collapse-incompatible. – Juan Perez May 26 '21 at 14:26
  • I see no contradiction, since the first observer measures nothing, whereas the second is studying more complex system. – Roger V. May 26 '21 at 14:29
  • @RogerVadim "A measurement of this observable yields |−⟩ 50% of the time for the Copenhagen interpretation, but never yields |−⟩ for the Many Worlds interpretation." is what I wrote in my question. Are you saying that that sentence is wrong? – Juan Perez May 26 '21 at 14:32
  • @JuanPerez you could also read Feynman's discussion of a sequence of stern-gerlach devices – Roger V. May 26 '21 at 15:25
  • @JuanPerez I'm trying to understand the logic of the question. Suppose we assert that $|a\rangle|A\rangle$ and $|b\rangle|B\rangle$ are the possible outcomes of a measurement, so that copenhagen is obliged to replace any superposition of those two states with either one or the other. If we assert that $X$ is measurable, doesn't that imply that copenhagen is not obliged to replace $|\pm\rangle$ with either $|a\rangle|A\rangle$ or $|b\rangle|B\rangle$? Don't these two assertions contradict each other within the copenhagen paradigm? – Chiral Anomaly May 27 '21 at 02:12
  • @ChiralAnomaly Wether Copenhagen results in a superposition, or in one collapsed state, depends on whether or not the second subsystem is considered a measuring device. In practice, because of the complexity, we can only measure X if the second subsystem is simple (not a measurer), so the distinction is irrelevant. But I see no reason why X shouldn't be possible to measure in theory, and i n that case we will see: If there is no collapse, Copenhagen is wrong. If there is, it's right. – Juan Perez May 27 '21 at 03:35
  • @JuanPerez In the language of decoherence theory, you are basically proposing that we could see a difference between MWI and a “true collapse” theory if we could capture all of the information that, from our limited human perspective, is lost to decoherence. I think everyone understands that this is true in principle but impossible in practice. Whether a seemingly impossible-to-measure difference counts as a different theory is either a semantic question or one for a philosopher of science. I suspect that a positivist would say it does not. – sasquires Jun 14 '21 at 03:18
  • @sasquires By that logic, if Alice says that it's possible for a human to go to Mars, and Bob says that it's physically impossible, both theories are in fact equal since we've yet to send a human to Mars. They are just different interpretations of the fact that no human has gone to Mars yet. And they'll remain that way only until we get an astronaut there. If this is what physicists mean when they say "interpretation", then I'm rather appalled, but I'll agree then that MWI and Copenhagen are just that. – Juan Perez Jun 15 '21 at 15:38
  • I made no statement about whether they count as different theories or not. I said that technical definitions of the term "theory" and "interpretation" belong in the domain of the philosophy of science. If someone actually found a way to do the experiment, then of course it would be extremely important for physics. – sasquires Jun 16 '21 at 04:00
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    The main problem here is that we all know that Copenhagen and MWI are ontologically different, and that this difference can in principle affect observations. But the implementation isn't just difficult (like going to Mars), but there are some fundamental reasons to think that it might be impossible, or at least that you can place such strong bounds on how much data that you would need to get statistically significant results that it would take longer than the age of the universe. – sasquires Jun 16 '21 at 04:03
  • Normally when someone proposes a thought experiment (such as, for example, Einstein's famous ones, like with a falling elevator and a ray of light), then there is an idea about how to do the implementation. Here there is not a single detail about the implementation of measuring $X$, and for very good reasons, which are that it may be impossible to measure interference effects between macroscopic observers (which, to satisfy the idea of collapse in Copenhagen, must be entangled with the quantum system in question in MWI). – sasquires Jun 16 '21 at 04:06

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