Experimentation on "Quantum Permutations"
Consider the following hypothetical experiments in quantum physics:
Experiment A:
Consider four 'shells' in a line (see here for inspiration for name), and there are four different colour peas/particles under them, say red, yellow, green, blue, and an observer can make a measurement by lifting up a given shell and observing which of the four types is underneath. An observer cannot simultaneously measure/observe what peas are under more than one shell at a time.
Suppose the observer finds the red pea under shell one, then the blue pea under shell three, but then when they observe the pea under shell one again they find not a red pea but a yellow pea!
Experiment B:
There are four coloured 'sources' containing each a particle of that colour. There are four 'targets', and the four particles 'go' from the sources to the targets. An observer can take a coloured particle, and can measure to what target it goes to. An observer cannot simultaneously measure/observe the targets of two sources.
Suppose the observer finds that the particle from the red source went to target 1, then found the blue particle went to target 3, but when they observe the particle from the red source again they find it going to target 2.
There are more rules here. 617
In particular, these measurements are all projective in the sense that if the observer finds the red pea under shell one, and check under shell one again they will find it again with probability 1, and furthermore, after observing the red pea under shell one, they will not immediately find the red pea under another shell.
Have any actual experiments in quantum physics taken place that exhibit the behaviour I am talking about in the above?
This question is related to the (purely mathematical) quantum permutations of <PERSON>; I am trying to figure out if the behaviour we see in these purely abstract objects has been observed in a lab.
It is worth noting that this cannot happen with three shells (peas red/blue/green), three targets/sources. Why? Mathematically, under suitable assumptions (aka <PERSON>'s definition), it is possible to show that all the measurements described above commute, and so quantum behaviour can not be exhibited.
Perhaps quantum physics says that if we measure what is under shell one, and it is red, then the peas under shells two and three, blue and green, are entangled... and so as soon as we measure that, e.g the green pea is under shell two, the 'wavefunction' for shell three collapses to pea blue and so quantum behaviour is not observed (i.e. we know that shells two and three contain, at this instant, peas green and blue, so the probability of now observing anything other than pea red under shell one is zero.
What would be cool also, and comprises a second question:
Consider Experiment B as perhaps something like three photons going from the three sources to three targets. Is it possible to set up detectors such that the number of fixed points (number of particles that go from source $i$ to target $i$), and the target for the particle from the first source can be simultaneously observed? If you can do this you can determine the permutation. This is a nice mathematical exercise but if an experiment can be arranged this would be almost a physical 'proof' that there is no quantum $S_3$.
Third question: is this consistent with the many worlds interpretation of QM? 617