The three measurements Qz, Qx, Qy are something like 'shifted Pauli matrices'. Same eigenvectors, but shift the eigenvalues from 1 and -1 to 0 and 1 respectively (for 'no' and 'yes' answers to the questions).
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There's a very influential toy model that also has this 'half the information' property: https://en.wikipedia.org/wiki/Spekkens_toy_model … Very simple model that reproduces some, but not all, features of quantum theory. It uses just the states on the six corners of the octahedron. So no magic.
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Now, there's a fascinating paper by van Enk that builds on the Spekkens toy model, by extending the 'half the information' property to more states: https://arxiv.org/abs/0705.2742 To do this, you need a good definition of information.
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Most obvious choice is the usual one from information theory, linked to the Shannon entropy: https://en.wikipedia.org/wiki/Entropy_(information_theory) … Turns out this isn't the right choice to reproduce QM, though. Instead you want a weirder entropy.
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Shannon is only one of a whole family of entropies, the Rényi entropies: https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy … Wrote some rough notes here that may or may not make sense http://keerlu.github.io/2018/07/16/renyi-entropy.html … Anyway the one we want is called the collision entropy.
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Mathematically this one isn't too bad. The constraint turns out to be that the sum of the squares of the four numbers in the boxes can't be more than half. Conceptually... why the hell is it this one?? I don't know.
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Apparently this has been independently advocated as a measure of information by Brukner and Zeilinger: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.83.3354 … Unfortunately I don't really follow the argument there, so I'm none the wiser.
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If you were expecting this to end with a satisfying conclusion... um, no, I'm still confused and have lots of questions. Some of which are here! So let me know if you have any helpful insights or references.pic.twitter.com/pmnWixChAG
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End of conversation
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