For finite Hilbert space dimension, there’s also a number of works by Christian Herrmann and coauthors on the complexity of quantum logic, such as this one.

]]>Of course it is still possible that 2=3 for this scenario, but there is no evidence for that as far as I know.

]]>But I’m skeptical that 2 can be different than 4 for I3322: The best upper bound we know is 0.250 875 38, which is equal to the best lower bound we know within numerical precision (see http://arxiv.org/abs/1006.3032 ). That paper shows also some inequalities in the 4422 scenario for which there is still a gap between the upper and lower bounds, so that is not ruled out.

]]>#37 and #38: we don’t know the answer to this question. If the answer is yes, then it would imply that 2 != 3, so resolving the question in either direction would be interesting.

]]>If one had such near representations of Higman’s group, then that would give the strategies with probabilities approaching one.

Does Higman’s group have such near representations?

Well, Connes’ is equivalent to every countable discrete group being hyperlinear. So yes if Connes’ is true. Moreover, Higman’s group is considered a good test case for this hyperlinear conjecture. William was aware of this and this is undoubtedly one reason that he chose HIgman’s group in his construction.

…plus ça change, plus c’est la même chose…

So if one follows Scott’s approach and finds optimal strategies in each dimension, one could then try to check, maybe using the 2nd order NPA hierarchy, their multiplicative behaviours on products of two generators. This might give some computational evidence for deciding if Higman’s group is hyperlinear.

A nice reference for these connections between hyperlinear groups and Connes’ embedding conjeccture is Capraro and Lupini’s:

arXiv:1309.2034v6

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