Markus P. Mueller, Lluis Masanes
It is sometimes pointed out as a curiosity that the state space of quantum theory and actual physical space seem related in a surprising way: not only is space three-dimensional and Euclidean, but so is the Bloch ball which describes quantum two-level systems. In this paper, we show how this observation can be turned into a rigorous mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that in some sense behave as "units of direction information", interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d=3 and quantum theory on two qubits (including entanglement, unitary time evolution and complementarity), and that it allows observers to infer local spatial geometry from probability measurements.
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http://arxiv.org/abs/1206.0630
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