Three-dimensionality of space and the quantum bit: an information-theoretic approach

It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this talk, I report on joint work with Lluis Masanes [1], where we attempt an information-theoretic analysis of this relationship, by proving a particular 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 behave in some sense as ``units of direction information,'' interacting continuously and reversibly in time. We prove that this uniquely determines spatial dimension d=3 and quantum theory on two qubits (that is, the complex Hilbert space formalism and unitary time evolution). Moreover, we prove that it allows observers to infer local spatial geometry from probability measurements. This applies and generalizes results obtained earlier with further collaborators [2,3]. \\[4pt] [1] M. P. Mueller and Ll. Masanes, Three-dimensionality of space and the quantum bit: how to derive both from information-theoretic postulates, arXiv:1206.0630\\[0pt] [2] G. de la Torre, Ll. Masanes, A. J. Short, and M. P. Mueller, Deriving quantum theory from its local structure and reversibility, Phys. Rev. Lett. 109, 090403 (2012)\\[0pt] [3] Ll. Masanes, M. P. Mueller, D. Perez-Garcia, and R. Augusiak, Entangling dynamics beyond quantum theory, arXiv:1111.4060