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

Abstract

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 paper, we suggest 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 carry "minimal amounts 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 and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.

13 + 22 pages, 9 figures. v4: some clarifications, in particular in Section V / Appendix C (added Example 39)

References in corpus (9)

• Reference frames, superselection rules, and quantum information
• The resource theory of quantum reference frames: manipulations and monotones
• The uncertainty principle determines the non-locality of quantum mechanics
• A derivation of quantum theory from physical requirements
• A generalized no-broadcasting theorem
• (Quantumness in the context of) Resource Theories
• Finite de Finetti theorem for conditional probability distributions describing physical theories
• Secret quantum communication of a reference frame
• Typical local measurements in generalised probabilistic theories: emergence of quantum bipartite correlations

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