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

TL;DR

This paper uses information theory to explain why our universe has three spatial dimensions and why quantum systems like qubits exhibit a three-dimensional state space, linking physical space and quantum information.

Contribution

It provides a novel information-theoretic proof that three spatial dimensions and quantum theory on two qubits are uniquely determined by minimal directional information and probabilistic interactions.

Findings

Spatial dimension d=3 is uniquely determined.

Quantum theory on two qubits is derived from minimal directional information.

Observers can infer local spatial geometry from probability measurements.

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.