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Shared symmetries of the hydrogen atom and the two-bit system

The hydrogen atom is the simplest system of atomic and molecular physics, while a two-qubit system is the simplest of quantum information. Remarkably, they share common symmetry aspects which are described in this paper, based on a correspondence between the four-dimensional unitary group and the six-dimensional rotational group with its non-compact extensions. Both systems involve 15 basic operators. Reductions to Lorentz and Poincare space–time group symmetries of a free particle are also discussed.