Two-qubit analogs of octonions and projective planes

TL;DR

This paper explores the mathematical connections between two-qubit systems in quantum information and advanced algebraic structures like octonions and projective geometries, revealing potential new insights for quantum theory and geometry.

Contribution

It develops novel links between two-qubit algebraic operators and complex geometric structures such as octonions and projective planes, expanding the mathematical framework of quantum information.

Findings

Identifies analogies between two-qubit operators and octonions

Connects two-qubit systems to projective geometry and Steiner systems

Suggests potential applications in quantum information and geometry

Abstract

A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its sub-algebras, which are important throughout the field of quantum information, with octonions, the projective plane of seven elements, and entities in projective geometry and design theory called partial Steiner systems of 10 and 15 elements. Although not the exact correspondence between a single spin and quaternions, the close analogy between these geometrical objects and the 15 operators of a two-spin system may be useful to both projective geometry and quantum information.