2004/03/18 by Timothy F. Havel, Chris Doran, Chris J. L. Doran
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Affine space #Algebra over a field #Algebraic and Geometric Analysis #Algebraic structures and combinatorial models #Bloch sphere #Clifford algebra #Eight-dimensional space #Euclidean distance matrix #Euclidean group #Euclidean space #Geometric algebra #Geometry #Group (periodic table) #Hilbert space #Mathematics #Physics #Pure mathematics #Quantum #Quantum mechanics #Qubit #Scalar (mathematics) #Seven-dimensional space #Tensor product #Unitary state #Vector space #quant-ph
paper · pdf · doi:10.1117/12.540929
published as Proc. SPIE, vol. 5436 (Quantum Information and Computation II, E Donkor, A. R. Pirich & H. E. Brandt, eds.), pp. 93-106 (2004) · 14 pages, 2 figures, in press (Proceedings of SPIE Conference on Defense & Security)
arxiv created 2004/03/18 · openalex publication_date 2004/08/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.