A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space

Timothy F. Havel; Chris J. L. Doran

doi:10.1117/12.540929

24 August 2004 A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space

Timothy F. Havel, Chris J. L. Doran

Proceedings Volume 5436, Quantum Information and Computation II; (2004) https://doi.org/10.1117/12.540929
Event: Defense and Security, 2004, Orlando, Florida, United States

Abstract

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.

Citation Download Citation

Timothy F. Havel and Chris J. L. Doran "A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space", Proc. SPIE 5436, Quantum Information and Computation II, (24 August 2004); https://doi.org/10.1117/12.540929

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