dorsal/arxiv
View SchemaA Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space
| Authors | Timothy F. Havel, Chris J. L. Doran |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0403136 |
| URL | https://arxiv.org/abs/quant-ph/0403136 |
| DOI | 10.1117/12.540929 |
| Journal | Proc. SPIE, vol. 5436 (Quantum Information and Computation II, E Donkor, A. R. Pirich & H. E. Brandt, eds.), pp. 93-106 (2004) |
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 special 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.
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"abstract": "Geometric algebra is a mathematical structure that is inherent in any metric\nvector space, and defined by the requirement that the metric tensor is given by\nthe scalar part of the product of vectors. It provides a natural framework in\nwhich to represent the classical groups as subgroups of rotation groups, and\nsimilarly their Lie algebras. In this article we show how the geometric algebra\nof a six-dimensional real Euclidean vector space naturally allows one to\nconstruct the special unitary group on a two-qubit (quantum bit) Hilbert space,\nin a fashion similar to that used in the well-established Bloch sphere model\nfor a single qubit. This is then used to illustrate the Cartan decompositions\nand subalgebras of the four-dimensional special unitary group, which have\nrecently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev.\nA 67, 042313, 2003] to study the entangling capabilities of two-qubit\nunitaries.",
"arxiv_id": "quant-ph/0403136",
"authors": [
"Timothy F. Havel",
"Chris J. L. Doran"
],
"categories": [
"quant-ph"
],
"doi": "10.1117/12.540929",
"journal_ref": "Proc. SPIE, vol. 5436 (Quantum Information and Computation II, E\n Donkor, A. R. Pirich \u0026 H. E. Brandt, eds.), pp. 93-106 (2004)",
"title": "A Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space",
"url": "https://arxiv.org/abs/quant-ph/0403136"
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