Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra.
| Published in | Science Research (Volume 3, Issue 5) |
| DOI | 10.11648/j.sr.20150305.11 |
| Page(s) | 240-247 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2015. Published by Science Publishing Group |
Qubits, Geometric Algebra, Clifford Translation, Berry Phase, Topological Quantum Computing
| [1] | A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That," January 2015. [Online]. Available: http://arxiv.org/abs/1502.02169. |
| [2] | A. Soiguine, Vector Algebra in Applied Problems, Leningrad: Naval Academy, 1990 (in Russian). |
| [3] | A. Soiguine, "Complex Conjugation - Realtive to What?," in Clifford Algebras with Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 285-294. |
| [4] | D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999. |
| [5] | C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge: Cambridge University Press, 2010. |
| [6] | A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available: http://arxiv.org/abs/1406.3751. |
| [7] | A. Soiguine, "A tossed coin as quantum mechanical object," September 2013. [Online]. Available: http://arxiv.org/abs/1309.5002. |
| [8] | X.-F. Qian, B. Little, J. C. Howell and J. H. Eberly, "Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields," Optica, pp. 611-615, 20 July 2015. |
| [9] | J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons, 2011. |
| [10] | D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Boston: Birkhauser, 2004. |
APA Style
Alexander Soiguine. (2015). Quantum State Evolution in C2 and G3+. Science Research, 3(5), 240-247. https://doi.org/10.11648/j.sr.20150305.11
ACS Style
Alexander Soiguine. Quantum State Evolution in C2 and G3+. Sci. Res. 2015, 3(5), 240-247. doi: 10.11648/j.sr.20150305.11
AMA Style
Alexander Soiguine. Quantum State Evolution in C2 and G3+. Sci Res. 2015;3(5):240-247. doi: 10.11648/j.sr.20150305.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.sr.20150305.11,
author = {Alexander Soiguine},
title = {Quantum State Evolution in C2 and G3+},
journal = {Science Research},
volume = {3},
number = {5},
pages = {240-247},
doi = {10.11648/j.sr.20150305.11},
url = {https://doi.org/10.11648/j.sr.20150305.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sr.20150305.11},
abstract = {Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra.},
year = {2015}
}
TY - JOUR T1 - Quantum State Evolution in C2 and G3+ AU - Alexander Soiguine Y1 - 2015/08/19 PY - 2015 N1 - https://doi.org/10.11648/j.sr.20150305.11 DO - 10.11648/j.sr.20150305.11 T2 - Science Research JF - Science Research JO - Science Research SP - 240 EP - 247 PB - Science Publishing Group SN - 2329-0927 UR - https://doi.org/10.11648/j.sr.20150305.11 AB - Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra. VL - 3 IS - 5 ER -
Information
Email: