| Title:
|
Möbius gyrovector spaces in quantum information and computation (English) |
| Author:
|
Ungar, Abraham A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
341-356 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball $\Bbb B^2$ of a Euclidean 2-space $\Bbb R^2$ is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball $\Bbb B^n$ of a Euclidean $n$-space $\Bbb R^n$ for all $n\ge2$. Similarly, the decomplexification of the complex $2\times 2$ qubit density matrix of QIC, which is parametrized by the real, 3-dimensional Bloch gyrovector, into an equivalent (in a specified sense) real $4\times 4$ matrix is presented. As in the case of Möbius addition, this decomplexification proves useful, enabling the resulting real matrix to be generalized into a corresponding matrix parametrized by a real, $n$-dimensional Bloch gyrovector, for all $n\ge 2$. The applicability of the $n$-dimensional Bloch gyrovector with $n=3$ to QIC is well known. The problem as to whether the $n$-dimensional Bloch gyrovector with $n>3$ is applicable to QIC as well remains to be explored. (English) |
| Keyword:
|
quantum information |
| Keyword:
|
Bloch vector |
| Keyword:
|
density matrix |
| Keyword:
|
hyperbolic geometry |
| Keyword:
|
gyrogroups |
| Keyword:
|
gyrovector spaces |
| MSC:
|
51M10 |
| MSC:
|
51P05 |
| MSC:
|
81P15 |
| MSC:
|
81P68 |
| idZBL:
|
Zbl 1212.51013 |
| idMR:
|
MR2426897 |
| . |
| Date available:
|
2009-05-05T17:11:43Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119727 |
| . |
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