Möbius gyrovector spaces in quantum information and computation

Abraham A. Ungar

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 341-356
  • ISSN: 0010-2628

Abstract

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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 𝔹 2 of a Euclidean 2-space ℝ 2 is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball 𝔹 n of a Euclidean n -space ℝ n for all n ≥ 2 . Similarly, the decomplexification of the complex 2 × 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 × 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 ≥ 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.

How to cite

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Ungar, Abraham A.. "Möbius gyrovector spaces in quantum information and computation." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 341-356. <http://eudml.org/doc/250484>.

@article{Ungar2008,
abstract = {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 $\mathbb \{B\}^2$ of a Euclidean 2-space $\mathbb \{R\}^2$ is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball $\mathbb \{B\}^n$ of a Euclidean $n$-space $\mathbb \{R\}^n$ for all $n\ge 2$. 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.},
author = {Ungar, Abraham A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces; quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces},
language = {eng},
number = {2},
pages = {341-356},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Möbius gyrovector spaces in quantum information and computation},
url = {http://eudml.org/doc/250484},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Ungar, Abraham A.
TI - Möbius gyrovector spaces in quantum information and computation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 341
EP - 356
AB - 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 $\mathbb {B}^2$ of a Euclidean 2-space $\mathbb {R}^2$ is presented. This decomplexification proves useful, enabling the resulting real Möbius addition to be generalized into the open unit ball $\mathbb {B}^n$ of a Euclidean $n$-space $\mathbb {R}^n$ for all $n\ge 2$. 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.
LA - eng
KW - quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces; quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces
UR - http://eudml.org/doc/250484
ER -

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