Möbius gyrovector spaces in quantum information and computation

Ungar, Abraham A.

Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 341-356.

Résumé

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.

Détail

Classification : 51M10, 51P05, 81P15, 81P68
Keywords: quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces