Möbius gyrovector spaces in quantum information and computation. (English) Zbl 1212.51013

Ungar, Abraham A.

Commentat. Math. Univ. Carol. 49, No. 2, 341-356 (2008).

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 \(\mathbb B^2\) 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 \(n\geq 2\). 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 \(n\geq 2\). 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.

MSC:

51M10	Hyperbolic and elliptic geometries (general) and generalizations
51P05	Classical or axiomatic geometry and physics
81P15	Quantum measurement theory, state operations, state preparations

Keywords:

quantum information; Bloch vector; density matrix; hyperbolic geometry; gyrogroups; gyrovector spaces