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

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.


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