Abstract
Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects that form the setting for the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for Euclidean geometry. It is thus interesting, in geometric quantum computation, to realize that the set of all qubit density matrices has rich structure with strong link to hyperbolic geometry. Concrete examples for the use of the gyro-structure to derive old and new, interesting identities for qubit density matrices are presented.
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Ungar, A.A. The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits. Foundations of Physics 32, 1671–1699 (2002). https://doi.org/10.1023/A:1021446605657
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DOI: https://doi.org/10.1023/A:1021446605657