Hilbert Projective Metric on a Gyrogroup of Qubit Density Matrices

Based on the fact that a gyrogroup provides an algebraic tool to study hyperbolic geometry, we consider the gyrogroup structure on the set of certain quantum mixed states presented by invertible density matrices. Some measurements like trace metric and fidelity have been commonly used to distinguish quantum states, but we study the Hilbert projective metric and its properties for a measurement of quantum states. Especially, we show the relationship between the Hilbert metric of qubit mixed states and the rapidity metric on the Einstein gyrogroup using their gyrogroup isomorphism. We also compute the Riemannian metric of qubit mixed states using such an isomorphism.