Introducing the Einstein Metric to Quantum Computation and Quantum Information Geometry

The Bures fidelity between two states of a qubit quantifies the extent of which the two states are distinguished from one another. It is generated by the so called Bloch vectors, which are elements of the closed unit ball of the Euclidean 3-space. We uncover a link between the Bures fidelity and Einstein's addition in the ball, Theorem 3. We show that in terms of Einstein's addition of relativistically admissible velocities, the Bures fidelity takes a simple, elegant form, (17). This, in turn, demonstrates that the Bures fidelity is regulated by the Beltrami ball model of the hyperbolic geometry of Bolyai and Lobachevski.