Distances of qubit density matrices on Bloch sphere

Abstract

We recall the Einstein velocity addition on the open unit ball \B of $\R^{3}$ and its algebraic structure, called the Einstein gyrogroup. We establish an isomorphism between the Einstein gyrogroup on \B and the set of all qubit density matrices representing mixed states endowed with an appropriate addition. Our main result establishes a relation between the trace metric for the qubit density matrices and the rapidity metric on the Einstein gyrogroup \B.Comment: I thank to my supervisor, Jimmie Lawson. This was accepted in Journal of Mathematical Physics at September 26, 201