Distances of qubit density matrices on Bloch sphere

We recall the Einstein velocity addition on the open unit ball B of R3 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.