  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 $\B$.
