  In this article we consider the extended version of a real counterpart of
qubit density matrices, called the M\"obius matrix, and we see that it is a
normalized Lorentz boost. Using the isomorphic gyrogroup structures between the
set $\PP$ of all Lorentz boosts and the Einstein gyrogroup on the open unit
ball $\B$ of $\R^n$ we give an explicit formula of the fidelity for M\"obius
matrices in terms of Lorentz gamma factors.
