  We derive the exact reduced density matrix for the electrons in an
analytically solvable electron-phonon model. Here, the electrons are described
as a Luttinger liquid that is coupled to Einstein phonons. We further derive
analytical expressions for the electron-phonon entanglement, its spectrum and
mutual information at finite and zero temperature as well as for excited
states. The entanglement entropy is additive in momentum for the quasi-particle
excited states, but not in the electron-phonon coupled eigenmodes of the
system.
