  In this work, the use of the Boltzmann collision operator for dissipative
quantum transport is analyzed. Its mathematical role on the description of the
time-evolution of the density matrix during a collision can be understood as
processes of adding and subtracting states. We show that unphysical results can
be present in quantum simulations when the old states (that built the density
matrix associated to an open system before the collision) are different from
the additional states generated by the Boltzmann collision operator. As a
consequence of the Fermi Golden rule, the new generated sates are usually
eigenstates of the momentum or kinetic energy. Then, the different
time-evolutions of old and new states involved in a collision process can
originate negative values of the charge density, even longer after the
collision. This unphysical feature disappears when the Boltzmann collision
operator generates states that were already present in the density matrix of
the quantum system before the collision. Following these ideas, in this paper,
we introduce an algorithm that models phonon-electron interactions through the
Boltzmann collision operator without negative values of the charge density. The
model only requires the exact knowledge, at all times, of the states that build
the density matrix of the open system.
