  The global-in-time existence and uniqueness of bounded weak solutions to a
spinorial matrix drift-diffusion model for semiconductors is proved. Developing
the electron density matrix in the Pauli basis, the coefficients (charge
density and spin-vector density) satisfy a parabolic $4\times 4$
cross-diffusion system. The key idea of the existence proof is to work with
different variables: the spin-up and spin-down densities as well as the
parallel and perpendicular components of the spin-vector density with respect
to the magnetization. In these variables, the diffusion matrix becomes
diagonal. The proofs of the $L^\infty$ estimates are based on Stampacchia
truncation as well as Moser- and Alikakos-type iteration arguments. The
monotonicity of the entropy (or free energy) is also proved. Numerical
experiments in one space dimension using a finite-volume discretization
indicate that the entropy decays exponentially fast to the equilibrium state.
