  Density Matrix Renormalization Group (DMRG) algorithm has been extremely
successful for computing the ground states of one-dimensional quantum many-body
systems. For problems concerned with mixed quantum states, however, it is less
successful in that either such an algorithm does not exist yet or that it may
return unphysical solutions. Here we propose a positive matrix product ansatz
for mixed quantum states which preserves positivity by construction. More
importantly, it allows to build a DMRG algorithm which, the same as the
standard DMRG for ground states, iteratively reduces the global optimization
problem to local ones of the same type, with the energy converging
monotonically in principle. This algorithm is applied for computing both the
equilibrium states and the non-equilibrium steady states, and its advantages
are numerically demonstrated.
