Matrix Product Density Operators: Simulation of finite-T and dissipative   systems
  We show how to simulate numerically both the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium. The method applies to both finite and inhomogeneous systems and it is based on two ideas: (a) a representation for density operators which extends that of matrix product states to mixed states; (b) an algorithm to approximate the evolution (in real or imaginary time) of such states which is variational (and thus optimal) in nature. 
F. Verstraete, J. J. Garcia-Ripoll, J. I. Cirac