  We propose the following definition of topological quantum phases valid for
mixed states: two states are in the same phase if there exists a time
independent, fast and local Lindbladian evolution driving one state into the
other. The underlying idea, motivated by K\"onig and Pastawski in 2013, is that
it takes time to create new topological correlations, even with the use of
dissipation. We show that it is a good definition in the following sense: (1)
It divides the set of states into equivalent classes and it establishes a
partial order between those according to their level of "topological
complexity". (2) It provides a path between any two states belonging to the
same phase where observables behave smoothly. We then focus on pure states to
relate the new definition in this particular case with the usual definition for
quantum phases of closed systems in terms of the existence of a gapped path of
Hamiltonians connecting both states in the corresponding ground state path. We
show first that if two pure states are in the same phase in the Hamiltonian
sense, they are also in the same phase in the Lindbladian sense considered
here. We then turn to analyse the reverse implication, where we point out a
very different behaviour in the case of symmetry protected topological (SPT)
phases in 1D. Whereas at the Hamiltonian level, phases are known to be
classified with the second cohomology group of the symmetry group, we show that
symmetry cannot give any protection in 1D in the Lindbladian sense: there is
only one SPT phase in 1D independently of the symmetry group. We finish
analysing the case of 2D topological quantum systems. There we expect that
different topological phases in the Hamiltonian sense remain different in the
Lindbladian sense. We show this formally only for the $\mathbb{Z}_n$ quantum
double models.
