Classification of phases for mixed states via fast dissipative evolution

Understand

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.