  Open system quantum dynamics can generate a variety of long-range entangled
mixed states, yet it has been unclear in what sense they constitute phases of
matter. To establish that two mixed states are in the same phase, as defined by
their two-way connectivity via local quantum channels, we use the
renormalization group (RG) and decoders of quantum error correcting codes. We
introduce a real-space RG scheme for mixed states based on local channels which
ideally preserve correlations with the complementary system, and we prove this
is equivalent to the reversibility of the channel's action. As an application,
we demonstrate an exact RG flow of finite temperature toric code in two
dimensions to infinite temperature, thus proving it is in the trivial phase. In
contrast, for toric code subject to local dephasing, we establish a mixed state
toric code phase using local channels obtained by truncating an RG-type decoder
and the minimum weight perfect matching decoder. We also discover a precise
relation between mixed state phase and decodability, by proving that local
noise acting on toric code cannot destroy logical information without bringing
the state out of the toric code phase.
