  We analyze the effect of decoherence, modelled by local quantum channels, on
quantum critical states and we find universal properties of the resulting mixed
state's entanglement, both between system and environment and within the
system. Renyi entropies exhibit volume law scaling with a subleading constant
governed by a "$g$-function" in conformal field theory (CFT), allowing us to
define a notion of renormalization group (RG) flow (or "phase transitions")
between quantum channels. We also find that the entropy of a subsystem in the
decohered state has a subleading logarithmic scaling with subsystem size, and
we relate it to correlation functions of boundary condition changing operators
in the CFT. Finally, we find that the subsystem entanglement negativity, a
measure of quantum correlations within mixed states, can exhibit log scaling or
area law based on the RG flow. When the channel corresponds to a marginal
perturbation, the coefficient of the log scaling can change continuously with
decoherence strength. We illustrate all these possibilities for the critical
ground state of the transverse-field Ising model, in which we identify four RG
fixed points of dephasing channels and verify the RG flow numerically. Our
results are relevant to quantum critical states realized on noisy quantum
simulators, in which our predicted entanglement scaling can be probed via
shadow tomography methods.
