  Certain pure-state symmetry-protected topological orders (SPT) can be used as
a resource for transmitting quantum information. Here, we investigate the
ability to transmit quantum information using decohered SPT states, and relate
this property to the "strange correlation functions" which diagnose quantum
many-body orders in these mixed-states. This perspective leads to the
identification of a class of quantum channels -- termed symmetry-decoupling
channels -- which do not necessarily preserve any weak or strong symmetries of
the SPT state, but nevertheless protect quantum many-body order in the
decohered mixed-state. We quantify the ability to transmit quantum information
in decohered SPT states through the coherent quantum information, whose
behavior is generally related to a decoding problem, whereby local measurements
in the system are used to attempt to "learn" the symmetry charge of the SPT
state before decoherence.
