I have often read that
thermal radiation contains no information.
Could someone elucidate this claim?
The statement looks puzzling to me because in apparent contradiction with the two following facts:
The spectrum of thermal radiation depends on the temperature $T$ of the emitting body. By collecting the thermal radiation of a perfect black body, a far away observer (coupled to a thermal bath) can recover the temperature $T$ to an arbitrary number of digits. That can be a lot of information.
If I collect thermal photons emitted from a body of temperature $T$, the their frequencies are sampled from the maximum entropy distribution consistent with the data "the emitting body is a temperature $T$". This is maximum entropy in the sense of Shannon. Thus this generates a maximal amount of information consistent with the temperature of the body.
This statement is of relevance for conversations about the fate of information that enters black holes. However, it's truth should be assessed and argued independently of that physical situation.