Thermal radiation contains no information? 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:


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*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.
 A: Here is another way of looking at this which might be helpful. 
When we say that a blackbody spectrum "contains no information", this means that the process of a body achieving thermal equilibrium with the radiation that surrounds it erases any information about the thermal history of the system. In other words, by the time the spectrum has formed, it is impossible to tell anything at all about how the body got to that temperature or where the radiation it equilibrated with came from: its history has been erased. 
A: First, what is information? It is the ability to do prediction. 
So if you collected up all the Hawking radiation (thermal, a macrostate) from a black hole evaporation say, you would not be able to predict/deduce anything about the inside of the black hole from it. i.e. thermal radiation has no information.  
This is totally unlike the radiation coming from a say, a star, (radio, light etc). If you collect this radiation (as scientists have) you can deduce a lot about the nature of the reactions going on inside the star. This radiation has information (microstates). 
There is more here
