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It seems to be part of the theory of Hawking radiation that the spectrum is black body and that it therefore cannot carry any information...thus creating a conflict with the conservation of information in QM. But Shannon's information is maximized by a random sequence. And any highly compressed signal looks random. So it seems that a given spectrum doesn't constrain information transmitted, whether the spectrum or is black body or something else. The power spectrum only determines the two-time correlation function. So why is it asserted that radiation with a black body spectrum cannot transmit information?

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    $\begingroup$ Doesn’t black-body radiation contain one piece of information, the temperature? A Planck distribution is completely specified by that one parameter. $\endgroup$
    – G. Smith
    Aug 24 '20 at 18:56
  • $\begingroup$ Indeed, my digital thermometer serves to obtain this information from black body radiation. $\endgroup$
    – my2cts
    Aug 24 '20 at 21:30
  • $\begingroup$ @G.Smith Don't forget $\hat k$. Blackbody radiation also contains a direction. $\endgroup$
    – JEB
    Aug 25 '20 at 0:58
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any highly compressed signal looks random

Yes, but two different signals, when (losslessly) compressed, result in two different outputs. The semiclassical theory of Hawking radiation says that two input states which form a black hole of the same mass will evolve into the same output state, which is $\rho_\beta = e^{-\beta H}/\text{ Tr } e^{-\beta H}$ where $\beta = 1/$temperature.

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  • $\begingroup$ I appreciate that the classical no-hair theorem says that the radiation is completely defined by it's temperature and the fact that it's black body. But since the black hole has a temperature and entropy in virtue of having some micro degrees of freedom I'm asking why the radiation must not also contain information and I'm claiming that simply saying a signal has some given spectrum doesn't not mean it only communicates the parameters of the spectrum. The spectrum is a statistical average and not a good measure of information. $\endgroup$ Aug 25 '20 at 6:27
  • $\begingroup$ Classically, a black hole has no micro degrees of freedom by no-hair theorem. Quantumly, no one knows, though it's suspected it does. If that's true the radiation isn't perfectly blackbody. $\endgroup$
    – Dwagg
    Aug 25 '20 at 13:33

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