Entropy, Relative entropy, Mutual information

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atyy
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The Bekenstein-Hawking entropy is expected to be, and has been shown to be in some cases, derived from counting states.

However, entropy is not defined for continuous probability densities, and so I have heard it said that relative entropy (of which the mutual information is a form) is more fundamental.

I think in classical statistical mechanics, the entropy is computed using canonical coordinates, since the phase space is continuous, which is one way to get round the need for discrete probability distributions.

In the context of string theory and quantum gravity, is entropy or the relative entropy more fundamental?


Some possibly related comments:
"the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit" http://arxiv.org/abs/1010.4038
 

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