Combing this with the previous inequality we find that the difference in informationcontent has to be smaller(!) than the change in area! So we see that in gravity, geometry and information are deeply connected and at the heart of it is positivity of relative entropy. 7/7

One (in certain situations) can relate the change in energy to the total area surrounding the System and in particular show that the change in Area has to be larger(!) than the change in energy. 6/x

Now that we kinda know what S does, why do I care? The nice thing is that if one of the states describes the system in thermal equilibrium, i.e. imagine a cup of coffee after it reached room temperature, there is a simple formula for the relative entropy.

Intuitively the relative entropy describes how probable it is to think that the system is in the state rho, if it actually is in the state sigma. If one performs N measurement of the system in the state rho, the probability to mistake the state for sigma falls exponentially with N and rate S 3/x

I am tired and bored, so I'll try this posting thing.
In my opinion, one of the deepest truths, which one unfortunately does not usually learn about in a physics curriculum, is the positivity of relative entropy. Although this is a rather abstract statement I want to try to explain it: 1/x