# The new Entropic Information approach

# Fine grained entropy notion and definition

We can define a fine-grained and coarse-grained entropy.

“Where fine grained entropy is the entropy of the density matrix calculated by the standard methods of quantum field theory in curved spacetime. In the literature, this is often simply called the von Neumann entropy”.

It is Shannon’s entropy with distribution replaced by density matrix. It is invariant under unitary time evolution.

While the coarse-grained entropy is defined as follows: we only measure simple observables and consider all possible density matrices which give the same result as our system Tr[*p̃*]= Tr[*p*]. We then choose the maximal von Neumann entropy over possible density matrices S(*p̃*).

It increases under unitary time evolution, i.e. thermodynamics entropy

The black holes entropic information formula expresses the gravitational fine-grained entropy of the black hole by a semiclassical gravity approach.

Indeed, in a semiclassical gravity view, matter is represented by quantum matter fields that propagate according to QFTCS.

*We have calculated the Von Neumann entropy, the gravitational fine-grained entropy of the black hole independently of the area of the horizon law.*

The black holes entropic information formula expresses the black hole gravitational fine-grained entropy down to the quantum level independently of the area horizon’s law permitting to the entropy of Hawking radiation to be entangled with the initial considered black hole seen as a whole quantum system.

The black holes entropic information formula must be put in relation to Ryu–Takayanagi formula being a general formula for the fine-grained entropy of quantum systems coupled to gravity.