# mathematical description

We know that the black hole entropic information formula is equal to the bekenstein bound

$$(kc^3ln(2)t_{evap}) \over 16π^2GM$$=$$2πkRE \over ℏc$$

We know following the work of Casini in 2008  about the Von Neumann entropy and the Bekenstein bound, that the proof of the Bekenstein bound is valid using quantum field theory.

For example, given a spatial region V, Casini defines the entropy on the left-hand side of the Bekenstein bound as

$$S_V = S (ρ_V) − S (ρ_V^0 ) =$$$$− t r ( ρ_V log (ρ_V) ) + t r ( ρ_V^0 log ( ρ_V^0))$$

Casini defines the right-hand side of the Bekenstein bound as the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state,

$$K_V = tr (K ρ_V) − tr (K ρ_V^0)$$

With these definitions, the bound reads SV ≤ KV, which can be rearranged to give:

$$tr (ρ_V log (ρ_V)) − tr (ρ_V log (ρ_V^0)) ≥ 0$$

This is simply the statement of positivity of quantum relative entropy, which proves the Bekenstein bound.

Diving into Casini’s work with the black holes entropic information formula, we obtain new enlightening about black hole fine-grained entropy.

The ingenious proposal of casini  is to replace 2 π R E, by:

$$K_V = tr (K ρ_V) − tr (K ρ_V^0)$$

Indeed, in Casini’s work, on the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity 2 π R E, where R is a characteristic length scale of the system and E is a characteristic energy.

This product has the same units as the generator of a Lorentz boost, and the natural analog of a boost in this situation is the modular Hamiltonian of the vacuum state $$K = − log (ρ_V^0)$$.

With these definitions, the bound reads

$$S_V ≤ K_V$$,

The version of the Bekenstein bound is SV ≤ KV, namely

$$S(ρ_V) − S(ρ_V^0 )≤$$$$Tr(K ρ_V) − Tr(K ρ_V^0)$$

is equivalent to

$$S_V ≡ S(ρ_V|ρ_V^0 ) ≡$$ $$Tr (ρ_V (log (ρ_V) − log (ρ_V^0))) ≥ 0$$

Where the black holes entropic information formula is equal to Sv where $$S (ρ_V)$$ is the Von Neumann entropy of the reduced density matrix $$ρ_V$$associated with V, V in the excited state ρ, and $$S (ρ_V^0)$$ is the corresponding Von Neumann entropy for the vacuum state $$ρ^0$$

$$S_V = S (ρ_V|ρ_V^0) =$$ $$S (ρ_V) − S (ρ_V^0 ) =$$ $$Tr (ρ_V (log (ρ_V) − log (ρ_V^0))) =$$  $$(kc^3ln(2)t_{evap}) \over 16π^2GM$$ ≥ 0

As black holes entropic information formula, is equal to Bekenstein universal bound

$$(kc^3ln(2)t_{evap}) \over 16π^2GM$$=$$2πkRE \over ℏc$$

as

the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state

$$K_V = tr (K ρ_V) − tr (K ρ_V^0)$$.

is equal to Bekenstein universal bound

we obtain:

$$K_V = tr (K ρ_V) − tr (K ρ_V^0) =$$ $$(kc^3ln(2)t_{evap}) \over 16π^2GM$$.

Finally, we obtain for black hole gravitational fine-grained entropy this equality:

$$K_V=tr (K ρ_V) − tr (K ρ_V^0)=$$ $$(kc^3ln(2)t_{evap}) \over 16π^2GM$$=$$S_v=S (ρ_V│ρ_V^0 )=$$$$S(ρ_V )- S(ρ_V^0 )=$$ $$Tr (ρ_V (log (ρ_V) - log (ρ_V^0)))=$$ $$2πkRE \over ℏc$$

Naive definitions of entropy and energy density in Quantum Field Theory suffer from ultraviolet divergences. In the case of the Bekenstein bound, ultraviolet divergences can be avoided by taking differences between quantities computed in an excited state and the same quantities computed in the vacuum state.

We must take note that the first version of the fine-grained entropy formula discovered by Ryu and Takayanagi is a general formula for the fine-grained entropy of quantum systems coupled to gravity.

The black holes entropic Information formula can calculate the entangled Hawking radiation down to the quantum system, describing black hole independently of the area law of the entropy of Bekenstein-Hawking. The black holes entropic information formula is equal to the universal bound originally found by Jacob Bekenstein. Which is equal by Casini’s work to the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state, itself equal to the Von Neumann entropy. The ultraviolet divergences can be avoided by taking differences between quantities computed in an excited state and the same quantities computed in the vacuum state; this must be put in relation to Ryu and Takayanagi conjecture, a general formula for the fine-grained entropy of quantum systems coupled to gravity