# The new Entropic Information approach

# The Ryu–Takayanagi conjecture

The Ryu–Takayanagi formula calculates the entropy of quantum entanglement in conformal field theories on Bekenstein-Hawking entropy of black holes in the context of Juan Martín Maldacena's holographic principle, in which conformal field theories on a surface form a gravitational theory in a closed volume .

“The Bekenstein–Hawking area law, while claiming that the area of the black hole horizon is proportional to the black hole's entropy, fails to provide a sufficient microscopic description of how this entropy arises. The holographic principle provides such a description by relating the black hole system to a quantum system which does admit such a microscopic description. In this case, the CFT has discrete eigenstates, and the thermal state is the canonical ensemble of these states. The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law. This turns out to be a special case of the Ryu–Takayanagi conjecture”

The first version of the fine-grained entropy formula was discovered by Ryu and Takayanagi. It was subsequently refined and generalized by several authors . Originally, the Ryu-Takayanagi formula was proposed to calculate holographic entanglement entropy in anti-de Sitter spacetime, but the present understanding of the formula is much more general. It requires neither holography, nor entanglement, nor anti-de Sitter spacetime. Rather it is a general formula for the fine-grained entropy of quantum systems coupled to gravity”.

The black-hole entropy is proportional to the area of its event horizon A.

* The black hole entropy horizon law which turns out to be a special case of the Ryu–Takayanagi conjecture*.

The black-hole entropy area relationship was generalized to arbitrary regions via the Ryu–Takayanagi formula, which relates the entanglement entropy of a boundary conformal field theory to a specific surface in its dual gravitational theory but the current understanding of the formula is much more general. As being a general formula for the fine-grained entropy of quantum systems coupled to gravity.