In 2006, Shinsei Ryu and Tadashi Takayanagi proposed a conjecture within holography (an approach to quantum gravity) that posits a quantitative relationship between the entanglement entropy of a conformal field theory and the geometry of an associated anti-de Sitter spacetime. The formula characterizes "holographic screens" in the bulk; that is, it specifies which regions of the bulk geometry are "responsible to particular information in the dual CFT". The authors were awarded the 2015 Breakthrough Prize in Fundamental Physics for "fundamental ideas about entropy in quantum field theory and quantum gravity", and awarded the 2024 Dirac Medal of the ICTP for "their insights on quantum entropy in quantum gravity and quantum field theories". The formula was generalized to a covariant form in 2007, and to mixed states in 2019.
Motivation The thermodynamics of black holes suggests certain relationships between the entropy of black holes and their geometry. Specifically, the Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area:
S BH = k B A 4 ℓ P 2 {\displaystyle S_{\text{BH}}={\frac {k_{\text{B}}A}{4\ell _{\text{P}}^{2}}}}
The Bekenstein–Hawking entropy S BH {\displaystyle S_{\text{BH}}} is a measure of the information lost to external observers due to the presence of the horizon. The horizon of the black hole acts as a "screen" distinguishing one region of the spacetime (in this case the exterior of the black hole) that is not affected by another region (in this case the interior). The Bekenstein–Hawking area law states that the area of this surface is proportional to the entropy of the information lost behind it. The Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system; however, there is another type of entropy that is important in quantum information theory, namely the entanglement (or von Neumann) entropy. This form of entropy provides a measure of how far from a pure state a given quantum state is, or, equivalently, how entangled it is. The entanglement entropy is a useful concept in many areas, such as in condensed matter physics and quantum many-body systems. Given its use, and its suggestive similarity to the Bekenstein–Hawking entropy, it is desirable to have a holographic description of entanglement entropy in terms of gravity.
Holographic preliminaries
The holographic principle states that gravitational theories in a given dimension are dual to a gauge theory in one lower dimension. The AdS/CFT correspondence is one example of such duality. Here, the field theory is defined on a fixed background and is equivalent to a quantum gravitational theory whose different states each correspond to a possible spacetime geometry. The conformal field theory is often viewed as living on the boundary of the higher dimensional space whose gravitational theory it defines. The result of such a duality is a dictionary between the two equivalent descriptions. For example, in a CFT defined on d {\displaystyle d} dimensional Minkowski space the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole. Important for the present discussion is that the thermal state of a CFT defined on the d {\displaystyle d} dimensional sphere corresponds to the d + 1 {\displaystyle d+1} dimensional Schwarzschild black hole in AdS space. 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.
Conjecture Consider a spatial slice Σ {\displaystyle \Sigma } of an AdS space time on whose boundary we define the dual CFT. The Ryu–Takayanagi formula states:
where S A {\displaystyle S_{A}} is the entanglement entropy of the CFT in some spatial sub-region A ⊂ ∂ Σ {\displaystyle A\subset \partial \Sigma } with its complement B {\displaystyle B} , and γ A {\displaystyle \gamma _{A}} is the Ryu–Takayanagi surface in the bulk. This surface must satisfy three properties:
γ A {\displaystyle \gamma _{A}} has the same boundary as A {\displaystyle A} .
γ A {\displaystyle \gamma _{A}} is homologous to A.
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