The Ryu–Takayanagi conjecture is a conjecture within holography that posits a quantitative relationship between the entanglement entropy of a conformal field theory and the geometry of an associated anti-de Sitter spacetime.
[1][2] 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".
[3] The conjecture is named after Shinsei Ryu and Tadashi Takayanagi, who jointly published the result in 2006.
Specifically, the Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area: The Bekenstein–Hawking entropy
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).
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.
The holographic principle states that gravitational theories in a given dimension are dual to a gauge theory in one lower dimension.
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.
dimensional Minkowski space the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole.
[8] Important for the present discussion is that the thermal state of a CFT defined on the
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.
[8] The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law.
of an AdS space time on whose boundary we define the dual CFT.
is the entanglement entropy of the CFT in some spatial sub-region
Furthermore, property (1) ensures that the formula preserves certain features of entanglement entropy, such as
[clarification needed] The conjecture provides an explicit geometric interpretation of the entanglement entropy of the boundary CFT, namely as the area of a surface in the bulk.
In their original paper, Ryu and Takayanagi show this result explicitly for an example in
is analogous to the corresponding CFT having a UV cutoff.
is the length of the CFT system, in this case the circumference of the cylinder calculated with the appropriate metric, and
In this case, the boundary CFT lives at coordinates
The minimal surface is easy to identify in this case, as it is just the geodesic through the bulk that connects
Remembering the lattice cutoff, the length of the geodesic can be calculated as If it is assumed that
Plugging in the length of the minimal surface calculated in (3) and recalling the central charge (2), the entanglement entropy is given by This agrees with the result calculated by usual means.