**Written by Dr. Ian T. Jardine in collaboration with Dr. A.W. Peet**

One of the key developments in theoretical physics in the last few decades was the discovery of the AdS/CFT correspondence. It says that quantum gravity in an asymptotically anti de Sitter (AdS) spacetime bulk is physically equivalent to a conformal field theory (CFT) on a spacetime conformal to boundary of the AdS, which has one fewer dimensions than the bulk. Anti de Sitter spacetime is a maximally symmetric solution to Einstein's equations with negative cosmological constant, and a conformal field theory is a quantum field theory with conformal symmetry. AdS/CFT is a very powerful type of duality because it trades strong coupling in AdS with weak coupling in the CFT and vice versa.

AdS/CFT duality holds not only for pure AdS geometries. Geometries can be more complex in the interior, as long as they have an asymptotically AdS region. This means that AdS/CFT duality can apply to asymptotically AdS black hole geometries as well as geometries with some broken symmetries. See the essay on black holes for some other information and background on this correspondence.

One important question is how gravity emerges from the CFT. Gravity and CFT are very different theories and it is not immediately obvious that these two theories are dual to one another. But there is a lot of evidence for AdS/CFT dualities, and so there must be some way to reorganize CFT to see gravity come out. This new organization would then require some additional structure, which we would then interpret as a new dimension in the gravity description. The challenge then is to find the correct way to organize our theory to see this new dimension.

AdS/CFT sees this emergent direction as a spatial direction. It is easier to see this type of emergent dimension, as the interpretation of time is the same in both theories. One may wish to consider a different kind of holography to get an emergent time direction instead. There has been work done on this, by conjecturing that de Sitter (dS) space, a maximally symmetric solution to Einstein's equations with positive cosmological constant, is dual to a CFT. However, there are a number of technical obstructions that make exploring and understanding this duality more difficult, such as the fact that the CFT in this case must be non-unitary.

Knowing how to fully reorganize the CFT to see gravity would mean solving the AdS/CFT correspondence completely. This is a very challenging and ambitious goal. So instead we often try to find easier paths, such as trying to pick out specific aspects of the full gravity theory. One simplifying approach is to focus on gravity in three dimensions, where there are no freely propagating gravity waves in the bulk. This makes it easier to compute quantities on both sides of the duality. Other approaches will focus on just getting out geometry, like points and curves, or fields that propagate on the geometry without worrying about the full quantum gravitational interactions. We can call this bulk reconstruction.

In 2006, Ryu and Takayanagi (RT) proposed a bulk dual for entanglement entropy in the AdS/CFT. Entanglement entropy is a quantity that measures the amount of entanglement between a region of space and its compliment. This definition was extended in 2007 by Hubeny, Rangamani, and Takayanagi to allow time dependence.

The dual is the area a minimal surface which hangs into the bulk which ends on the boundary of the region in question. Think of this like a soap bubble sitting on the ground. In this analogy, the ground would be the boundary of the spacetime, the bulk would be everything above the ground, the minimal surface would be the surface of the bubble and the patch of ground inside the bubble would be the region of the CFT whose entanglement entropy we are computing.

Importantly, this minimal surface is sensitive to the bulk geometry, giving us hope that it can be used to reconstruct the geometry. The connection between entanglement and geometry was put forward as a general feature of holography by van Raamsdonk in 2010. Moreoever, AdS/CFT isn't just true for string theories. The results of Heemskerk, Penedones, Polchinski, and Sully in 2009 showed that any semiclassical geometry should be dual to a state in a CFT with appropriate conditions and vice versa. So this relationship between gravity and entanglement is present in general.

One major concrete realization of this connection occurred in 2013 by the authors Faulkner, Guica, Hartman, Myers, and van Raamsdonk. They were able to use the RT relationship to derive the linearized Einstein equation. That is, one can find the equations for gravity at leading order just by having RT. Although this is not the full equations of gravity, it shows just how powerful the connection between entanglement and geometry truly is.

One could instead ask how much of the bulk can we reconstruct if we do not have the full CFT to work with. Purely from the gravity point of view, the part of the bulk that would be reconstructable would be the areas within causal contact of the region of interest, the causal wedge. However, it has been argued by various papers, beginning with Wall in 2012, that the maximum that can be reconstructed is in fact the entanglement wedge. Roughly, this is a region in the bulk that is bounded by the RT surface.

This entanglement wedge covers a larger volume than the causal wedge. This is a general property of the geometry of negatively curved spaces; there is less "spacetime" away from the boundary. So the RT surface will sit further in, as it is generally minimal. Therefore the causal wedge will be contained within it.

This all sounds very hopeful that we can reconstruct the bulk from entanglement. However, it turns out that there are some problems. One main problem are areas that are behind event horizons, i.e. inside a black hole. They cannot be fully reconstructed. This means that this approach cannot shed light on problems like the black hole information problem. However, there is evidence to suggest that there can even be problem reconstructing regions outside of black holes using minimal surfaces. There were called entanglement shadows by Balasubramanian, Chowdhury, Czech, and de Boer in 2014. This is why holography hasn't been solved yet and cannot be solved by entanglement entropy alone. So what other tools can we use to probe the bulk when entanglement entropy fails?

As it turns out, entanglement entropy is not the only measure of entanglement of a system. The study of quantities and measurements that encode the information about a quantum system is known as quantum information theory. It has many applicable uses outside of holography, most notably applied to quantum computing. A great textbook on this large field can be found in the Further Reading section.

This then begs the question: what are these other quantities and measurements holographically dual to in the bulk? What aspects of physics do they captrue that entanglement entropy cannot? Which set of probes are the best for our bulk reconstruction purposes? All these have partial answers, but there is still a lot to know.

There are many different quantum information measures to discuss. A generalization of the entanglement entropy is the Renyi entropies. These are a series of entropies that, all together, completely characterize a quantum mechanical state. The entanglement entropy turns out to be a particular limit of this series. Another important measure is the relative entropy, which will encode the differences between two states. This can be used in conjunction with mutual information, which encodes the shared information between two states. There are of course many, many more. The main point here is that quantum information theory gives us a large toolbox to use for characterizing states.

We can now ask how these quantum information quantities can be used in our holographic correspondence. As it turns out, the Renyi entropies also obey an area law. Additionally, the Renyi entropies can be computed using a method called the replica trick. Taking the limit of this carefully also gives a calculational tool for the entanglement entropy.

Relative entropy has many uses in the bulk. It can be used measure the distance between two states, having an important monotonicity property. It was used by Blanco, Casini, Hung, and Myers in 2013 to define relationship between the change of entanglement entropy and the change of the modular Hamiltonian (a quantity derived from the state itself), called the entanglement first law. This is in analogy to the thermodynamic relationship between entropy and energy and it has been used to derive a number of results. This includes the Faulkner et. al. result mentioned earlier and many different results for gravitational energy bounds.

Another important aspect to quantum information theory is the setup of quantum error correction. This setup of a quantum system sees encoding information with extra, redundant information so that the original information can still be recovered even if part of the system is lost. This found itself in holography, first discussed by Almheiri, Dong, and Harlow in 2014. They showed that the symmetry of representing bulk information in the CFT over different regions was manifestation of quantum error correction.

Another quantity that shows up in quantum information theory is the notion of complexity of a state. This is the number of fundamental quantum transformations (unitary gates) that are needed to bring some reference state to the state in question. This quantity has been explored in the holographic context, first with Susskind in 2014, where it was originally seen as a volume of a section of the bulk. Since then, there is has since been a number of alternate suggestions for the true bulk dual and some discussion of whether this quantity is well defined in the CFT.

One way to explore questions like this is to consider a specific model where computations can be done. One set of toy models for holography has been tensor networks, the first connection was made by Swingle in 2009. As the name implies, one connects a number of tensors with a set number of legs together, keeping at least one leg out for the bulk. The remaining, uncontracted legs are then the boundary. Exactly what these tensors are and how many legs they have will depend on the model in question. These have seen use in exploring complexity, quantum error correction, and entanglement entropy for holography. However, these do have the drawback of being intrinsically discrete, which can be a benefit for studying the intrinsically discrete information quantities like complexity, but obviously is very different from a continuous theory like conformal field theories.

All of this should lead to a picture that quantum information has an important place in the AdS/CFT correspondance. In fact, the study of holography has lead to a number of novel results for information theoretic quantities. For example, the mutual information for holographic states was shown to be monogamous in 2011 by Hayden, Headrick, and Maloney. These results further suggest that the entanglement structure of CFTs is where gravity is hidden.

If it is the entanglement that seems to hold gravity, the natural question to ask is if there is a way to reorganize the CFT in terms of its entanglement? One proposed space that reflects this reorganization is called kinematic space, first discussed by Czech, Lamprou, McCandlish, and Sully in 2015. This is the space of geodesics, which corresponds to the minimal surfaces that are dual to the entanglement entropy of the CFT in three dimensions.

We can go a bit further than just matching geodesics to points in the kinematic space. Any curve in a geometry can be cut into bits by geodesics and use these to reconstruct curves and surfaces in the bulk. This is similar to how we split segments of a curve into infinitesimal pieces for differential geometry, so similarly previous method is called integral geometry. Additionally, we can identify diamond shaped regions in kinematic space to conditional mutual information (another quantum information quantity) in the CFT. This provides a definition of causality for kinematic space as well. So kinematic space is the natural setting to view how the entanglement and geometry influence one another.

If kinematic space is a spacetime, one could wonder what it looks like. This is a very tricky question. When we consider pure AdS, that is AdS that does not have any heavy fields in it, and consider only a spatial slice of pure AdS, then kinematic space is de Sitter or dS. This holds for any number of spatial dimensions. In the case of 2+1 dimensions we can include time back in and find the full kinematic space to be dS${}_2\times$dS${}_2$.

One could wonder about geometries that are not pure AdS. Less is known about more general geometries. One direction of interest to us has been using kinematic space to explore more complicated geometries, such as black holes or conical defects. These geometries are not as symmetric as the pure AdS and so determining the geometry of the kinematic space is difficult. We are hopeful however that kinematic space will help elucidate aspects of the deep connections between entanglement and geometry.

Quantum Computation and Quantum Information

by Michael Nielsen and Isaac Chuang, one of the standard introductory quantum information textbook approachable by seniour undergraduatesA First Course in String Theory

by Barton Zwiebach, a textbook introducing string theory to well prepared senior undergraduatesIntroduction to the AdS/CFT Correspondence

by Horatiu Nastase, a technical textbook introducing AdS/CFT to high-energy theory graduate studentsLectures on Gravity and Entanglement

by Mark van Raamsdonk, a review article that for graduate students in the field