Tadashi Takayanagi

While the conventional gauge/gravity correspondence is applicable only for a universe with a negative cosmological constant, we have found a gauge/gravity correspondence, with a positive cosmological constant like the real universe for 3-dimensional spacetime. By introducing a new quantum information quantity called "pseudo-entropy," we succeeded in describing gravitational spaces, which depend on imaginary time in the gauge-gravity correspondence, from the viewpoint of quantum information. Furthermore, we discovered "wedge holography," in which a gravitational theory placed in a wedge-shaped universe corresponds to a quantum field theory localized at its tip. Using related methods, we constructed a model describing the evaporating black holes and derived the Page curve which implies that the evaporation process is a unitary.

Below we describe the following four highlights of my research achievements so far: (1) Gauge/gravity correspondence of the three dimensional de Sitter universe, (2) Gauge/gravity correspondence and pseudo-entropy, (3) Wedge holography, and (4) Moving mirror models and Page curves.

(1) Gauge gravity correspondence in a 3-D dodged-jitter universe
The gauge/gravity correspondence, also called the AdS/CFT correspondence, is the correspondence between the d+1-dimensional anti-de Sitter (AdS) universe and the d-dimensional conformal field theory (a special case of quantum field theory with scale symmetry, called CFT). This makes it possible to understand the problem of quantum gravity theory, a difficult problem in theoretical physics, by connecting it to a theory of quantum many-body systems that does not include gravity. However, the anti-de Sitter universe has a negative cosmological constant, while the real universe is considered to be a de Sitter universe with a positive value based on observations and other factors. Therefore, the generalization of the gauge-gravity correspondence to the de Sitter universe (called dS/CFT correspondence) is an extremely important issue. Till now, little progress has been made because the corresponding conformal field theory is not known. However, in this work, we have identified a conformal field theory that is equivalent to a 3-dimensional de Sitter universe, by utilizing the correspondence between 3-dimensional gravity theory and Chern-Simons gauge theory, and the relationship between Chern-Simons theory and a conformal field theory called the 2-dimensional WZW model. This is given by the conformal field theory obtained by taking the parameter called the level of the SU(2)WZW model as a complex number, and it is confirmed that the partition function of the 3-dimensional de Sitter universe predicted from the conformal field theory and that from the gravity theory perfectly agree. This result was published in Physical Review Letter and was selected as Editor's Suggestion. In addition, it was featured in a Viewpoint article in Physics, the outreach journal of the American Physical Society.

(2) Gage-Gravity Correspondence and Pseudo-Entropy
　In the gauge-gravity correspondence, the entanglement entropy of conformal field theories is calculated by the entanglement entropy of conformal field theories. As discovered by this fellow in 2006. This idea has been a recent trend in string theory. In collaboration with Yoshifumi Nakata, a researcher of quantum information theory, and others, the Fellow has found that a generalization of entanglement entropy, called "pseudo-entropy," is equal to the area of the minimal surface in a more general universe that incorporates the time evolution of imaginary time. We also gave an operational interpretation that for a certain class of quantum states, the pseudo-entropy is equal to the amount of entanglement entropy that can be extracted from the intermediate states between the initial and final states. In a subsequent study, we found that pseudo-entropy acts as an order parameter in classifying the quantum phases of the quantum Ising model, and his results were published in Physical Review Letter.

(3) Wedge holography
　The gauge-gravity correspondence is a correspondence in which the gravity theory in the anti-de Sitter universe is equivalent to a conformal field theory in a space-time space one dimension lower than the anti-de Sitter universe. The fellow found that the gravity theory in the wedge-shaped region is equivalent to the conformal field theory in a two-dimensional lower spacetime corresponding to the tip of the wedge. Free energy, correlation functions, entanglement, and entropy are shown to coincide with each other in this correspondence. This result was published in Physical Review D and was selected as Editor's suggestion.

(4) Moving mirror model and page curves
　The moving mirror model is a model of quantum field theory with a time-dependent boundary, and it is known that Hawking radiation from a black hole can be described if the mirror trajectory is chosen appropriately. When a black hole evaporates with Hawking radiation, it becomes seemingly mixed after evaporation, and in the time evolution, the unitarity, which is the basis of quantum theory, appears to be violated. This is the information loss problem of black holes. To avoid this problem, the entanglement entropy of radiation must follow a special profile called the Page curve. We analyzed a two-dimensional moving mirror model using conformal field theory, and succeeded to calculate the time evolution of entanglement entropy analytically. Using this method, we analytically showed the time evolution of entanglement entropy follows an ideal Page curve. Although this model look like a quantum field theory without gravity, it can be regarded as a system where a conformal field theory interacts with an evaporating black hole by partially using the gauge/gravity correspondence. Therefore, the above result provides evidence that unitarity cannot be violated by the evaporation of a black hole. This work was published in Physical Review Letter.

One of the primary concerns of theoretical physics is to gain complete understanding of the physical principles of gravity at micro-level, that is, to elucidate the quantum gravity theory. A holographic derivation of entanglement entropy published in 2006 by Dr. Tadashi Takayanagi and Dr. Shinsei Ryu (Ryu–Takayanagi formula) has been one of the most important breakthrough discoveries since Dr. Juan Maldacena proposed AdS/CFT (anti-de-Sitter/conformal field theory) correspondence in 1997. So significant was their derivation that it changed the whole surface of the research trends in this field. In an interview conducted after receiving the 2014 Kyoto Prize, Dr. Edward Witten cited the Ryu–Takayanagi formula as the first on the list of “highlights in the past fourteen years.” Owing to its microscopic explanations based on AdS/CFT correspondence, the Ryu–Takayanagi formula has secured a critical position in theoretical physics.
 
For 14 years since announcing it, Dr. Takayanagi has continuously evolved the Ryu–Takayanagi formula to make leading contributions to the theoretical elucidation of the holographic principle and its application. Dr. Takayanagi and Dr. Matthew Headrick, for example, discovered that a strongly subadditive inequality of von Neumann entropy corresponds to a triangle inequality of the minimum surface, a geometric characteristic of the Ryu–Takayanagi formula. Revealing that the profound nature of entropy is reflected in a geometric structure of gravity theory, this discovery has significantly catalyzed the subsequent development in this field. Furthermore, over the past several years, it has become clear that quantum error correction codes of the quantum information theory are playing important roles in the microscopic mechanism of the holographic principle. The Ryu–Takayanagi formula plays an essential role in the elucidation of this mechanism as well.
 
The research topic that Dr. Takayanagi has proposed aims to further advance the novel idea of a close relationship between “quantum information” and the “quantum gravity theory,” which has become widespread after the publication of the Ryu–Takayanagi formula, and to bring about a breakthrough in the elucidation of the quantum gravity theory, understanding of which is still in the early stages. Some of the relevant and recent research findings by Dr. Takayanagi include: “entanglement of purification” (an amount of entanglement entropy extended to its mixed state) is equal to the cross-section of a certain wormhole in AdS/CFT correspondence, and that, when calculating the path integral of the conformal field theory by discretizing it, geometric features of anti-de Sitter space (AdS) appear by themselves if the most efficient discretization process is required. It is expected that a new phase of research will be developed based on these findings.
 
The series of research efforts by Dr. Takayanagi, which have resulted in the discovery of an entanglement entropy formula in the holographic principle and its subsequent progress, have marked fundamental achievements in the quantum gravity theory and superstring theory. His research also facilitates collaboration between the condensed matter theory and quantum information theory. In addition to these impressive fruits of research, Dr. Takayanagi is an excellent academic advisor, under whose tutelage numerous young researchers have been produced. Dr. Takayanagi is a world-leading researcher of superstring theory and, with support from the InaRIS Fellowship Program, is expected to be a driving force for theoretical physics research in Japan over the next decade.