We speak to the eminent researcher after whom the ‘Ryu-Takayanagi formula’ is named.
BY DEBDUTTA PAUL
Tadashi Takayanagi is a professor at Yukawa Institute for Theoretical Physics, Kyoto University. After completing his Ph.D. at Tokyo University in 2002, he worked for four years as a postdoc at Harvard and Kavli Institute for Theoretical Physics (KITP). He was an assistant professor at Kyoto University and an associate professor at Kavli IPMU, Tokyo University before he moved to his current position in 2012. He was awarded the Yukawa-Kimura Prize in 2011, the Nishinomiya-Yukawa Memorial Prize in 2013, the New Horizons in Physics Prize in 2014, and the Nishina Memorial Prize in 2016. In August this year, Takayanagi was awarded the ICTP Dirac Medal. Takayanagi’s main research contributions have been studies of quantum entanglement by using holography and quantum field theories. In particular, he discovered a holographic formula for entanglement entropy in 2006 with Shinsei Ryu. He has contributed to developing this idea to relate gravitational spacetimes and quantum information from various methods. Professor Takayanagi (TT) spoke to Debdutta Paul (DP) during his visit to ICTS-TIFR for the Quantum Information, Quantum Field Theory and Gravity program.
The full text of the interview is reproduced below. The answers are lightly edited, and long paragraphs have been split for readability. The questions and initials are in bold.
DP: Professor Takayanagi, welcome to ICTS. How do you explain to a high school student interested in physics and mathematics about your field of research?
TT: We are interested in understanding the origin of the universe. We expect that the currently very large universe originally started from a very tiny universe, which we call the Planck-scale universe, which is a microscopic universe. To understand the laws of physics that can describe this microscopic universe, we have to think very much, and from our current knowledge of physics, we cannot answer this question: How did the universe begin? We need a theory of quantum gravity. ‘Quantum’ means a theory that has a microscopic degree of freedom. We want to understand quantum gravity, and that’s the goal of my research. One key idea to understand quantum gravity is actually related to a completely different subject, which is called quantum information. This connection between quantum information and quantum gravity is actually the main topic of my research, and I made important contributions to that connection.
Let me talk about quantum information. It is microscopic information. We can talk about just normal information — like maybe you can imagine some coin. It has a head or tail. Mathematically, we can describe the head as a zero (0) and the tail as a one (1). So, it is 0 or 1, and we can deal with such binary information in computers. We use classical computers — the usual computers. But, if we go to the quantum world, and computers become something called quantum computers, they contain microscopic information. It’s more complicated. So, we can have such a 0 situation or a 1 situation — head or tail — but there are also mixtures of that. That happens in the quantum world. We call this the linear superposition of quantum states, which arises from the fundamental property in the quantum world, so-called ‘wave-particle duality’, and that’s based on the idea of quantum mechanics. So, how information exists is complicated. Such a theory is called quantum information theory, which is a very deep theory, and we can treat how much information we can extract from some complicated quantum system and how we can use that information to do something interesting like teleportation or so on. But interestingly, this quantum information theory is directly related to quantum gravity.
In gravity, we have a universe, and we can talk about its size, especially when thinking of some area of some surface. If we have a very large universe, then its surface area becomes very large. But if the universe is very small, the area becomes very small. My colleague Shinsei Ryu and I contributed to this subject. We found the formula that relates this area of the universe to the amount of quantum information. More precisely, by quantum information, I mean some correlation between two bodies, like there are two coins: one coin and a second coin, and if one coin is a head, then the second coin is also a head; if one coin is a tail, then the other coin is also a tail. So, there are correlations called ‘quantum entanglement’, and we showed that the amount of quantum entanglement is proportional to the area of the universe.
DP: You’re talking about the ‘Ryu-Takayanagi formula’, right?
TT: Exactly!
DP: Can you explain it a bit and how it’s applied to black holes?
TT: Yes, first of all, this formula is based on the idea called ‘holography’, the most important example of the holographic principle, called the ‘AdS/CFT correspondence’, found by Juan Maldacena in 1997, and this idea is quite surprising. If you have some quantum system, maybe you can think about some metals or semiconductors or maybe an insulator. Let’s assume such a theory is a very highly interacting, very complicated theory. Such a theory, actually, can be equivalently described by gravity on some manifold, some spacetime called ‘Anti-de Sitter space’ (AdS): that’s a special universe which is a little bit different from our universe. Some curved spacetime, okay? And if we think of gravity in such a spacetime, many things like energy, entropy, etc., agree with each other. Using that framework, we did the calculation of the amount of quantum entanglement. For example, if we divide a system into two parts, like separated systems A and B, how are A and B correlated with each other? This correlation is called ‘quantum entanglement’. Such a quantity, which characterises the amount of entanglement, is called ‘entanglement entropy’. We showed that entanglement entropy is equal to just the surface area in the Anti-de Sitter universe. More precisely, it is the area of the surface which surrounds the subsystem A and which has the minimal area. So this equation relates geometry — spacetime geometry — to quantum information. So, if we have a lot of quantum information, then our universe becomes very large, but if you have a very small universe, just one or two bits of information, the universe is very microscopic.
Since you asked about black holes, there is a very interesting, recent development. In black holes, there is a very famous paradox. We call them black holes because they are such heavy objects that anything will be strongly attracted, and even light rays cannot come out of the black hole region. We cannot see the light from the black hole, so it looks black. But actually, Hawking showed that a black hole is not exactly black because it has a temperature. So, it is a little bit of a hot object, and it emits radiation. So, if you have some iron and if you put it on the stove, it looks red and emits radiation. Similarly, black holes emit some radiation, and light rays come out of black holes. A black hole is originally a very heavy object but loses its mass because energy comes out of the black hole. And in the end, the black hole disappears. This process is called ‘black hole evaporation’, but then, there is a basic question, a paradox. In the black hole, we have a lot of information. We can form a black hole from a very heavy star. If it becomes very compact, it becomes a black hole. Originally, the star had a lot of information, and the observer could see inside the star. So, originally, there was a lot of information included in the black hole. Such information inside the black hole is called ‘black hole entropy’. But, this entropy, in the end, disappears because the black disappears and just turns into radiation. From radiation, we expect that we cannot get any information — so information included originally in the black hole disappears. In quantum theory, information should not be lost. So, the black hole evaporation contradicts the basic theory of physics. And this is called the ‘black hole information paradox’.
Owing to recent progress, by generalising our formula to the case where we can have gravity, which describes black holes and also radiation, we can explicitly show from the calculation of entanglement entropy that, actually, the black hole will not lose the information inside it. That’s one of the latest, very big progress. This was discovered in the paper by Penington and the other paper by Almheiri, Engelhardt, Marolf, and Maxfield.
DP: And is this applicable to our universe where we actually see black holes? Can we test this theory?
TT: Yes, that is a very interesting thing. So, in our universe, we have many black holes, and each black hole has some temperature and emits radiation. But, it’s probably not so easy to collect all the radiation practically. Still, if we can do this using some satellite or such, I mean, by observation, then we may, in principle, confirm that we can reproduce information inside the black hole from radiation. The recent progress on quantum entanglement and radiation which I explained just before shows we can, in principle, reproduce information if we really collect all the radiation. But even if we collect all the radiation, how we can reproduce the original state requires a very highly complicated procedure. In principle, it is possible, but even if we use a full-fledged quantum computer, which is still not available at present. Such a reconstruction of the state does not seem to be possible within a reasonable time, unfortunately. We know this from many discussions, but anyway, the point is that there is no paradox. So, it does not contradict any basic laws in physics. That is a very important point, and it has actually come out of recent progress in this field.
DP: Okay, so, you have been able to solve the problem.
TT: Yes, at least in one of the frameworks of the black hole information problem. By computing entropy, there are no contradictions at all. I’m not saying that we completely resolved the problem, but at least one crucial problem was solved.
DP: So today, what is the fundamental question you are trying to solve?
TT: I mentioned the calculation of entropy, but more than that, we really need to reproduce the original quantum states. That is a very non-trivial point. The calculation of entropy can be done by generalising our formula. It’s a formula, called the ‘Island formula’, by the scientists in 2019, and using that formula, we can explicitly confirm that information inside the black hole can be reproduced from the radiation, in principle. But, if we want to ask how we can really reproduce the information, it’s a difficult question, and many researchers are working on this topic.
DP: And this is one of the open questions?
TT: Yes, it’s one of the open questions.
DP: Is there an application of your research to condensed matter physics?
TT: Yes! In condensed matter physics, characterising some complicated quantum states with complicated interactions is very difficult. So, people usually use a numerical approach using some supercomputer to simulate quantum states under some approximations. In that context, entanglement entropy is relatively easy to calculate. It’s a quantity that they can compute first. So, the knowledge of quantum entanglement is quite useful. Using our gravity formula, we can calculate this quantity, entanglement entropy, in many different quantum states and can compare that with condensed matter systems. That actually helps some analysis of condensed matter physics. The gravity models correspond to very highly interacting condensed matter systems using the holographic relation. Such a condensed matter system is very difficult to analyse using a direct approach — we need highly powerful supercomputers. But on the gravity side, we can calculate it by just using tabletop computers, while sometimes it’s just analytical calculations. So, many difficult problems turn out to be very simple problems. An explicit example which I have worked on before is something called ‘strange metals’. It is a metallic state, so there is conductivity, but it’s a very complicated state; it’s highly interacting, and because of that, it has properties different from those of usual metals. Such a metal also has anomalous or unusual behaviours, such as in its specific heat. But how much unusual? This is a very interesting question, and by the holographic and our formula of entanglement entropy, we can make some predictions about this problem.
DP: So, even though gravity is not important in these systems, the calculations carry over?
TT: Yes! Originally, there is no gravity at all — it is a metallic system. But we can go to a higher dimension, use an equivalent description, and gravity appears. Using this gravity picture, we can compute entropy in a very geometric way. So, originally, the quantum system is described by something called the quantum wavefunction, which is very complicated algebraically if we want to calculate the entanglement entropy. But if we look at this geometric picture, everything is just geometry, and it’s much easier. So, the algebraic problem becomes a geometric problem. It is actually a miraculous aspect of holography and the holographic entanglement.
DP: This is the AdS/CFT correspondence?
TT: Yes, exactly.
DP: Right, but our world is a de Sitter (dS) space?
TT: Yes, our world has a positive cosmological constant, so we cannot directly apply this AdS/CFT correspondence, but there is an idea called dS/CFT correspondence, first proposed by Andrew Strominger in 2001. Again, we consider de Sitter (dS) space, which is equivalent to some quantum matter but is now very non-trivial and exotic. In dS/CFT, the dS space is supposed to be equivalent to some matter situated at future infinity — this is another example of holographic principle, which means that some bulk region is equivalent to some boundary region. In the AdS space, the boundary includes the time direction. But in dS space, the boundary is at a future infinity. So, there is no time. In summary, de Sitter space gravity is somehow expected to be equivalent to some matter without the time direction. This is a very difficult situation because we need to understand why time coordinate emerges from something without it. Though dS/CFT is not well-explored, there are actually quite a few people who are interested in it nowadays. I am also working on that, and I hope there will be major developments in the near future.
DP: We’d love to know about your journey, starting from how you came into it to your present adventures.
TT: When I was a graduate student, I was in a string theory group at the University of Tokyo, but most people were interested in quantum field theories, especially conformal field theories (CFT). So, I was working in that direction. But during that time, the pioneering papers on AdS/CFT correspondence appeared. So, I gradually became interested in the AdS/CFT correspondence, but during my Ph.D time, I was not working well on this topic as few people around me working on it at that time. Instead, I focused more on the analysis of dynamical phenomena in string theory, such as tachyon condensation. Then I moved to the US. I was a postdoc at Harvard, where most people were quite interested in AdS/CFT or black holes. Stimulated by this, I also got interested in such gravitational aspects related to holography, so I decided to work on the AdS/CFT correspondence, and especially when I next moved to KITP, Santa Barbara as a postdoc, I met with my condensed matter colleague, Shinsei Ryu. We talked about the interdisciplinary connection between condensed matter and string theory, and then we realised that quantum entanglement would be a key concept. So, we started to discuss it. Holography is a very attractive idea in string theory. In condensed matter physics, at that time, entanglement was a very hot topic, and remarkable progress had been made. So, we decided to combine these two ideas. That’s the reason we ended up with the formula.
DP: Professor Takayanagi, thank you so much for your time.
The author thanks Spenta Wadia for inputs.
This is an excellent blogpost! With very few questions you got a beautiful summary of AdS/CFT, quantum entanglement, and how complex algebraic aspects can at least be closely guessed in another dimension using geometric aspects of gravity. Thanks!