Emergence of Spacetime from Quantum Entanglement: The Impact of the Ryu-Takayanagi Formula

In the study of quantum gravity theory, several notable achievements are known. One of these is a discovery made by two Japanese researchers, condensed matter theorist Shinsei Ryu and superstring theorist Tadashi Takayanagi, who were both postdocs at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, in 2006. This achievement, now known as the "Ryu-Takayanagi formula," subsequently garnered interest from many researchers around the world and gave rise to a major field of study. It is a research trend based on the idea of "It From Qbit" and "all things are quantum information," where quantum entanglement gives rise to spacetime and matter.

The work of Ryu and Takayanagi is based on the duality of superstring theory known as the AdS/CFT correspondence. This is a theory published in 1997 by a person named Juan Maldacena. Here, AdS refers to anti-de Sitter spacetime, and it points to its quantum gravity theory. Also, CFT refers to conformal field theory in a flat spacetime with one fewer spatial dimension. As shown in Figure 1, the AdS/CFT correspondence is the claim that quantum gravity in anti-de Sitter spacetime, which is a curved spacetime, is equivalent to the theory of flat matter fields existing on its infinite spatial boundary.

In this Figure 1, space is inside a 2-dimensional circle, and its boundary is the 1-dimensional circumference at the edge. It is said that two theories with different spacetime dimensions are actually equivalent. If we view the internal 3-dimensional spacetime as being generated from the boundary surface as a 2-dimensional film, we can see that this is a correspondence like a hologram in quantum optics. This is a concrete realization of the "holographic principle" proposed by Gerard 't Hooft and Leonard Susskind before the AdS/CFT correspondence. The entropy of a black hole is not proportional to its volume, but is written as the area of the event horizon divided by four times the gravitational constant G. In the holographic principle, this entropy proportional to the area is captured by the interpretation that the degrees of freedom inside the black hole are replaced by the degrees of freedom on the horizon surface.

Ryu and Takayanagi further deepened this discussion. In the conformal field theory of matter fields living on the infinite boundary, they considered two adjacent regions as shown on the left of Figure 2, and examined the entanglement entropy between them. As a result, they proposed that the entanglement entropy can be written as the area of the minimal surface in the anti-de Sitter space that has that boundary, as shown on the right of Figure 2.

In the following, we adopt the natural unit system where

$$
c=\hbar=k_B=1
$$

At this time, the Ryu-Takayanagi formula becomes

$$
S=\frac{A}{4G}
$$

Here, the left side is the entanglement entropy of the conformal field theory, and the right side is the area A of the minimal surface divided by 4G. It was found that quantum entanglement is described by something called area within spacetime. In other words, it was shown that "quantum entanglement → area as the geometry of spacetime." In particular, the fact that the form of the right side of the formula itself, including the coefficient, matches the form of black hole entropy surprised many physicists.

By the way, before the discovery of the Ryu-Takayanagi formula, in 1995, a researcher named Ted Jacobson had already made the claim that the Einstein equations could be derived from the form of black hole entropy. A local inertial coordinate system exists at every point in any curved spacetime. This means that a nearly flat coordinate system can be taken in the vicinity of each point. Around that point, gravity vanishes due to the equivalence principle. In this sense, Jacobson says, let us consider the vacuum locally in this coordinate system. And if we consider an observer undergoing uniform acceleration within that coordinate system, the zero-point oscillations of the field turn into a heat bath at the temperature T below, which is proportional to the acceleration κ.

$$
T=\frac{\kappa}{2\pi}.
$$

This is called the Unruh effect.

The horizon for an observer accelerating in flat spacetime is called the "Rindler horizon," and this corresponds to a black hole horizon at infinite mass. In such a situation, let us treat the energy-momentum tensor of matter accompanied by a very thin concentration of energy flow perturbatively. And we take a setting where this energy flow is absorbed by the Rindler horizon. At this time, from the corresponding first law of thermodynamics, the change in thermal energy of matter δQ is given by the temperature T of the Unruh heat bath multiplied by the entropy change δS.

$$
\delta Q= T \delta S.
$$

And we can write this δQ using the energy-momentum tensor T of matter and the vector k tangent to the Rindler horizon.

$$
\delta Q \rightarrow T_{\mu\nu} k^\mu k^\nu.
$$

Here, Jacobson identifies this S with the area of the Rindler horizon divided by 4G, similar to black hole entropy, and with the Ricci tensor of spacetime curvature,

$$
\delta S=\frac{1}{4G}\delta A \rightarrow R_{\mu\nu} k^\mu k^\nu
$$

He established such a correspondence. Since the tangent vector to the Rindler horizon in a local inertial frame can be chosen in various ways, for this correspondence to hold even with that degree of freedom, he used a scalar quantity F to state that

$$
R_{\mu\nu} -g_{\mu\nu} F=8π G T_{\mu\nu}
$$

Jacobson argued that this relationship holds. Then, by imposing the conservation law of the covariant energy-momentum tensor of matter, the Einstein equations are obtained.

$$
R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R +\Lambda g_{\mu\nu} =8π G T_{\mu\nu}.
$$

This means that the physical laws of the volume of curved spacetime emerged from area. In other words, Jacobson demonstrated the relationship: "Area as spacetime geometry → Volume of curved spacetime." By combining this with the Ryu-Takayanagi formula's "Quantum entanglement → Area as spacetime geometry," we get "Quantum entanglement → Volume of curved spacetime," which leads to the emergence of spacetime from quantum entanglement. This is called the "emergence of spacetime from quantum entanglement." In short, the Ryu-Takayanagi formula opened the way to viewing spacetime as quantum information. It also gave rise to a major global research trend, currently being actively discussed, that "everything is quantum information." Ryu and Takayanagi have received numerous international awards for this achievement.