FEATURE

Kavli IPMU Principal Investigator

Yasunori Nomura
Research Area： Theoretical Physics

Quantum Gravity and Quantum Information

Understanding the nature of space and time hasbrought many revolutions to physics. In old days,
the introduction of a coordinate system which isoften attributed to Descartes led to Newtonianmechanics through contributions by many scientists,
including Galileo. At the beginning of the 20th
century, Albert Einstein proposed special relativity,
which, in a sense, unifies space and time. According
to this theory, space and time are closely related,causing phenomena such as a contraction oflength and a delay of time when an object moves
at a speed close to that of light. These effects are
precisely measured in present-day experiments,
leaving no room to doubt the theory. The resulting
concept of spacetime, which treats space and timein a unified manner, has been playing a fundamentalrole in contemporary physics until today.
One of the most important properties ofspacetime was revealed by Einstein himself about 10
years after the announcement of special relativity.
According to this new theory called generalrelativity, spacetime is not a fixed entity, as one
might imagine from the term coordinate system ;rather, it is a dynamical object that can be bent
or distorted. General relativity says that it is thesebending and distortion of spacetime that we callgravity. For example, gravity between the Sun andthe Earth arises because the existence of the Sun
bends spacetime around it. The Earth moves around4

the Sun because of this bending of spacetime.
This revolutionary theory of spacetime makes
an important prediction: because spacetime
is dynamical, its distortions can propagate
independently of the existence of matter. For
example, when heavy bodies such as black holes
move violently, they disturb spacetime greatly.
This effect can then propagate to places far
away from the location of the original bodies aswave of spacetime distortions. This wave is called
gravitational wave, and it was first detected directly
in 2016, almost 100 years after the formulation ofgeneral relativity. This was an extremely important
experimental discovery, and for this achievement the2017 Nobel Prize in Physics was awarded.

Theory of Quantum Gravity
As we have seen, our understanding of spacetime
has dramatically evolved over the last few hundredyears. However, we know that this is not the end.
The reason is that our world obeys quantum
mechanics.
Quantum mechanics was discovered at the
beginning of the 20th century in the process ofstudying microscopic objects such as atoms, butlater it was found to be a more fundamental
principle governing the world. Quantum mechanics
makes a number of strange predictions. For

Kavli IPMU News No. 43 September 2018

example, an object such as an electron cannot
exist in one place; rather, its location spreads
probabilistically. (Since this spread is usually so smallthat we cannot perceive it directly in our daily life.)
It is also possible that two worlds exist parallelly inprobability space, the state we call a superposition.
While these phenomena sound extremely strange
from the sense of everyday life, the correctnessof the theory has been repeatedly demonstrated
experimentally. In fact, the accuracy of theseexperiments has dramatically improved in recent
years, so that predictions of quantum mechanics
have now been confirmed precisely at a detailed
level. Furthermore, scientists are advancing thedevelopment of a computer̶ quantum computer̶
which has capability completely different from theconventional one and which positively utilizes theprinciple of quantum mechanics.
Let us return the story to spacetime. Generalrelativity explains all gravitational phenomenathat have been observed until now. This includes
minuscule deviations from the predictions of
Newtonian gravity, the existence of gravitationalwave, and the expansion of the universe. Thereis almost no doubt about the correctness of thetheory at scales we can currently observe. Generalrelativity, however, does not contain the effect ofquantum mechanics, which is a basic principle of
the world we live in. This is not an experimentallyurgent problem. The scale at which the quantum
effect of spacetime becomes important directly canbe estimated theoretically, which is about 10 -33 cm,
the distance called the Planck length. This is about
16 orders of magnitude smaller than the smallest
length probed so far by the worldʼs most powerfulaccelerator. Does this mean that it is meaningless toinvestigate the quantum effect on spacetime?
There are a number of reasons, at least
theoretically, why this is not the case. One is the
problem of infinity. Quantum mechanics and generalrelativity do not get along very well. Assumingthat the theory of gravity at our scales is generalrelativity, its simple extrapolation below the Planck

length ̶where quantum effects on spacetime andgravity become important̶ leads to an innumerable
amount of infinities, destroying the predictivityof the theory. This means that in order to obtaina complete theory including quantum effects on
spacetime and gravity, it is not enough to simply
consider general relativity and quantum mechanics
together. The currently most promising theory tosolve this problem is string theory, whose structure
and consequences are vigorously studied by many
researchers.
Another reason to study quantum mechanicaltheory of spacetime and gravity̶ commonlyreferred to as quantum gravity̶ is that it brings
about a revolutionary picture of spacetime. As wewill see in this article, it is not true that the effectof quantum gravity becomes important only at the
Planck length; on the contrary, it has increasingly
become clear that it also plays a decisive role inphysics at long distances such as the physics ofblack holes and multiverse cosmology. Moreover,this relatively recent progress is creating a newresearch area that unifies two fields of quantum
gravity and quantum information sciences, whichhave been developed separately so far. Below I will
give a brief overview of this new development.

Quantum Mechanics of Spacetime andthe Holographic Principle
The beginning was the physics of black holes. In
a theory with gravity, when the density of matterbecomes larger than a certain limit, it produces aregion around it from which nothing can escapedue to its strong gravity. The boundary of thisregion is called a horizon, and the inside is a blackhole.
Black holes had been known to have thefollowing intriguing property: in general relativity,
the total area of horizons does not decrease. For
example, when a black hole swallows an object, its
area increases. (Here and below, when we refer tothe area of a black hole, it means the area of thehorizon of the black hole.) Also, when two black

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A2
A3
A1
A3 ≥ A1 + A2
Figure 1: The area, A3, of the black hole formed after the merging of two black holes isequal to or larger than the sum of the areas, A1 and A2, of the original black holes.

holes merge into one ̶ this phenomenon was
actually observed by the detection of gravity wave
in 2016!̶the area of the produced black hole isequal to or larger than the sum of the areas of theinitial two black holes; see Figure 1.
The fundamental importance of this fact was
realized by Jacob Bekenstein in his revolutionary
paper in 1973. Bekenstein asked what happens tophysical quantities when an object is thrown intoa black hole. The answer is clear for energy. Theenergy of the thrown object becomes (a part of)
the mass of the black hole, so that the total energy
is conserved. But what about entropy? According
to the second law of thermodynamics, entropy
should not decrease. However, the entropy of thethrown object seems to simply disappear when it isabsorbed into the black hole. Does this mean thatthe second law of thermodynamics is violated?
What Bekenstein suggested in his 1973 paper isthat the entropy of the thrown object is converted
into the entropy of the black hole. What is theentropy of the black hole? He proposed that thearea of the horizon is the one. Recall that the
total sum of horizon areas does not decrease ingeneral relativity. This is exactly the property thatentropy must have. Indeed, by defining the entropyof a black hole in this way, it can be shown thatthe entropy of the black hole after the object is

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absorbed is always equal to or larger than the sumof the entropies of the initial black hole and thethrown object.
The idea that a black hole has entropy was
initially received with skepticism. If a black holehas entropy, the first law of thermodynamics
requires it to also have temperature. However,
if it has temperature, then it must be radiating.
Isnʼt it the case that nothing can come out of a
black hole? This apparent contradiction, however,
was resolved in 1974 when Stephen Hawking
performed a calculation which included the effectof quantum mechanics into general relativity. While
his calculation was not based on a complete theoryof quantum gravity, he could still show that a blackhole indeed emits radiation because of quantum
effects. This established the thermodynamics ofblack holes.
A strange thing here is that the entropy of a blackhole is given by its area. Since a black hole is the
final state of the evolution of any initial state, theentropy of the black hole must be indicating thelargest possible entropy that the region can have. In
modern days, we know that entropy is given by thelogarithm of the number of possible quantum statesthat the system can take. If space is composed ofsome simple constituents as in ordinary materials(e.g., if space can be approximated by a lattice with

Kavli IPMU News No. 43 September 2018a spacing of order the Planck length), then thelargest entropy of a region must be proportional
to its volume. However, the discovery of Bekensteinand Hawking says that it is proportional to thesurface area.
This implies that in a quantum theory with gravity,
the number of fundamental degrees of freedom isgiven by that of spacetime with one dimension lessthan that of the original, dynamical spacetime. For
example, while spacetime we live in seems to havethree spatial dimensions (ignoring possible small
extra dimensions), the true theory describing it
must be formulated in spacetime with two spatial
dimensions and one time dimension. This is possible
because if you try to fill matter at each point inspace (e.g., at each site of the Planck-size lattice),
black holes form long before it is completely filled,
and putting further matter only increases the sizeof the black holes. Namely, it is merely a fictionthat the space we live in has full three dimensional
degrees of freedom.
The idea that the theory of quantum gravity is
formulated in spacetime that has lower dimensions
than the apparent, dynamical spacetime is called theholographic principle. The lower dimensional theory
formulated in this way̶ called the holographic
theory̶ does not have the discrepancy between thenumber of true and apparent degrees of freedom,
so it does not have gravity: the dynamical spacetime
in higher dimensions and its associated gravity are
only emergent. The fact that the holographic theory
does not contain gravity at the fundamental levelmeans that it can be treated as a regular quantum
system, and hence can give a rigorous definition ofquantum gravity. This seems to be a pretty crazyconclusion. It would be natural if one cannot believe
such a thing.

AdS/CFT Correspondence
However, it has been shown ̶though only insome special cases ̶that quantum gravity indeed
satisfies the holographic principle! This discovery

was made by studying structures of string theory.
In 1997, Juan Maldacena proposed, based on
numerous evidences, that quantum gravity
describing physics in spacetimes that asymptoticallyapproach a certain space called Anti-de Sitter(AdS) space is equivalent to conformal field theory(CFT) formulated in non-gravitational spacetimethat has dimensions one less than the gravitational
asymptotically AdS space. This relationship is called
the AdS/CFT correspondence.
The AdS/CFT correspondence is an extremely
powerful mechanism despite the fact that it applies
only to special spacetimes. First, CFT is a classof quantum field theory and is mathematicallywell defined. This implies that quantum gravity inspacetimes that are not so distant from the one welive in (asymptotically AdS space) is defined for the
first time without relying on perturbation theory.
Also, many theories describing the nature (e.g.,
QCD describing nuclear force and theories used incondensed matter physics) are well approximatedby CFT at strong coupling. In general, it is extremely
difficult to solve such a theory with strong coupling,
but by the AdS/CFT correspondence we can solve it
approximately using general relativity in one higher
dimensions. And above all, this correspondence
gives a concrete example of the holographic
principle, showing how dynamical spacetime withgravity is generated from non-gravitational theory
in lower dimensions.
The mechanism of generating dynamical
spacetimes has been studied by many researchers,
and in this process a very important discovery
was made by Shinsei Ryu and Tadashi Takayanagi.
What they found is the following relation between
the geometry of AdS and quantum states in CFT;
see Figure 2. Consider a quantum state on an
equal time slice in CFT. Let us divide this slice into
– . Then, thea subregion A and its complement A– is given＊
entanglement entropy between A and A
by the area of the surface in AdS that is anchored to＊See, Kavli IPMU News No. 31, p. 28.

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Featureholographic
dimension

time
space

A
Figure 2: Entanglement entropy of a state in CFT between a–spatial region A and its complement A is given by the area ofthe surface in AdS which is anchored to the boundary of A
and whose area takes an extremal value.

the boundary of A and has the minimal area (moregenerally the surface which extremizes the area
in AdS). Here, the entanglement entropy between– is a quantity representing the amount of
A and A
quantum entanglement between the degrees of–.
freedom in A and A
The importance of this relationship, called the
Ryu-Takayanagi formula, is not just that it relates
two quantities. First, the fact that a geometricobject in the gravitational bulk̶ a minimal area
surface in AdS ̶ is related to a quantum information
theoretic quantity in the holographic theory̶ anentanglement entropy̶ implies that the mechanismof generating dynamical spacetime is intrinsicallyquantum mechanical. Also, taking the view thatthe theory of quantum gravity is rigorously defined
only holographically, we may say that what weperceive as gravitational spacetime is nothing otherthan quantum information theoretic propertiesof quantum states in a more fundamental, nongravitational theory. In fact, by using the RyuTakayanagi formula, one can show that the Einstein

8equation in higher dimensional AdS background is
an automatic consequence of the relation between
the geometry of emergent spacetime and quantum
information in the holographic theory.
The AdS/CFT correspondence gives us a basic
picture of how dynamical spacetime may begenerated. We could draw the following analogy.
Consider liquid water in place of spacetime. We
want to know its microstructure such as watermolecules and forces acting between them. For
this purpose, it is barely useful to study phenomenathat occur as perturbation in the limit that liquidwater is regarded as continuum, e.g., water waves.
These are described by the Navier-Stokes equationof fluid dynamics, which corresponds to the Einsteinequation in our context, regarding spacetime ascontinuum. In contrast, AdS/CFT can tell us theidentity of water molecules and forces between
them, i.e., CFT. A big difference between water andspacetime, however, is that in the case of spacetime
its constituents somehow live in one lower
dimensions.

Kavli IPMU News No. 43 September 2018

Beyond AdS/CFTAs we have seen above, AdS/CFT is extremely
powerful, but the actual spacetime we live in is not
an asymptotically AdS space. On the other hand,
the holographic principle is expected to apply
more generally in quantum gravity as can be seenfrom the consideration of black holes. How thendoes holography work in general spacetimes? This
question has been investigated by several researchgroups. Below, I will briefly describe the work wehave done.
The starting point is to identify which of theproperties of AdS/CFT are specific to AdS space andwhich are more generally applicable in quantum
gravity. For example, we expect that, from various
considerations, the fact that the holographic
theory is local quantum field theory like CFT is due
to a special feature of asymptotically AdS space.
Then, which of the properties of AdS/CFT are
manifestations of the more general holographic
principle?
Based on a number of suggestions, we have
hypothesized that the relationship between
quantum information in a lower dimensional
holographic theory and the geometry of higher
dimensional quantum gravity, as represented by
the Ryu-Takayanagi formula, is general. Since theholographic theory for general spacetimes is not
yet known, this requires us to extract essential
elements of the theory using entropy bounds
obtained by extending the result of Bekenstein and
Hawking. I omit the details of this procedure dueto the limitation of pages. In any case, the idea is to
examine the validity of the hypothesis by analyzing
what consequences it leads to.
In a sense, this strategy corresponds, e.g., tothe one Max Planck took when he examinedthe consequences of his quantum hypothesis
upon the creation of quantum mechanics. At
that time, there was an experimental result̶theblack body spectrum ̶to which he compared
the consequences. In the case of quantum

gravity, however, there is no corresponding direct
experimental result to which we can compare
theoretical results. Then how can we test the validityof the hypothesis?
Our idea is to use general relativity as
experiment. Specifically, under the assumptionthat the holographic relationship between quantum
information and generated geometries is valid in
general spacetimes, we have derived the properties
that the dynamical spacetimes must satisfy in orderfor the quantum information theoretic quantities
to be consistently interpreted holographically. The
result was a series of monotonicity properties andinequalities, which were not known before. If theoriginal assumption is correct, then these properties
must be satisfied for general spacetimes, and indeed
we were able to prove them using general relativity.
An important point is that if general relativity did
not possess features that are needed to prove theproperties we found, then the original assumption
could have been rejected. In other words, thehypothesis that quantum information theoretic
properties of states in the holographic theory are
the origin of dynamical spacetime in cases beyond
AdS/CFT has passed a nontrivial test. This increases
our confidence about the holographic principle
and relation between quantum information andquantum gravity.

SummaryThis article has described the relationship basedon the holographic principle between quantum
information and dynamical spacetime, including
AdS/CFT. This has been investigated not only
by our group but also by other researchers, andimportant discoveries have been made day-to-day.
These discoveries are providing fundamentally new
answers to the question of what spacetime reallyis. Hopefully, this new research area emerging at
the intersection of quantum gravity and quantum
information will reveal a deep and fundamental
picture of spacetime.

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