# A Comment on “Traversable wormhole dynamics on a quantum processor”

Galina Weinstein  Reichman University, The Efi Arazi School of Computer
Science, Herzliya; University of Haifa, The Department of Philosophy, Haifa,
Israel.

###### Abstract

There has been a great buzz surrounding the latest _Nature_ paper,
”Traversable wormhole dynamics on a quantum processor”. The _Nature_ paper
discusses an experiment in which Google’s Sycamore quantum processor is used
to simulate a sparsified version of an SYK model. It is shown that the
simplified model preserves the key gravitational characteristics of the
original SYK model and that it is sufficient to produce a traversable wormhole
behavior. The experiment does not create an actual wormhole. Rather, the team
of researchers shows an equivalence between a gravity picture and a quantum
information picture. This paper gives an account of the experiment and
addresses philosophical questions arising from the theoretical and
experimental work.

##  1 Introduction

In 1993 Charles H. Bennet and other authors from IBM Thomas J. Watson Research
Center invented quantum teleportation. They began their paper, “Teleporting an
unknown quantum state via dual classical and Einstein-Podolsky-Rosen
Channels”, in which they suggested teleportation, by saying: “The existence of
long range correlations between Einstein-Podolsky-Rosen (EPR) pairs of
particles raises the question of their use for information transfer”. In his
_Autobiographical notes_ , ”Einstein himself used the word ’telepathically’ in
this context. It is known that instantaneous information transfer is
definitely impossible. Here, we show that EPR correlations can nevertheless
assist in the ’teleportation’ of an intact quantum state from one place to
another, by a sender who knows neither the state to be teleported nor the
location of the intended receiver” [1].

Bennet et al. described teleportation as follows:

1\. Consider an EPR pair of particles. One EPR particle (particle $2$) is
given to Alice while the other (particle $3$) is given to Bob. Alice wishes to
teleport an unknown state $\ket{\phi}$ of a particle (particle $1$) to Bob.
The EPR pair at this stage contains no information about the state vector
$\ket{\phi}$ of particle $1$. Particle $2$ is the “ancilla”.

2\. An entanglement between the EPR pair and particle $1$ is brought about in
the next step: Alice performs a complete measurement of the von Neuman type on
the joint system consisting of particle $1$ and particle $2$ (her EPR
particle). She couples particle $1$ with particle $2$ and then obtains four
states.

3\. These states also involve the states of Bob’s particle $3$ since we are
dealing with an EPR pair. Hence, each of these possible states for Bob’s EPR
particle $3$ is related to the original state $\ket{\phi}$ of particle $1$
which Alice sought to teleport.

4\. An accurate teleportation can be achieved only by having Alice tell Bob
the classical outcome of her measurement, after which Bob applies the required
action to transform the state of his particle $3$ into a replica of
$\ket{\phi}$.

5\. Bob, therefore, converts his EPR particle $3$ into a replica of Alice’s
original state $\ket{\phi}$ of a particle $1$.

6\. Alice is left with the coupled states of particles $1$ and $2$ and thus
without any trace of the original $\ket{\phi}$.

Unlike the quantum correlation of Bob’s EPR particle $3$ and Alice’s particle
$2$, the result of Alice’s measurement is purely classical information, not
transformed “telepathically” and it can be transmitted, copied, and stored at
will in any physical medium. This information need not be destroyed or
canceled to bring the teleportation process to a successful conclusion.

In 2017 Leonard Susskind and Ying Zhao suggested that one could teleport a
unit of quantum information (a qubit) between the two sides of a microscopic
Einstein Rosen (ER) bridge, connecting an entangled EPR pair. The conditions
for teleportation render the ER bridge traversable so that a qubit entering
one end of the ER bridge will, after a suitable time, appear at the other end.

Susskind and Zhao explain that although we cannot send information faster than
the speed of light, a qubit is teleported through the microscopic so-called
”baby” ER bridge, connecting the entangled EPR pair. However, for such a
microscopic system the ER bridge is too small to have a classical space-time
geometry.

On the other hand, on the macroscopic level according to the ER = EPR
relation, the existence of an ER bridge between two black holes (joined by a
throat) implies that they are entangled and entanglement between two black
holes implies that they are connected by an ER bridge. We can, therefore,
hypothetically apply the same reasoning, of teleportation as communication of
quantum information through space-time wormholes, when the entangled EPR
systems are a pair of entangled macroscopic black holes. Then we may hope to
follow the geometry of the wormhole as the teleported system passes through it
[19].

Yet Susskind and Zhao are aware that [19]:

> “The operations we’ve described would be very hard to do, if not impossible,
> for real black holes, but we can imagine laboratory settings where similar
> things may be possible. Suppose that in our lab we have two non-interacting
> large shells of matter, each of which has been engineered by condensed
> matter physicists to support conformal field theories [CFT] of the kind that
> admit gravitational duals. If we make the two shells out of entangled matter
> we can produce the shells in the thermofield double state for some
> temperature”.

Would there really be a hidden little wormhole connecting the shells? Assuming
a model in which the entire laboratory is embedded in a world that satisfies
ER=EPR, then in this model laboratory, there would be such a wormhole. But we
are not yet speaking of our real world - I will discuss this matter further in
section 9.

To confirm that there is a hidden wormhole, Alice and Bob can merge themselves
with the matter forming the shells and eventually be scrambled into the CFT
thermal state - I will discuss scrambling in section 2. In the dual
gravitational picture, they would each fall into their respective black holes,
and if conditions were right, they would meet before being destroyed at the
singularity; all this taking place in some space outside ordinary spacetime.
Unfortunately, they would not be able to inform the exterior world that either
the wormhole is real or that they successfully met. However, quantum
teleportation allows Alice and Bob to confirm the existence of the wormhole
without jumping in. Alice may convince Tom to jump into the wormhole, which
she and Bob can render traversable by introducing a temporary coupling. When
Tom emerges out of the degrees of freedom of Bob’s shell, he will recall
everything he encountered and can confirm that he really did traverse the
wormhole [13].

Susskind and Zhao’s paper extends Susskind’s 2016 idea of quantum
teleportation through Schwarzschild wormholes (or ER bridges). Teleportation
through a Schwarzschild wormhole violates classical general relativity,
according to which Schwarzschild wormholes are non-traversable because they do
not allow us to send a signal from one asymptotic region to the other when
suitable positive energy conditions are obeyed. According to the ER = EPR
relation, an ER bridge between two black holes is created by EPR-like
correlations between the microstates of the two black holes [18].

In the same year, Ping Gao, Daniel Louis Jafferis, and Aron C. Wall showed
that the ER = EPR relation allows the ER bridge to be traversable. Gao,
Jafferis and Wall suggested that in order for the ER bridge to become
traversable, entanglement alone cannot be used to transmit information and we
need quantum teleportation: Alice and Bob share an entangled pair of qubits
$A$ and $B$. Alice is given a qubit $Q$ to be transmitted to Bob. Alice
performs a joint measurement of $Q$ and $A$ and sends the result to Bob as
classical bits of information. Bob will perform a unitary operation on his
qubit $B$, which depends on Alice’s results, and gets qubit $Q$. “Of course in
the limit that Alice’s measurement is essentially instantaneous and classical,
the traversable window will be very small […] – just enough to let the single
qubit $Q$ pass through. Therefore, we propose that the gravitational dual
description of quantum teleportation understood as a dynamical process is that
the qubit passes through the ER=EPR wormhole of the entangled pair, $A$ and
$B$, which has been rendered traversable by the required interaction” [5].

This finding comes with implications for Hawking’s black hole information
paradox because hypothetically, an observer can enter a Schwarzschild black
hole and then escape to tell about what they have seen. This suggests that
black hole interiors might exist and that what goes in must come out and we
can learn about the information that falls inside black holes. Can Hawking’s
chronology protection conjecture protect against this situation?

Suppose Alice throws qubit $Q$ into black hole $A$. She then measures a
particle of its Hawking radiation, $a$, and transmits the result of the
measurement through the external universe to Bob, who can use this knowledge
to operate on $b$, a Hawking particle coming out of black hole $B$. Bob’s
operation reconstructs $Q$, which appears to pop out of $B$, a perfect match
for the particle that fell into $A$. The new traversable ER = EPR wormhole
allows information to be recovered from black holes. Consequently, Gao,
Jafferis, and Wall explain that another possible interpretation of their
result is to relate it to the recovery of information from evaporating black
holes [5]:

What is essential here is that “Assuming that black hole evaporation is
unitary, it is in principle possible to eventually recover a qubit which falls
into a black hole, from a quantum computation acting on the Hawking radiation.
Assuming that you have access to an auxiliary system maximally entangled with
the black hole, and that the black hole is an efficient scrambler of
information, it turns out that you only need a small (order unity) additional
quantity of Hawking radiation to reconstruct the qubit” [5].

In the traversable wormhole picture, the qubit ”may be identified with the
system that falls into the black hole from the left and gets scrambled, the
auxiliary entangled system is on the right, and the boundary interaction
somehow triggers the appropriate quantum computation to make the qubit
reappear again, after a time of order the scrambling time” [5]. I will
thoroughly discuss this protocol in sections 2, 3, 4, 5, and 6.

Gao, Jafferis, and Wall conclude their article by saying that their example
”provides a way to operationally verify a salient feature of ER=EPR that
observers from opposite sides of an entangled pair of systems may meet in the
connected interior”: If, after the observers jump into their respective black
holes, a coupling is activated, then the ER bridge can be rendered
traversable, and the meeting inside may be seen from the boundary [5]. The
above traversable wormhole does not require energy matter that violates the
averaged null energy condition because the negative energy-matter in the ER =
EPR configuration is similar to the Casimir effect. Further, any infinite null
geodesic which makes it through the wormhole must be chronal, i.e. the
wormhole does not violate Hawking’s chronology protection conjecture. In
addition, the wormhole does not violate the generalized second law of
thermodynamics.

Susskind sheds some light on the ER = EPR traversable wormhole [18]: “It is
especially interesting that if Tom encounters objects during his passage from
Alice’s side to Bob’s side, his experiences may be recorded in his memory.
This would allow him to report the conditions in the wormhole to laboratory
observers. Quantum teleportation through the wormhole is a real game-changer;
it provides a direct way to observe the interior geometry of a wormhole. One
can no longer claim that life behind the horizon is unphysical, meaningless,
unobservable, or scientifically unfalsifiable”. Susskind points out that
“quantum teleportation of a single qubit through a channel of a single bell
pair is already an experimental reality. ER=EPR allows it to be interpreted as
teleportation through a Planckian wormhole”. In 2017 Susskind claimed that
this opens a possibility for seeing quantum gravity in a lab equipped with
quantum computers. I will discuss quantum gravity in the lab in section 10.

Susskind asked in 2017 [18]: “Can laboratory experiments of this type be
carried out? I don’t see why not. Instead of shells supporting conformal field
theories [CFT], a more practical alternative might be quantum computers
simulating the CFTs. Entangling two identical quantum computers into a
thermofield double state should be feasible. To teleport a genuine sentient
Tom through the wormhole would require an enormous number of qubits, but with
a few hundred logical qubits one can teleport a register, composed of say ten
qubits - enough for a primitive memory”. The operations needed for the kind of
laboratory teleportation offered above, said Susskind in 2017, “are fairly
complex (in the computational sense) and are therefore difficult”, but
Susskind did not see “anything forbidding them once quantum computers become
available”.

A version of Susskind’s suggestion was carried out on Google’s Sycamore
processor by Jafferis and a team of researchers [9]. This paper gives a
detailed account (with no equations) of the recent wormhole experiment and in
sections 5, 8, 9, and 10, addresses philosophical questions arising from the
theoretical and experimental work.

##  2 Quantum chaos and scrambling

Let us begin with the quantum _butterfly effect_ , which is essential for the
understanding of the experiment. The butterfly effect implies scrambling [8].
_Quantum scrambling_ is the quantum analog of chaotic dynamics in classical
systems. Scrambling describes many-body dynamics which, though ultimately
unitary, scatter initially localized quantum information across all of the
system’s available degrees of freedom. Black holes are the fastest scramblers
in the universe and are therefore the most chaotic bodies in the cosmos [20];
[2].

More specifically, quantum information present in a small local area of space
spreads out, and we must search a large region to recover the information.
This is the scrambling of the quantum information while the system evolves.
Heisenberg’s operators evolve in a way that reminds the chaotic butterfly
effect: they were first local, and now they are spread over many regions in
space. This is the butterfly effect in quantum systems. The models for the
onset and dynamics of quantum chaos are called the _Sachdev-Ye-Kitaev_ (SYK)
models.

It should be stressed that what is meant by black holes here and thereafter is
_eternal black holes_ or _anti-de Sitter space_ (Ads) black holes. The main
feature of black holes in AdS is that their Hawking radiation is reflected
back by the boundary. For small enough black holes, this is not really
important, since after all the entire black hole could evaporate before the
radiation gets to the boundary, but as the Schwarzschild radius of the black
hole approaches the AdS radius we eventually reach a point where the radiation
is being reflected back into the black hole as fast as it is being emitted. At
this point, the black hole never evaporates, so large enough black holes in
AdS are therefore eternal [7].

If two external black holes are close to each other, this would give rise to
an interaction between them connecting the two separated boundaries of AdS2.
The interaction can produce a throat that connects the two sides of the black
hole so that the final geometry is a horizonless _eternal traversable
wormhole_. The interaction creates _negative energy_ in the bulk. Imagine a
particle that starts from one boundary, living in the first system; it then
moves to the other side, living in the second system. ”This is somewhat
reminiscent of two weakly coupled oscillators where the excitation moves from
one oscillator to the other, except that here it is doing this through the
bulk geometry”. The same property holds in the two copies of SYK models
coupled by simple interaction. ”The dynamics of the two coupled SYK systems
looks like that of a traversable wormhole” [14].

What this means is that the _eternal traversable wormhole_ is dual to two
copies of the original _conformal field theory_ (CFT) in the _thermofield
double_ (TFD) state. The left and right external bulk regions of the eternal
black hole are joined through a wormhole and are thus dual to the TFD state
[14]. The TFD state is an entangled pure state between two identical copies of
the quantum system (CFT):

$\frac{1}{\sqrt{Z}_{\beta}}\ket{TFD_{\beta}}=e^{-\beta(H_{L}+H_{R})}\ket{n}_{L}\otimes\ket{n}_{R}.$
(1)

Tracing out one of the two copies $H_{L}$ (the SYK Hamiltonian applied to the
left system) or $H_{R}$ (the SYK Hamiltonian applied to the right system)
gives a thermal state (with Majorana fermions). In other words, the reduced
density matrix at inverse temperature $\beta$ is obtained by tracing out
either side. The $\ket{n}_{L,R}$ is the thermofield double state at an
infinite temperature.

The SYK models lead to scrambling and spreading of the information among the
quantum many-body system. But the SYK models possess gravity duals. They are
also a paradigm for quantum holographic matter and the gravitational
interpretation through the holographic principle or duality (the AdS/CFT
correspondence or gauge/gravity duality); the equivalence between two
descriptions of the same system: quantum gravity in $(d+1)$ dimensions, on the
one hand, and quantum field theory in $d$ dimensions, on the other. The AdS
description is often called the _bulk_ theory while the CFT is called the
_boundary_ theory. In AdS/CFT, the two theories are equivalent, and we should
use either one description or the other but not both [7].

The above characteristics of the SYK Hamiltonian for $N$ fermions have led to
realizing holographic physics in the laboratory, what is called _quantum
gravity in the lab_. I will further discuss quantum gravity in the lab in
section 10.

An SYK model becomes extremely chaotic at the very beginning of its
development. In the SYK model, the _out-of-time-order correlation_ (OTOC)
functions are used to diagnose quantum chaos, and measure the growth of
operators in space, unitarily evolving (in the Heisenberg interpretation of
quantum mechanics) as a function of time. With chaotic time evolution, the
butterfly effect will cause most of the OTOC functions in the average to decay
exponentially [8].

In the semi-classical limit (in quantum systems with many degrees of freedom),
this scrambling of information and operator growth due to chaotic behavior is
exponential and is measured using the quantum Lyapunov exponent. But unlike
the classical Lyapunov exponent, there exists a bound on the quantum Lyapunov
exponent. This is additionally measured by the _butterfly velocity_ , the very
equivalent measure of the classical chaotic butterfly effect. The quantum
Lyapunov exponent and the scrambling rate are the ones that characterize the
beginning and appearance of quantum chaos in this system [15].

It should be noted that an interesting characteristic of the SYK model, which
is related to the quantum Lyapunov exponent and the OTOC, is that the model
exhibits maximally chaotic behavior. It means that like eternal black holes,
the SYK model is a very fast scrambler of information. There is another
important quantity called, _Loschmidt echo_ , which is intimately tied to
quantum chaos. The echo is defined as the probability that the chaotic system
would return to its initial state.

As said above, we characterize quantum scrambling and quantum chaos by
measuring the OTOC function. However, OTOCs do not generally discriminate
between _quantum scrambling_ and the effects of both ordinary _quantum
decoherence_ and experimental noise: quantum scrambling and classical noise
lead the OTOC to decay exponentially with time. It is a major problem if
quantum scrambling is indistinguishable from quantum decoherence and noise,
where the information in a system is lost to the environment [12];[21].

Isolated systems are idealized models but unfortunately, realistic systems are
open systems and are in interaction with the environment. Suppose there is a
system of $n$ qubits. This system is not an isolated and closed system. The
$n$ qubits are interacting with many interfering particles in the complex
environment. It is almost impossible to follow the dynamics of each particle,
so what we have here is a system that is many-body system, and decoherence
induced by the environment. As the system evolves, the $n$ qubits get
entangled with the many-body system of the environment, and there are more
disturbances and perturbations and more degrees of freedom. Decoherence
happens naturally to quantum computers since like scrambling, qubits can’t be
perfectly isolated from the environment.

It was found that a quantum teleportation protocol enables one to
differentiate between scrambling and decoherence. Thus using teleportation one
can verify scrambling behavior even in the face of decoherence and
experimental imperfection [2];[21].

##  3 SYK models and holography

The SYK Hamiltonian is a model for quantum chaos and holography. That is,
there is correspondence between the SYK model and scrambling/quantum chaotic
behavior on the one hand, and eternal black holes, on the other. This dual
possibility led a team of researchers to the realization that they might be
able to create a model of teleportation through a traversable wormhole. They
discovered that a process called _unscrambling_ comes after scrambling in a
wormhole. The discovery of a process of scrambling followed by unscrambling
has boosted the possibility of realizing a quantum mechanism called _size
winding_ in the lab. This process completely goes against everything we know
from classical chaos and irreversibility. The size-winding mechanism is
reminiscent of _Poincare’s Recurrence Theorem_ of classical physics. But in
the dual gravitational interpretation, size-winding leads to the interesting
conclusion that a particle can pass through a wormhole (a holographic
wormhole).

The protocol is the following: on the left side of the wormhole, the
information is scrambled. Since the two sides, right and left of the wormhole
are connected (coupled), the information, i.e., qubits, is unscrambled and
pops up on the right side. Two essential things enable traversability: the two
sides of the wormhole must be entangled before sending the information and the
two sides must be coupled after sending the message.

It was thought that it was possible to study the dynamics of a wormhole,
through which a qubit can pass, by simulating the SYK model of $N$ Majorana
fermions. It was suggested that realizing the holographic SYK model on the
Google Sycamore chip might open a window to an understanding of the quantum
gravity of holographic traversable wormholes.

The SYK models of a quantum many-body system simulate the scrambling-
unscrambling method. According to the holographic principle, systems that are
not gravitational but are entangled will exhibit properties that are identical
to quantum gravity. Hence, reasoned the team of researchers, an experiment
implementing the entanglement of qubits can be performed in the laboratory to
test theories of quantum gravity. This experiment consists of two entangled
systems of $n$ qubits, on the right and $n$ qubits on the left. In this
protocol obviously, the dynamics of the system are chaotic (quantum
mechanically) and is described by the SYK model.

One inserts a qubit (the message) on the left side of the system ($L$
subsystem), and it evolves in time. The qubit is entangled with one of the
qubits on the $L$ subsystem. It means that the qubit begins to spread among
the $n$ qubits (a small number of qubits) on the left side, and in all parts
of the subsystem. After a certain time, the qubit is entangled with the qubits
of the $L$ subsystem. But then the qubit suddenly reappears, is unscrambled,
and recoheres on the other side, the right side ($R$), very far from the $L$
side, where it was scrambled.

There is something that caused the original qubit, which entered on the far-
left side, to suddenly be focused on the far-right side at a future time, even
though it was completely mixed up on the left side.

It is bizarre from the quantum mechanical point of view, says the team of
researchers, but what makes things less weird is that it may be explained or
interpreted using the paradigm of quantum gravity and the holographic
principle: a traversable wormhole protocol is equivalent to the above quantum
information protocol [3].

We start from two separated black holes, the so-called scramblers. The
teleported signal reappears on the right side when the two black holes are
connected by a wormhole. In the quantum system, we speak of the method of
scrambling-unscrambling. But with respect to a wormhole, the above mechanism
is called _teleportation-by-size_ , a protocol of quantum teleportation
through the wormhole, i.e., information transmission is dependent on operator-
size growth.

So, argues the team of researchers, if we imagine that the two sides of the
system represent two sides of the eternal black holes ($L$ and $R$) that are
connected by a wormhole, then the explanation for the phenomenon is _simpler_.
A teleported message is sent through an emergent wormhole: it is injected into
$L$ and arrives at $R$ later due to a coupling operator [3].

##  4 Perfect size winding

The above scenario requires _perfect size winding_. The team of researchers
first describes size-winding purely from the boundary point of view and then
applies it to the traversable wormholes (in the bulk).

In the Heisenberg picture, near the _scrambling time_ (just before the onset
of the chaotic behavior) for the SYK model, a thermal operator $P$ is inserted
at a negative time into the left boundary (the left side $L$). Recall that the
growth of the size of an operator is a basic manifestation of quantum chaos
and complexity of the system. The operator-size distribution is winding in the
clockwise direction. A coupling is applied between the two subsystems $L$ and
$R$. The $LR$ coupling unwinds the complex winding of the operator size
distribution, it winds the size distribution in the opposite direction,
accurately reversing the winding direction. The thermal operator $P$ from the
left side will be exactly mapped to its right side. We obtain a
counterclockwise size distribution corresponding to a thermal operator P
inserted on the other boundary (the right side $R$) at a positive time [3].
The team of researchers stresses: ”We explicitly show size-winding of thermal
operators near the scrambling time for the SYK model, and we conjecture that
the phenomenon can also be found in other holographic systems” [17].

Perfect size winding provides a necessary condition for traversable wormhole
behavior. It occurs in the ground state, the state of lowest possible energy
where the temperature is zero (low temperature through the wormhole).

The team of researchers expects systems with a holographic dual to exhibit
perfect size winding [17]. In other words, the SYK model is dual to a
traversable wormhole only in the low-temperature regime, and it exhibits
perfect size winding in the low-temperature limit. But this applies to large
$N$ Majorana fermions interacting with large $q$ other Majorana fermions
(teleportation of $q$ fermions).

The team of researchers then pondered: What is the most simplified Hamiltonian
that preserves the gravitational physics of the original SYK model? How many
qubits do we need to simulate this Hamiltonian on a quantum device? It was
shown that $N=10$ was sufficient to produce the traversable wormhole behavior.
The team employed learning techniques to construct a sparsified version of the
SYK model. _Sparsification_ reduces the complexity of the system.

A simplified learned Hamiltonian was constructed. Its ground state was close
to a TFD state. Techniques from machine learning (and a kind of approximation
called _Trotterization_) were applied to optimize the procedure. The
techniques were performed on a classical computer. The sparsification
procedure reduced the SYK model to a sparse $N=10$ SYK model. ”We choose
$q=4$” fermions interacting with $N$ other fermions of the simplified version
of the SYK Hamiltonian, ”and demonstrate gravitational physics at sufficiently
small $N$”, where $N=7$. Since ”The wormhole teleportation protocol also
introduces a pair of entangled qubits, i.e., a reference qubit that is
entangled with the injected qubit”, then ”the total circuit has $9$ qubits”.
Hence, the sparsified SYK model was experimentally realized with $N=9$ qubits
[9].

It should be mentioned that at about the same time, Susskind and a team of
researchers were working on what seems like a bigger project, a sparse SYK
model that recovers the global physics of ordinary SYK models. In particular,
at low temperatures, their model exhibits a gravitational sector that is
maximally chaotic. The sparsity of the model, so writes the team,
”considerably reduces the cost of quantum simulation algorithms”. This, so
claims the team, makes their sparse SYK model ”the most efficient currently
known route to simulate a holographic model of quantum gravity”. The team of
researchers add: ”On a practical level, sparse systems typically admit much
more efficient computer simulations—both classical and quantum. By
significantly reducing the resources needed to simulate black holes in
holographic models of quantum gravity, these results bring us closer to the
goal of studying ’quantum gravity in the lab’” [22].

##  5 Majorana fermions versus transmons

The sparse Hamiltonian is doubled to give left $H_{L}$ and right $H_{R}$
Hamiltonians with $N$ Majorana fermions on each side. Each side is a
simulation of the SYK model, the learned Hamiltonian.

The wormhole experiment was realized with superconducting qubits on the Google
Sycamore. I would like to emphasize that I am not speaking now of claims
related to quantum supremacy. The Sycamore consists of an array of $54$
superconducting qubits called _transmons_ (transmission-line shunted plasma
oscillation qubits). The transmon is closely related to the charge qubits or
Cooper–Pair–Box (CPB) (Cooper pairs that are tunneling in a Josephson
junction). The transmon fixes the weakness of the CPB and as compared to the
CPB, it greatly reduces charge noise sensitivity in the qubit [11]. The
wormhole experiment was reportedly done on Google’s $72$-qubit quantum
processor.

That said, the team of researchers is speaking of the Majorana SYK model with
$N$ fermions with which they produce evidence of gravitational physics in the
sparsified SYK system: ”To encode $7$ Majorana fermions on the left system and
$7$ Majorana fermions on the right system, we require $7$ qubits (two fermions
per qubit)” [9]. The team of researchers also writes: ”we assume that the
total number of qubits (or fermions) on each side is $n$, and the number of
message qubits (or fermions) that are transmitted by the state transfer or
operator transfer protocols is $m$” [17].

This is problematic because it is not at all clear whether one superconducting
transmon qubit represents two Majorana fermions or rather, one transmon qubit
represents one Majorana fermion.

##  6 The quantum information picture

The practical steps of the teleportation protocol (step-by-step) in the
quantum information picture (without gravity) are as follows (based on [3]):

1) Two identical copies of the quantum system are prepared: a system of $7$
qubits on the left (side $L$) and a system of $7$ qubits on the right (side
$R$). The two subsystems are entangled in the TFD state; that is, we have
entangled Bell pairs shared between $L$ and $R$.

2) We evolve all the qubits on the side $L$ “backward in time” by acting with
the inverse of the time-evolution operator ($\exp^{+iHt}$).

3) A qubit $Q$ (the message) is injected into $L$ at a certain time (swapped
into side $L$: a SWAP gate). Now we evolve subsystem $L$ “forward in time”
using the time-evolution operator ($\exp^{-iHt}$). As a result, $Q$ is
entangled with a reference qubit $P$; $Q$ is then scrambled with $P$ and among
the $7$ qubits on the subsystem $L$ (the carrier qubits).

4) The teleportation protocol requires classical communication between the two
sides. So, we now weakly couple side $L$ to side $R$ (at $t=0$), applying a
coupling operator ($\exp^{i\mu V}$, where $V$ is the interaction term and
$\mu$ represents the coupling interaction). The coupling is applied suddenly:
All the $7$ qubits on side $L$ are now coupled to the $7$ qubits on the side
$R$.

5) We now evolve side $R$ “forward in time” using the time-evolution operator
($\exp^{-iHt}$). Side $R$ is subsequently measured. The qubit $Q$ (the
message) reappears unscrambled, it arrives unscathed at $R$ and there is no
need to decode it (a final SWAP gate: extract qubit $Q$ from $R$). The message
has been teleported while being first scrambled and then unscrambled. The
teleported qubit is highly error-protected [6].

The team of researchers distinguishes between two mechanisms of transmission
with the wormhole circuit [3]:

1) The low-temperature teleportation: If $\mu<0$, the qubit $Q$ experiences a
time advance and is rescued on the side $R$. This is wormhole teleportation.

2) On the other hand, when $\mu>0$ the qubit is entangled with the qubits of
side $L$ but is not unscrambled and its destiny is oblivion.

##  7 The gravity picture

According to Occam’s razor, the simplest explanation for the above mechanism
is teleportation-by-size, i.e., holographic teleportation through a ”baby”
wormhole. Thus in the gravity picture, a message has been teleported through a
semi-classical holographic traversable wormhole [3]. Holographically, the
above coupled $LR$ quantum system (the CFT side of the duality) is dual to a
wormhole that connects the two sides of the eternal black hole (the AdS side
of the duality). The $LR$ coupling renders the wormhole traversable; if
$\mu<0$, the coupling operator generates a negative energy shockwave in the
bulk, modifying the geometry of the wormhole and allowing traversability. When
$\mu>0$ the coupling generates a positive energy shockwave and the qubit falls
into the singularity. The team of researchers writes: ”we observe increased
teleportation when the interaction introduces a negative energy shockwave
rather than a positive one. The asymmetric signature is consistent with the
physical interpretation that the qubit underwent teleportation through the
wormhole” [9].

The point is that for very low temperatures, the information does not vanish
and the original entanglement between $Q$ and $T$ does not get destroyed by
chaotic perturbations. How is this possible? Although there is scrambling and
quantum chaotic behavior, the weak coupling interaction between $L$ and $R$
entangles $L$ and $R$, and the qubit $Q$ is unscrambled. This is perfect size
winding which causes teleportation around the scrambling time. In the perfect
size winding protocol of scrambling followed by unscrambling the teleported
qubit is highly error-protected [6]. I further discuss this issue in section
10.

##  8 Why should we believe the gravity picture?

In the new experiment performed with the Sycamore chip, the team of
researchers shows that their coarse-grained SYK model preserves key properties
of the traversable wormhole physics: perfect size winding, coupling
interaction on either side of the wormhole that is consistent with a negative
energy shock wave, a Shapiro time delay, causal time-order of signals emerging
from the wormhole, and scrambling [9].

Besides sending a single qubit from left to right, another qubit is inserted
from right to left. The result is time-ordered teleportation, which is
interpreted as a demonstration of gravitational teleportation. At time
$-t_{0}$, a qubit $Q$ is swapped into $L$. Simultaneously, a qubit $R$ is
swapped into $R$. At the time $t_{1}$, the team of researchers performs a
measurement and compares the two processes.

They found that the presence of $R$ delayed the arrival of the signal
traveling left to right, and interpreted this delay observed in the learned
Hamiltonian as due to a _Shapiro time delay_. It is also demonstrated that in
the high-temperature regime, non-gravitational teleportation occurs, and there
is no size winding [9].

Working with collaborators from Caltech, Fermilab, and Harvard, the quantum
system was subjected to numerous tests to determine if it showed quantum
gravitational behavior. The above signatures were verified on classical
computers, so claims the team of researchers, confirming that the dynamics of
the quantum system were consistent with a quantum gravity interpretation and
the holographic principle [23].

##  9 Scientific explanation is not truth

We usually proceed from the success of an experiment to the conclusion that
our explanation is likely to be approximately true, or true. We think that if
an explanation is the best among the competing explanations of the experiment,
then it is probably true. But it should be stressed that the fit between the
_simplified_ SYK model and the explanation in terms of an emergent wormhole
does not mean that the latter explanation is _literally true_. Neither does it
mean that holographic wormholes _exist_ or that they are _real_. What is meant
by saying that this explanation is the simplest among the other hypotheses is
mainly that, it is the best fit for the experimental setup, and that
holographic teleportation fits the teleportation mechanism at the basis of the
said experiment.

The point is that according to the ER = EPR hypothesis, the gravity picture is
equivalent to the quantum information picture, and ”The traversable wormhole
expressed as a quantum circuit, equivalent to the gravitational picture in the
semiclassical limit of an infinite number of qubits” [9]. But although the
analogy between the experimental setup and the emergent geometry is
suggestive, it does not follow from the experiment that the wormhole
gravitational picture is real. We can only say that teleportation-by-size is
the hypothesis that explains the experiment best. This is so even if it
explains the evidence. ”Truth” requires a step beyond the judgment that the
holographic wormhole hypothesis fits the experimental setup and the data and
is better than all of its rivals.

##  10 Quantum gravity in the lab

Advocates of the ”quantum gravity in the Lab” program argue: ”The ‘quantum
gravity in the lab’ program does not need to wait for large error-corrected
quantum computers. Progress can be made even in the Noisy Intermediate-Scale
Quantum (NISQ) era” [17]. There is a problem with this statement.

As is well known, quantum computers are prone to many errors and the Sycamore
quantum device has a large error rate [10]. In this state of affairs, ”If, at
any point in time, a small error occurs, the chaotic dynamics will not undo
themselves, and the particle will not make it through the wormhole” [23].

At large times, a small perturbation can destroy the correlations between the
two sides $L$ and $R$ of the quantum system that would otherwise exist without
the perturbation. Although the qubits of the Sycamore processor are cooled
down to cryogenic temperatures and are held in an ultra-high vacuum chamber,
the entangled qubits can decohere quickly due to interaction (entanglement)
with the environment (incoherent errors). The team of researchers writes: ”In
general, errors can include coherent errors [crosstalk errors and qubit phase]
and incoherent sources of noise; in simulations, we assume fully incoherent
errors and observe agreement with experimental data” [9].

A team of researchers trained a quantum neural network (in a quantum machine
learning context). An appropriate ansatz can mitigate coherent errors for only
a small number of qubits ($18$) on the Sycamore quantum device [16].
Proponents of the ”quantum gravity in the Lab” program show that ”with some
caveats we can use a finite fraction of the fermions” [17]. So in order to
reduce the coherent errors, ”the total circuit has $9$ qubits” [9]. Recall
that in practice, only $7$ qubits were used to simulate a ”wormhole-like
teleportation”. The other two qubits served as the teleported qubits [9] (see
sections 4 and 6). Using machine learning, the team of researchers was able to
make the quantum model simple enough to preserve the key gravitational
properties, so that it could be realized with a circuit with $164$ two-qubit
gates [9]. A more complex model would increase the number of gates, and
consequently also the error rate.

In the _Caltech_ press release, it is said that the team of researchers found
a quantum system, a “baby” SYK-like model, prepared to preserve the key
properties of a gravitational wormhole. To achieve this, the team had to first
reduce the SYK model to a simplified form, a feat they achieved using machine
learning tools _on conventional computers_. They employed learning techniques
to find and prepare a simple SYK-like quantum system that could be encoded in
the Sycamore quantum architecture, and that would preserve the key
gravitational property: the negative energy shockwave. The greatest
achievement was simplifying the microscopic description of the SYK quantum
system and studying the resulting effective model that the team found on the
Sycamore quantum processor. The team of researchers found it “curious and
surprising how the optimization on one characteristic of the model [the
negative energy shockwave or LR coupling] preserved the other characteristics”
[4].

Reducing the SYK model to a simplified form is an achievement that is to be
celebrated. But this demonstrates the amazing capabilities of conventional
computers. It is important to stress that no wormhole was created in the lab,
and moreover, no one has ever observed or found any evidence of any wormhole.

## Acknowledgement

This work is supported by ERC advanced grant number 834735.

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# Distance inequalities of generalized $k$-intersection bodies

Julián Haddad  Departamento de Análisis Matemático, Universidad de Sevilla,
Sevilla, España<EMAIL_ADDRESS>
###### Abstract.

It was shown in [11] that for every origin-symmetric star body
$K\subseteq\mathbb{R}^{n}$ of volume $1$, every even continuous probability
density $f$ on $K$ and $1\leq k\leq n-1$, there exists a subspace
$F\subseteq\mathbb{R}^{n}$ of codimension $k$ such that

$\int_{K\cap F}f\geq c^{k}(d_{\rm ovr}(K,\mathcal{BP}_{k}^{n}))^{-k}$

where $d_{\rm ovr}(K,\mathcal{BP}_{k}^{n})$ is the outer volume ratio distance
from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a
universal constant. The upper bound $d_{\rm ovr}(K,\mathcal{BP}_{k}^{n})\leq
c^{\prime}\sqrt{n/k}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}$ was
established in [13] for every convex body $K$. In this note we show that there
exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous
probability density $f$ supported on $K$ such that for every subspace $F$ of
codimension $k$,

$\int_{K\cap F}f\leq\left(c\sqrt{\frac{n}{k\log(n)}}\right)^{-k}.$

As a consequence we obtain a lower bound for $d_{\rm
ovr}(K,\mathcal{BP}_{k}^{n})$ with $K$ a convex body, complementing the upper
bound in [13]. This is

$c\sqrt{n/k}(\log(n))^{-1/2}\leq\sup_{K}d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})\leq
c^{\prime}\sqrt{n/k}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}.$

The case $k=1$ was obtained previously in [5, 6].

###### Key words and phrases:

Radon transform, slicing inequality, $k$-intersection bodies, outer volume
ratio distance

###### 2010 Mathematics Subject Classification:

52A20, 44A12

##  1\. Introduction

The problem known as the slicing inequality for general measures asks which is
the greatest possible value $S_{n}$ such that for every origin-symmetric
convex body $K$ (convex compact with non-empty interior) and every even
continuous probability density $f$ on $K$, there exists a hyperplane
$F\subseteq\mathbb{R}^{n}$ with

$\int_{K\cap F}f\geq S_{n}.$

Koldobsky investigated this problem in [9, 10, 11] and showed that
$S_{n}\geq\frac{1}{2\sqrt{n}}$, although the constant can improve if $K$ is
restricted to certain classes of convex bodies.

A star body is a set $K\subseteq\mathbb{R}^{n}$ of the form

$K=\\{0\\}\cup\\{x\in\mathbb{R}^{n}\setminus\\{0\\}:|x|\leq\rho_{K}(x/|x|)\\}$

where the radial function $\rho_{K}:S^{n-1}\to(0,\infty)$ is continuous and
strictly positive, and $|\cdot|$ is the euclidean norm. Every convex body
containing the origin in the interior is also a star body. The radial distance
between two star bodies $K,L$ is $\max_{v\in
S^{n-1}}|\rho_{K}(v)-\rho_{L}(v)|$.

Given a star body $K\subseteq\mathbb{R}^{n}$, the intersection body of $K$ is
the star body ${\rm I}K$ defined by $\rho_{{\rm I}K}(v)=\left|K\cap
v^{\perp}\right|_{n-1}$, where $|\cdot|_{k}$ denotes the $k$-dimensional
volume. Taking the closure in the radial metric of the sets ${\rm I}K$ with
$K\subseteq\mathbb{R}^{n}$ a star body, we get the class of intersection
bodies, which we denote by $\mathcal{I}_{n}$.

For any class of star bodies $\mathcal{C}$ the outer volume ratio distance
from a star body $K$ to $\mathcal{C}$ is defined by

$d_{\operatorname{ovr}}(K,\mathcal{C})=\inf\left\\{\left(\frac{\left|D\right|_{n}}{\left|K\right|_{n}}\right)^{1/n}:K\subseteq
D,D\in\mathcal{C}\right\\}.$

In [11] the following inequality was proven:

######  Theorem 1.1.

Let $K$ be an origin-symmetric star body in $\mathbb{R}^{n}$. Then for any
even continuous non-negative function $f$ we have

$\int_{K}f\leq 2d_{\rm
ovr}(K,\mathcal{I}_{n})\left|K\right|_{n}^{1/n}\max_{v\in S^{n-1}}\int_{K\cap
v^{\perp}}f.$

In particular, if $K$ is an origin-symmetric star body in $\mathbb{R}^{n}$ of
volume $1$ and $f$ is an even continuous probability density in $K$ then

(1) $\max_{v\in S^{n-1}}\int_{K\cap v^{\perp}}f\geq(2d_{\rm
ovr}(K,\mathcal{I}_{n}))^{-1}.$

Here $d_{\operatorname{ovr}}(K,\mathcal{I}_{n})$ is the outer volume ratio
distance between $K$ and the class of intersection bodies. Since origin-
symmetric ellipsoids are intersection bodies, if $K$ is an origin-symmetric
convex body, John’s theorem implies that
$d_{\operatorname{ovr}}(K,\mathcal{I}_{n})\leq\sqrt{n}$, hence the bound
$S_{n}\geq\frac{1}{2\sqrt{n}}$.

The upper bound is $S_{n}\leq c/\sqrt{n}$, in other words, there exists an
origin-symmetric convex body $K$ of volume $1$ and an even continuous
probability density $f$ on $K$, such that

$\max_{v\in S^{n-1}}\int_{K\cap v^{\perp}}f\leq c\frac{1}{\sqrt{n}}.$

This estimate was obtained in [5] up to a factor $\sqrt{\log(\log(n))}$, and
this factor was later removed in [6]. It follows from Theorem 1.1 that the
largest possible distance from a convex body to $\mathcal{I}_{n}$ has order
$c\sqrt{n}$, this is,

$c\sqrt{n}\leq\sup_{K}d_{\operatorname{ovr}}(K,\mathcal{I}_{n})\leq\sqrt{n}$

where the supremum runs over all origin-symmetric convex bodies
$K\subseteq\mathbb{R}^{n}$, and $c>0$ is a universal constant. A similar
argument was used to show in [1] (lower bound) and [12] (upper bound) that

$c\sqrt{n/p}\leq\sup_{K}d_{\operatorname{ovr}}(K,L_{p}^{n})\leq
c^{\prime}\sqrt{n/p}$

for $p\geq 1$, where $L_{p}$ is the class of convex bodies $L$ such that
$(\mathbb{R}^{n},\|\cdot\|_{L})$ embeds in $L_{p}$ as a Banach space (see [8]
for details). Also, in [4] the convex body and measure constructed in [5] were
adapted to a different slicing inequality from [3] to obtain the lower bound

(2) $\sup_{K}d_{\operatorname{ovr}}(K,L_{-p}^{n})\geq cn^{\frac{1}{2p}},$

for $p\in[1,n)$, where the supremum runs over all origin-symmetric convex
bodies $K\subseteq\mathbb{R}^{n}$, $c>0$ is a universal constant and
$L_{-p}^{n}$ is the class of star bodies $L$ for which
$(\mathbb{R}^{n},\|\cdot\|_{L})$ embeds in $L_{-p}$. For integer $k\in[1,n)$,
$L_{-k}$ is the class of $k$-intersection bodies (see [8] for details).

A version of Theorem 1.1 with hyperplanes replaced by lower dimensional
subspaces (Theorem 1.2 below) was established also in [11]. Denote by
$G_{n,n-k}$ the Grassmanian of subspaces of $\mathbb{R}^{n}$ of codimension
$k$. The class of convex bodies responsible for the $k$-codimensional problem
is the class of generalized $k$-intersection bodies, denoted by
$\mathcal{BP}_{k}^{n}$, which can be defined as follows: Given two star bodies
$K,L$ and $p>0$, their radial $p$-sum is the star body $K\tilde{+}_{p}L$ whose
radial function is

$\rho_{K\tilde{+}_{p}L}^{p}=\rho_{K}^{p}+\rho_{L}^{p}.$

For $k\in[1,n]$, the class $\mathcal{BP}_{k}^{n}$ is defined as the closure in
the radial metric of finite radial $k$-sums of origin-symmetric ellipsoids.

######  Theorem 1.2 ([11, Corollary 1]).

Let $K$ be an origin-symmetric star body in $\mathbb{R}^{n}$. Then for any
even continuous non-negative function $f$,

(3) $\int_{K}f\leq c^{k}d_{\rm
ovr}(K,\mathcal{BP}_{k}^{n})^{k}\left|K\right|_{n}^{k/n}\max_{F\in
G_{n,n-k}}\int_{K\cap F}f,$

where $c>0$ is a universal constant. In particular, if $K$ is an origin-
symmetric star body in $\mathbb{R}^{n}$ of volume $1$ and $f$ is an even
continuous probability density in $K$ then

(4) $\max_{F\in G_{n,n-k}}\int_{K\cap F}f\geq c^{-k}(d_{\rm
ovr}(K,\mathcal{BP}_{k}^{n}))^{-k}.$

An upper bound for the distance to ${\mathcal{BP}}_{k}^{n}$ was obtained in
[13] (the power of the logarithmic term was later corrected in [11,
Proposition 1]).

######  Theorem 1.3.

Let $K$ be an origin-symmetric convex body in $\mathbb{R}^{n}$, and let $1\leq
k\leq n-1$. Then

(5) $d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})\leq
c\sqrt{n/k}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}$

where $c>0$ is a universal constant.

Moreover, since ${\mathcal{BP}}_{k}^{n}\subseteq L_{-k}^{n}$ (this was proven
in [7] and [15], see also [8, Theorem 4.23]), inequalities (2) and (5) imply

(6)
$cn^{\frac{1}{2k}}\leq\sup_{K}d_{\operatorname{ovr}}(K,L_{-k}^{n})\leq\sup_{K}d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})\leq
c\sqrt{n/k}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}.$

The two estimates are tight for large values of $k$ (at least proportional to
$n$). For low values of $k>1$ the lower and upper bounds are far apart as
$n\to\infty$. In this paper we use the same technique as in [5] to prove that
Theorem 1.2 is sharp up to a logarithmic term, thus obtaining the correct (up
to logarithms) order of
$\sup_{K}d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})$. Our main theorem
is the following:

######  Theorem 1.4.

There exists a universal constant $c>0$ such that for every $n\geq 2$ and
$1\leq k\leq n-1$, there exists an origin-symmetric convex body
$K\subseteq\mathbb{R}^{n}$ and an even continuous probability density $f$ in
$K$ such that for every subspace $F\subseteq\mathbb{R}^{n}$ of codimension
$k$,

$\int_{F}f\leq\left(c\sqrt{\frac{\log(n)k}{n}}\right)^{k}.$

As a consequence of Theorems 1.4 and 1.2 we deduce that the estimate (5) is
also asymptotically sharp, up to logarithmic terms.

######  Corollary 1.5.

There exists a universal constant $c>0$ such that for every $n\geq 2$ and
$1\leq k\leq n-1$, there exists an origin-symmetric convex body
$K\subseteq\mathbb{R}^{n}$ such that

$d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})\geq
c\sqrt{n/k}(\log(n))^{-1/2}.$

Combining Corollary 1.5 and Theorem 1.3 we obtain

$c\sqrt{n/k}(\log(n))^{-1/2}\leq\sup_{K}d_{\operatorname{ovr}}(K,{\mathcal{BP}}_{k}^{n})\leq
c^{\prime}\sqrt{n/k}\left(\log\left(\frac{en}{k}\right)\right)^{3/2}$

where the supremum runs over all origin-symmetric convex bodies
$K\subseteq\mathbb{R}^{n}$, and $c,c^{\prime}>0$ are universal constants.

It should be noted that our proof is a simplification of that used in [5] and
[6] since we do not intend to suppress any logarithmic term. We thank prof. A.
Koldobsky for suggesting this approach, and for many other useful discussions
related to this article.

The author was supported by Grant RYC2021-032768-I, funded by the Ministry of
Science and Innovation/State Research Agency/10.13039 /501100011033 and by the
E.U. Next Generation EU/Recovery, Transformation and Resilience Plan.

##  2\. Proof of Theorem 1.4

Theorem 1.4 shall be proven at the end of the section after some auxiliary
results. Throughout the section $c,c^{\prime},c^{\prime\prime}$ will denote
universal constants that may change from one line to the next one. We denote
$\varphi(t)=e^{-t^{2}}$, the euclidean unit ball of $\mathbb{R}^{n}$ is
denoted by $\mathbb{B}_{n}$, the canonical vectors of $\mathbb{R}^{n}$ are
$e_{1},\ldots,e_{n}$ and $\Gamma$ is the gamma function. For $F\in G_{n,n-k}$
and $x\in\mathbb{R}^{n}$, the euclidean distance from $x$ to $F$ is denoted by
$d(F,x)$.

First we recall two elementary inequalities from [4] and [5]:

######  Lemma 2.1 ([4, Lemma 2.4]).

For $0\leq\mu\leq\lambda$ we have

$\lambda^{\mu}\Gamma(\lambda-\mu)\geq\Gamma(\lambda).$

######  Lemma 2.2 ([5, Lemma 3.1]).

Let $Y_{1},\ldots,Y_{N}$ be independent, identically-distributed random
variables attaining values in the interval $[0,1]$. Let $p\in[0,1]$ satisfy
$p\geq\mathbb{E}Y_{i}$. Then,

$\mathbb{P}\left(\frac{1}{N}\sum_{i=1}^{N}Y_{i}\geq 3p\right)\leq e^{-pN}.$

The next elementary lemma concerns the parametrization of the sphere with
respect to the so-called bispherical coordinates. It can be found in [14,
formula (3.9)].

######  Lemma 2.3.

Consider the decomposition $\mathbb{R}^{n}=\mathbb{R}^{m}\times\mathbb{R}^{k}$
with $m+k=n$ and $m,k\in[1,n]$. If $f:S^{n-1}\to\mathbb{R}$ is a measurable
function, then

$\int_{S^{n-1}}f(v)dv=\int_{S^{m-1}}\int_{S^{k-1}}\int_{0}^{\pi/2}f(\cos(\alpha)x,\sin(\alpha)y)\cos(\alpha)^{m-1}\sin(\alpha)^{k-1}d\alpha
dydx$

where integration on the sphere $S^{l}$ is with respect to the $l$-dimensional
Hausdorff measure. The cases $k=1$ and $m=1$ are also included, the measure in
$S^{0}=\\{\pm 1\\}$ being the counting measure.

######  Lemma 2.4.

Let $n\geq 1$ and $\theta$ be a random point uniformly distributed in
$S^{n-1}$. Let $F$ be any subspace of codimension $k$ with $1\leq k\leq n$,
then

$\mathbb{E}(\varphi(d(F,n\theta)))\leq n^{-k/2}.$

(Notice that there is no constant in the right-hand side.)

###### Proof.

By rotational invariance, we may assume without loss of generality that $F$ is
generated by $e_{k+1},\ldots,e_{n}$. We decompose
$\mathbb{R}^{n}=\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ and notice that
$d(F,(x,y))=|x|$.

To find the expectation we integrate, use Lemma 2.3 and then the change of
variables $t=\cos(\alpha)$, $dt=-\sqrt{1-t^{2}}d\alpha$.

(7) $\displaystyle\mathbb{E}(\varphi(d(F,n$ $\displaystyle\theta)))$ (8)
$\displaystyle=\frac{1}{\kappa_{n}}\int_{S^{n-1}}\varphi(nd(F,\theta))d\theta$
(9)
$\displaystyle=\frac{1}{\kappa_{n}}\int_{S^{k-1}}\int_{S^{n-k-1}}\int_{0}^{\pi/2}\varphi(n\cos(\alpha))\cos(\alpha)^{k-1}\sin(\alpha)^{n-k-1}d\alpha
dxdy$ (10)
$\displaystyle=\frac{\kappa_{k}\kappa_{n-k}}{\kappa_{n}}\int_{0}^{\pi/2}\varphi(n\cos(\alpha))\cos(\alpha)^{k-1}\sin(\alpha)^{n-k-1}d\alpha$
(11)
$\displaystyle=\frac{\kappa_{k}\kappa_{n-k}}{\kappa_{n}}\int_{0}^{1}\varphi(nt)t^{k-1}(1-t^{2})^{\frac{n-k-2}{2}}dt,$

where $\kappa_{m}=\frac{2}{\Gamma(m/2)}\pi^{m/2}$ is the $m-1$ dimensional
volume of $S^{m-1}$.

To estimate the constant, we use Lemma 2.1.

(12) $\displaystyle\frac{\kappa_{k}\kappa_{n-k}}{\kappa_{n}}$
$\displaystyle=\frac{2\Gamma(n/2)}{\Gamma(k/2)\Gamma\left(\frac{n-k}{2}\right)}$
(13) $\displaystyle\leq 2\Gamma(k/2)^{-1}\left(\frac{n}{2}\right)^{k/2}.$

For the integral we use a change of variables $s=nt$.

(14)
$\displaystyle\int_{0}^{1}\varphi(nt)t^{k-1}(1-t^{2})^{\frac{n-k-2}{2}}dt$
$\displaystyle=n^{-k}\int_{0}^{n}\varphi(s)s^{k-1}\left(1-\left(\frac{s}{n}\right)^{2}\right)^{\frac{n-k-2}{2}}ds$
(15) $\displaystyle\leq n^{-k}\int_{0}^{n}\varphi(s)s^{k-1}ds$ (16)
$\displaystyle=n^{-k}2^{k/2-1}\Gamma(k/2)$

Combining (12) and (14) we get the result. ∎

The Grassmanian $G_{n,n-k}$ of $k$-codimensional subspaces of $\mathbb{R}^{n}$
can be constructed as the quotient space of the group of rotations $O_{n}$, by
the subgroup of rotations that fix a given $k$-codimensional subspace. As
such, $G_{n,n-k}$ inherits naturally a metric $d$, which is the quotient of
the operator-norm metric. This is defined as

(17) $d_{G}(F_{1},F_{2})=\min\\{\|I-T\|_{\rm op}:T\in O_{n},T(F_{1})=F_{2}\\}$

for $F_{1},F_{2}\in G_{n,n-k}$.

First we need the following simple lemma.

######  Lemma 2.5.

Let $\theta\in S^{n-1}$ be fixed, then the function $F\in G_{n,n-k}\mapsto
d(F,\theta)\in\mathbb{R}$ is Lipschitz with constant $1$.

###### Proof.

Let $F_{1},F_{2}\in G_{n,n-k}$. By the definition of $d_{G}$, there exists
$T\in O_{n},T(F_{1})=F_{2},\|T-I\|_{\rm op}=d_{G}(F_{1},F_{2})$. We have

(18) $\displaystyle|d_{G}(F_{1},\theta)-d_{G}(F_{2},\theta)|$
$\displaystyle=|d_{G}(F_{2},T\theta)-d_{G}(F_{2},\theta)|$ (19)
$\displaystyle\leq|T\theta-\theta|$ (20) $\displaystyle\leq\|T-I\|_{\rm op}$
(21) $\displaystyle\leq d_{G}(F_{1},F_{2}),$

and the lemma follows. ∎

We shall use the following result by Szarek [16].

######  Theorem 2.6 ([16, Proposition 8]).

For every $0\leq\delta\leq\sqrt{2}={\rm diam}(G_{n,n-k})$ there exists a
$\delta$-net in the Grassmanian $G_{n,n-k}$ with respect to the distance
$d_{G}$, of cardinality less than $(c/\delta)^{k(n-k)}$.

######  Proposition 2.7.

There exist a universal constant $c>0$ and $N=cn^{\frac{k}{2}+4}$ points
$\theta_{1},\ldots,\theta_{N}$ such that

$\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta))\leq 4n^{-k/2}$

for every $F\in G_{n,n-k}$.

###### Proof.

Let $\delta=n^{-k/2-1}$ and consider the corresponding $\delta$-net
$\mathcal{F}$ of cardinality $(c^{\prime}n^{k/2+1})^{k(n-k)}\leq
e^{c^{\prime\prime}n^{4}}$. Take $\theta_{1},\ldots,\theta_{N}$ independent
random points uniformly distributed in $S^{n-1}$. By Lemma 2.4 we have
$\mathbb{E}(\varphi(d(F,n\theta_{i})))\leq n^{-k/2}$. Applying Lemma 2.2, for
every $F\in\mathcal{F}$ we have

(22)
$\displaystyle\mathbb{P}\left(\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta_{i}))\geq
3n^{-k/2}\right)$ $\displaystyle\leq e^{-n^{-k/2}N}=e^{-cn^{4}}$
$\mathbb{P}\left(\exists
F\in\mathcal{F},\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta_{i}))\geq
3n^{-k/2}\right)\leq e^{c^{\prime\prime}n^{4}}e^{-cn^{4}}$

which is less than $1$ for all $1\leq k\leq n$, $n\geq 2$, for some universal
constant $c>0$ depending on $c^{\prime\prime}$. We get the existence of
$\theta_{1},\ldots,\theta_{N}$ such that

$\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta_{i}))\leq 3n^{-k/2}$

for every $\forall F\in\mathcal{F}$.

By Lemma 2.5, the function
$F\mapsto\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta_{i}))$ is Lipschitz with
constant $n$. Then for any $F\in G_{n,n-k}$,

$\frac{1}{N}\sum_{i=1}^{N}\varphi(d(F,n\theta_{i}))\leq
n\delta+3n^{-k/2}=4n^{-k/2}$

and the proposition follows. ∎

Let $\gamma_{n}(x)=(2\pi)^{-n/2}e^{-|x|^{2}/2}$ denote the standard Gaussian
probability measure.

######  Definition 2.8.

Let $\theta_{1},\ldots,\theta_{N}$ be the points obtained in Proposition 2.7.
Define the function $f_{0}(x)=\frac{1}{M}\sum_{i=1}^{M}\gamma_{n}(x-p_{i})$,
where

$\\{p_{1},\ldots,p_{M}\\}=\\{n\theta_{1},-n\theta_{1},n\theta_{2},-n\theta_{2},\ldots,n\theta_{N},-n\theta_{N}\\},$

and $M=2N$.

Define the convex set $K_{0}$ as the convex hull of the points
$p_{1},\ldots,p_{M},$ and $\pm ne_{1},\ldots,\pm ne_{n}$.

The set $K_{0}$ and the function $f_{0}$ will satisfy the necessary properties
of Theorem 1.4, after normalization and rescaling. These properties will be
stated separately in the following three propositions.

######  Proposition 2.9.

Let $f_{0}$ be as in Definition 2.8. Let $F$ be any subspace of codimension
$k$ with $1\leq k\leq n-1$, then

$\int_{F}f_{0}=(2\pi)^{-k/2}\frac{1}{M}\sum_{i=1}^{M}\varphi(d(F,p_{i}))\leq
4(2\pi n)^{-k/2}.$

###### Proof.

Let $q_{i}$ be the orthogonal projection of $p_{i}$ onto $F$. Using the change
of variables $y=x-q_{i}$,

(23) $\displaystyle\int_{F}f_{0}$
$\displaystyle=\frac{1}{M}\sum_{i=1}^{M}(2\pi)^{-n/2}\int_{F}e^{-\frac{1}{2}|x-p_{i}|^{2}}dx$
(24)
$\displaystyle=\frac{1}{M}\sum_{i=1}^{M}(2\pi)^{-n/2}\int_{F}e^{-\frac{1}{2}|x-q_{i}|^{2}-\frac{1}{2}d(F,p_{i})^{2}}dx$
(25)
$\displaystyle=(2\pi)^{-k/2}\frac{1}{M}\sum_{i=1}^{M}e^{-\frac{1}{2}d(F,p_{i})^{2}}(2\pi)^{-(n-k)/2}\int_{F}e^{-\frac{1}{2}|y|^{2}}dy$
(26)
$\displaystyle=(2\pi)^{-k/2}\frac{1}{M}\sum_{i=1}^{M}\varphi(d(F,p_{i})).$

The inequality follows from Proposition 2.7. ∎

######  Proposition 2.10.

The convex body $K_{0}$ in Definition 2.8 satisfies

$\left|K_{0}\right|_{n}^{1/n}\leq c\sqrt{\log(n)k},$

where $c>0$ is a universal constant.

###### Proof.

By a result of Gluskin [2, Theorem 2], a polytope inside $\mathbb{B}_{n}$
which has less than $m$ vertices, has volume at most

$\left(c\frac{\sqrt{\log(m/n)}}{n}\right)^{n},$

where $c>0$ is a universal constant. Since $K\subseteq n\mathbb{B}_{n}$,

(27) $\displaystyle\left|K\right|_{n}^{1/n}$ $\displaystyle\leq
c\sqrt{\log((M+2n)/n)}$ (28) $\displaystyle\leq
c\sqrt{\log((cn^{\frac{k}{2}+4}+2n)/n)}$ (29) $\displaystyle\leq
c\sqrt{\log(n)(k/2+3)}$

and the theorem follows with $c>0$ independent of $k$ and $n$, provided $n\geq
2$. ∎

######  Proposition 2.11.

Let $K_{0},f_{0}$ be as in Definition 2.8, then

$\int_{3K_{0}}f_{0}\geq 3/4.$

###### Proof.

Since $K_{0}$ contains the vectors $\pm ne_{i}$, it contains
$\sqrt{n}\mathbb{B}_{n}$. Then $3K_{0}$ contains
$K_{0}+2\sqrt{n}\mathbb{B}_{n}$ and

(30) $\displaystyle\int_{3K_{0}}f_{0}$
$\displaystyle\geq\int_{K_{0}+2\sqrt{n}\mathbb{B}_{n}}f_{0}$ (31)
$\displaystyle\geq\frac{1}{M}\sum_{i=1}^{M}\int_{p_{i}+2\sqrt{n}\mathbb{B}_{n}}\gamma_{n}(x-\delta_{p_{i}})dx$
(32) $\displaystyle=\int_{2\sqrt{n}\mathbb{B}_{n}}\gamma_{n}(x)dx\geq 3/4,$

where the last inequality follows from the Markov-Chebyshev inequality and the
fact that $\int_{\mathbb{R}^{n}}|x|^{2}d\gamma_{n}(x)=n$. ∎

###### Proof of Theorem 1.4.

Consider the set $K=K_{0}/\left|K_{0}\right|_{n}^{1/n}$ and the function
$f:\mathbb{R}^{n}\to\mathbb{R}$ defined by

$f(y)=\left|3K_{0}\right|_{n}\left(\int_{3K_{0}}f_{0}\right)^{-1}f_{0}(\left|3K_{0}\right|_{n}^{1/n}y)$

for $y\in K$, and $f(0)=0$ otherwise.

Then clearly $\left|K\right|_{n}=1$ and $\int_{K}f=1$, so $f$ is an even
probability density in $K$, which is also continuous in $K$.

By Propositions 2.9, 2.10 and 2.11, for any $F\in G_{n,n-k}$,

(33) $\displaystyle\int_{F}f$
$\displaystyle\leq\left|3K_{0}\right|_{n}\left(\int_{3K_{0}}f_{0}\right)^{-1}\int_{F}f_{0}(3\left|K_{0}\right|_{n}^{1/n}y)$
(34)
$\displaystyle=\left|3K_{0}\right|_{n}^{k/n}\left(\int_{3K_{0}}f_{0}\right)^{-1}\int_{F}f_{0}$
(35) $\displaystyle\leq 3^{k}\left(\sqrt{\log(n)k}\right)^{k}\frac{4}{3}\times
4(2\pi n)^{-k/2}$ (36)
$\displaystyle\leq\left(c\sqrt{\frac{\log(n)k}{n}}\right)^{k}.$

∎

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