exchange, $f\longleftrightarrow\not\hskip 13.00005pt(1-f)$, discussed in Sec.
5.4 and Sec. 6.1 arises only at order $O(1)$, as shown explicitly in (214).

Additionally, we observe that in all the cases considered here, we find that
the average $K$-local entropy studied in [20] and the average $G$-local
entropy derived here satisfy:

$\langle S_{KA}\rangle-\langle S_{GA}\rangle=\tfrac{1}{2}\log N+O(1)\,.$ (216)

The difference can be attributed to the entanglement in the magnetic degrees
of freedom probed by $K$-local observables, as discussed in (208).

##  7 Discussion

This paper introduces a mathematical framework for symmetry-resolved
entanglement with a non-abelian symmetry group $G$. The framework relies on
the notion of subsystem defined operationally in terms of a subalgebra of
observables (Sec. 2). In the case of a non-abelian symmetry, this framework
identify symmetry-resolved observables that are $G$-invariant, symmetry-
resolved states that can be prepared and measured by these observables, and a
decomposition of the Hilbert space that generalizes the familiar notions of
direct sum over abelian charges and of tensor product over independent
subsystems (Sec. 3). The framework is general and does not require a built-in
notion of locality or the choice of a Hamiltonian. Once we introduce a notion
of locality, a distinction between $K$-local and $G$-local observables arises:
kinematical observables in a many-body system are naturally local but, in
general, are not invariant under transformations of the group $G$. On the
other hand, invariant observables are generally non-local. $G$-local
observables are both local and $G$-invariant. In various physical settings,
they are the only accessible observables defining a symmetry-resolved
subsystem. Symmetry-resolved entanglement entropy is defined here as the
entropy $S_{GA}$ of symmetry-resolved states restricted to a symmetry-resolved
subsystem (Sec. 4). To illustrate this general framework, we considered the
example of a system invariant under the group $G=SU(2)$, computed the exact
average entanglement entropy and its variance for random symmetry-resolved
states (Sec. 5) and studied its Page curve in the thermodynamic limit (Sec.
6).

In the case of an abelian symmetry, such as the group $G=U(1)$, the
construction presented here reduces to known results [4, 5, 6, 7] because the
relation (102) significantly simplifies the symmetry-resolved decomposition.
In particular, in the analysis of symmetry-resolved entanglement [12, 13, 14],
the block-diagonal form of the reduced density matrix with $U(1)$ symmetry
plays a key role. In fact, the notion of entanglement asymmetry associated
with a projected density matrix has been proposed as a measure of symmetry
breaking [60]. In the non-abelian case, the block-diagonal form (77) arises
once one identifies the generalization (74) of symmetry-resolved subsystems.
The relevant expressions for the $SU(2)$ case are (152) and (143), which are
non-trivial for excited states with $j\neq 0$. It would be interesting to use
this formulation to extend the analysis of entanglement asymmetry as a probe
of $SU(2)$ symmetry breaking [60].

The Page curve was first introduced in [1] as a measure of the typical
entanglement entropy of a random pure state without any constraints. The
distribution of the typical entropy in the presence of an additive constraint
(corresponding to an abelian symmetry) was introduced in [2], where an exact
formula for the average and the variance is given. In this paper, we derived
exact formulas for the average and variance of the distribution of the
entanglement entropy of random pure states with non-abelian symmetry group
$G$, (42)–(43)–(78). When the results of [2] for an abelian symmetry are
applied to the thermodynamic limit of a many-body system, the average over the
distribution reproduces a volume law $V\sim N$ for the entanglement entropy
[4, 5, 6, 7]. Moreover, a $\sqrt{V}$ correction arises at half-system size
because of the abelian constraint. This correction was first observed
numerically [100] and explained in terms of energy conservation [101], and
then derived from number conservation [4]. A global non-abelian symmetry, such
as $SU(2)$, also results in a leading-order volume-law entanglement entropy,
together with a square-root-volume correction at half-system size, as first
found in [20] and derived in Sec. 6 from the asymptotics of the exact formulas
for the typical entropy (See (212) for a comparison). The coefficient (183) of
the volume-law scaling depends on the spin density $j/V$, which can be
generalized to the densities $q_{k}/V$ for the non-abelian charges, i.e., the
$\textrm{rank}(G)$ Casimir operators of a general compact semisimple Lie group
$G$. Moreover, at order $O(1)$ in the thermodynamic limit, an asymmetry under
subsystem exchange arises, $f\longleftrightarrow\not\hskip 13.00005pt(1-f)$,
where $f=V_{A}/V$ is the subsystem-volume fraction. This is a new feature of
non-abelian symmetry-resolved entanglement that we discuss in Sec. 4.2 for a
general group $G$, and illustrate in (173) for a small system and in (214) for
a many-body system in the thermodynamic limit. While for a global symmetry,
this phenomenon is subleading in the thermodynamic limit, it would be
interesting to investigate its consequences in the case of a local symmetry
and its impact on the estimate of the Page time in black hole evaporation [23,
24, 25, 102, 103, 104].

Figure 5: Quantum polyhedra provide a concrete example of $SU(2)$ symmetry-
resolved states that can be probed only using $G$-local observables that
measure their intrinsic geometry. Each spin corresponds to a quantum plane of
fixed area, and the $SU(2)$ invariance in the coupling of angular momenta
corresponds to the closure of the faces of the polyhedron [97, 98].

The results of this paper are general and do not depend on the choice of a
specific Hamiltonian. They depend only on the structure of the Hilbert space
and on the representation of the symmetry group $G$. This group can be
understood as a symmetry of the dynamics, such as the Hamiltonian (2)
discussed in the introduction as a motivation. Entanglement in eigenstates of
a specific quantum-chaotic Hamiltonian with abelian symmetry group $G=U(1)$ is
studied in [8, 9] for the spin-$1/2$ $XXZ$ chain and in [10, 11] for $XXZ$
model and the mixed-field Ising model with a constrained energy window.
Remarkably, they show that the distribution of the entanglement entropy of
mid-spectrum energy eigenstates, computed numerically, agrees with the
analytical distribution found in [2]. We conjecture that the distribution
$P(S_{GA})$ with average and variance (42)–(43)–(78) found in this paper (See
Fig. 3(a)) matches the distribution of the entanglement entropy of energy
eigenstates of quantum-chaotic Hamiltonians with non-abelian symmetry group
$G$, restricted to $G$-local subsystems. It would be interesting to test this
conjecture numerically, extending the state-of-the-art exact diagonalization
of [8, 9, 10, 11] to quantum-chaotic Hamiltonians with a non-abelian symmetry,
such as the random Heisenberg model (2) and the Heisenberg model on a lattice
with local interactions [17], as done in [19, 20] for $SU(2)$. This conjecture
is of direct relevance to recent developments in non-abelian eigenstate
thermalization [18, 19, 20], thermodynamics with noncommuting conserved
charges [81, 82, 26, 83], and quantum many-body scars [87, 88].

Another new phenomenon that arises in the non-abelian case is the distinction
(88) between $K$-local and $G$-local subsystems. The notion of $G$-local
observables plays an effective role in systems with a gapped Hamiltonian and
selection rules that restrict the accessible observables to multiplets in the
energy spectrum (See Fig. 1). $G$-local observables play also a fundamental
role in systems with an intrinsic symmetry that constrains the accessible
observables [44, 45, 46, 47, 48, 49], as it is the case for quantum reference
frames where one adopts an intrinsic perspective [105, 106, 107, 108]. Another
example where $G$-local observables play a central role is the case of the
quantum geometry of a polyhedron [97, 98]. The $G$-local observables
$\vec{S}_{a}\cdot\vec{S}_{b}$ described in Fig. 1 measure the angle between
the faces of the quantum polyhedron, shown in Fig. 5 for different sectors of
spin $j$.

It would be interesting to derive also the exact formulas for the average and
the variance of the probability distribution $P(S_{KA})$ for $K$-local
observables (See Fig. 3). In [20], the asymptotics of the average was computed
for a spin system using a combination of analytical and numerical techniques,
and in (216) we observed that the averages of $S_{KA}$ and $S_{GA}$ differ at
order log-volume in the thermodynamic limit.

The framework introduced in this paper applies both to groups $G$ that act
globally on the system—a rigid symmetry—and to groups $G^{\times N}$ that act
locally at the $N$ nodes of a lattice, i.e., a lattice gauge symmetry [38, 39,
40]. In this case, we expect that the new features of non-abelian symmetry-
resolved entanglement play a central role, and it would be interesting to
explore their effect on the Page curve in lattice gauge theory [109] and spin-
network states in loop quantum gravity [41, 42, 43, 110, 111, 112, 113].
Finally, the framework introduced here assumes a finite-dimensional Hilbert
space. It would be interesting to extend this analysis to quantum field theory
[34, 35, 36, 37] where, in a finite volume and at fixed energy, the
microcanonical sectors of the Hilbert space have finite dimension.

## Acknowledgments

We thank Lucas Hackl and Marcos Rigol for many useful discussions. EB
acknowledges support from the National Science Foundation, Grant No.
PHY-2207851. This work was made possible through the support of the ID# 62312
grant from the John Templeton Foundation, as part of the project “The Quantum
Information Structure of Spacetime” (QISS). The opinions expressed in this
work are those of the authors and do not necessarily reflect the views of the
John Templeton Foundation. PD thanks the Cross-Consortium visiting program of
the QISS Project for supporting his stay at the IGC, where the first part of
this work was done. EB also thanks the Cross-Consortium visiting program of
the QISS Project for supporting his stay at the IQOQI. RK thanks the Blaumann
Foundation for support to participate in the QISS 2023 School where
preliminary results of this project were first presented.

##  Appendix A Dimension of $SU(2)$ intertwiner spaces

In this section we give a derivation of the dimension of the Hilbert spaces
$\mathcal{H}_{GA}^{(\ell)}$, $\mathcal{H}_{GB}^{(j,\ell)}$, and
$\mathcal{H}_{G}^{(j)}$ based on the calculation of the dimension of the
$SU(2)$ invariant spaces

$\nu(j_{1},\ldots,j_{L})=\mathrm{dim}\,\mathrm{Inv}_{G}(\mathcal{H}^{(j_{1})}\otimes\cdots\otimes\mathcal{H}^{(j_{L})})\
.$ (217)

These invariant spaces, also known as intertwiners, are well studied in loop
quantum gravity [41, 42], where they are the fundamental building blocks for
describing quantum geometries [97, 98]. Using techniques typical of
intertwiner calculations, we write $\nu$ in integral form

$\nu(j_{1},\ldots,j_{L})=\int_{SU(2)}\chi^{(j_{1})}(g)\cdots\chi^{(j_{L})}(g)\;\mathrm{d}g\,,$
(218)

where $\mathrm{d}g$ is the Haar measure for the group $SU(2)$ and
$\chi^{(j_{i})}$ is the character of the representation with spin $j_{i}$. We
compute the integral using the class-angle parametrization of the group,
$\chi^{(j)}(g)=\frac{\sin\left(2j+1\right)\theta}{\sin\theta}$, where $\theta$
is the class angle of $g$ and
$\int\mathrm{d}g=\frac{2}{\pi}\int_{0}^{2\pi}(\sin\theta)^{2}\,\mathrm{d}\theta$.

This paper uses the dimension $\nu$ with $N$ spin-$1/2$ and one spin-$j$.
After a few manipulations with elementary trigonometric identities, we find

$\nu(\tfrac{1}{2},\ldots,\tfrac{1}{2},j)=\frac{2^{N}}{\pi}\int_{0}^{2\pi}(\sin\theta)\,(\cos\theta)^{N}\,(\sin\left(2j+1\right)\theta)\;\mathrm{d}\theta\,.$
(219)

Using the residue theorem, we evaluate the integral (219) as a contour
integral on the unit circle of the complex plane. The only contribution to the
integral comes from the pole in the origin

$\textstyle\nu(\tfrac{1}{2},\ldots,\tfrac{1}{2},j)=-\frac{1}{2}\mathrm{Res}_{z=0}\left(\frac{1}{z}\left(\frac{1}{z}-z\right)\left(\frac{1}{z^{2j+1}}-z^{2j+1}\right)\left(\frac{1}{z}+z\right)^{N}\right)\
.$ (220)

We expand the binomial
$\left(\frac{1}{z}+z\right)^{N}=\sum_{k=0}^{N}\binom{N}{k}z^{2k-N}$ and read
the residue from the coefficient of $\frac{1}{z}$. If $N+2j$ is odd, the
integral is vanishing. If $N+2j$ is even, we find

$\nu(\tfrac{1}{2},\ldots,\tfrac{1}{2},j)=\frac{2j+1}{\frac{N}{2}+j+1}\binom{N}{\frac{N}{2}+j}\,.$
(221)

This is the expression also found, for instance, in [19, 20] and derived via
combinatorial methods.

Using the intertwiner methods discussed above, we can now compute the
dimension $\nu$ with $N-N_{A}$ spin-$1/2$, one spin-$j$, and one spin-$\ell$
which appears in the formula for the non-abelian symmetry-resolved
entanglement entropy. We find

$\displaystyle\nu(\tfrac{1}{2},\ldots,\tfrac{1}{2},j,\ell)$
$\displaystyle=\frac{2^{N-N_{A}+1}}{\pi}\int_{0}^{2\pi}(\cos\theta)^{N-N_{A}}\,\big{(}\sin(2j+1)\theta\big{)}\big{(}\sin(2\ell+1)\theta\big{)}\,\mathrm{d}\theta$
(222)
$\displaystyle=\binom{N-N_{A}}{\frac{N-N_{A}}{2}+j-\ell}-\frac{\frac{N-N_{A}}{2}-j-\ell}{\frac{N-N_{A}}{2}+j+\ell+1}\binom{N-N_{A}}{\frac{N-N_{A}}{2}+j+\ell}\
.$ (223)

We note that this formula reduces to the expression (221) for $\ell=0$ or
$j=0$.

To summarize, we have

$\displaystyle D_{j}=$
$\displaystyle\textstyle\;\dim\mathcal{H}_{G}^{(j)}=\nu(\frac{1}{2},\ldots,\frac{1}{2},j)\,,$
(224) $\displaystyle d_{\ell}=$
$\displaystyle\textstyle\;\dim\mathcal{H}_{GA}^{(\ell)}=\nu(\frac{1}{2},\ldots,\frac{1}{2},\ell)\,,$
(225) $\displaystyle b_{j\ell}=$
$\displaystyle\textstyle\;\dim\mathcal{H}_{GB}^{(j,\ell)}=\nu(\frac{1}{2},\ldots,\frac{1}{2},j,\ell)\,.$
(226)

##  Appendix B Laplace approximation and discontinuities

In this section, we review the asymptotic expansion of integrals using the
Laplace method [114] and extend this method to the presence of
discontinuities. We derive an asymptotic expansion for $N\gg 1$ of integrals
of the form

$I=\int_{K}h(x)\,\mathrm{e}^{Ng(x)}\,\mathrm{d}x\,,$ (227)

where $K\subset\mathbb{R}$ and $h(x)$ and $g(x)$ are two real-valued functions
defined on $K$ such that the integral is well defined for large enough $N$. We
first assume that the functions $h$ and $g$ are smooth on $K$. Then, we
present the formula for the case with $h$ continuous but with a discontinuous
first derivative at a point. We assume that the function $g$ has a global
maximum at $x_{0}$ in $K$ where the gradient vanishes, $g^{\prime}(x_{0})=0$,
and the function also vanishes at the maximum $g(x_{0})=0$. The integral is
approximated asymptotically by

$I=h(x_{0})\,\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\left(1+\frac{C_{1}}{N}+o\left(\frac{1}{N}\right)\right)\
,$ (228)

where

$C_{1}=-\frac{1}{2}\frac{h^{\prime\prime}(x_{0})}{h(x_{0})g^{\prime\prime}(x_{0})}+\frac{1}{8}\frac{g^{\prime\prime\prime\prime}(x_{0})h(x_{0})}{h(x_{0})g^{\prime\prime}(x_{0})^{2}}+\frac{1}{2}\frac{g^{\prime\prime\prime}(x_{0})h^{\prime}(x_{0})}{h(x_{0})g^{\prime\prime}(x_{0})^{2}}-\frac{5}{24}\frac{g^{\prime\prime\prime}(x_{0})^{2}h(x_{0})}{h(x_{0})g^{\prime\prime}(x_{0})^{3}}\
.$ (229)

The approximation (228) relies on three main observations.

  1. 1.

Only the immediate neighborhood of $x_{0}$ contributes to the integral.

  2. 2.

We expand the function $g(x)$ around $x_{0}$, and we approximate the
exponential part of the integrand with a narrow Gaussian
$e^{-\frac{N}{2}g^{\prime\prime}(x_{0})(x-x_{0})^{2}}$. This confirms the
first assumption.

  3. 3.

We expand the remaining part of the integrand around $x_{0}$ and extend the
integral to the whole real line.

The integrals of terms of the expansion with the Gaussian ($x_{0}$ is a
maximum, so $g^{\prime\prime}(x_{0})<0$) are given by

$M^{(p)}=\int_{-\infty}^{\infty}(x-x_{0})^{p}\,\mathrm{e}^{N\frac{g^{\prime\prime}(x_{0})}{2}(x-x_{0})^{2}}\mathrm{d}x=\begin{cases}0&p\text{
odd,}\\\
\sqrt{2\pi}(p-1)!!\left(-\frac{1}{Ng^{\prime\prime}(x_{0})}\right)^{\frac{p+1}{2}}&p\text{
even.}\end{cases}$ (230)

If we are interested in the leading order or (228), expanding the integrand up
to order $O(1)$ and combining with $M_{0}$ is sufficient

$h(x_{0})M^{(0)}=h(x_{0})\,\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\ .$
(231)

However, to compute the next-to-leading order (229), we have to expand the
integrand up to order $O(x-x_{0})^{6}$. The relevant terms are

$\displaystyle h^{\prime\prime}(x_{0})M^{(2)}$
$\displaystyle=\textstyle-\frac{1}{N}\frac{1}{2}\frac{h^{\prime\prime}(x_{0})}{g^{\prime\prime}(x_{0})}\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\
,$ $\displaystyle\textstyle
N\left(\frac{1}{24}g^{\prime\prime\prime\prime}(x_{0})h(x_{0})+\frac{1}{6}g^{\prime\prime\prime}(x_{0})h^{\prime}(x_{0})\right)M^{(4)}$
$\displaystyle=\textstyle\frac{1}{N}\left(\frac{1}{8}\frac{g^{\prime\prime\prime\prime}(x_{0})}{g^{\prime\prime}(x_{0})^{2}}+\frac{1}{2}\frac{g^{\prime\prime\prime}(x_{0})h^{\prime}(x_{0})}{h(x_{0})g^{\prime\prime}(x_{0})^{2}}\right)\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\
,$ $\displaystyle
N^{2}\frac{1}{72}g^{\prime\prime\prime}(x_{0})^{2}h(x_{0})M^{(6)}$
$\displaystyle=\textstyle-\frac{1}{N}\frac{5}{24}\frac{g^{\prime\prime\prime}(x_{0})^{2}}{g^{\prime\prime}(x_{0})^{3}}\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\
.$

Collecting them together, we obtain (229).

In the derivations in Sec. 6, we need to compute integrals of the form (227)
with a function $h$ that is not continuous in $x_{0}$. For simplicity, we will
assume that $g(x)$ is still smooth in $x_{0}$. We follow the same strategy,
expanding the integrand around $x_{0}$ and separating the integrals in
$x<x_{0}$ and $x>x_{0}$. The split Gaussian integrals are such that

$\int_{x_{0}}^{\infty}(x-x_{0})^{p}\,\mathrm{e}^{N\frac{g^{\prime\prime}(x_{0})}{2}(x-x_{0})^{2}}\mathrm{d}x=(-1)^{p}\int_{-\infty}^{x_{0}}(x-x_{0})^{p}\,\mathrm{e}^{N\frac{g^{\prime\prime}(x_{0})}{2}(x-x_{0})^{2}}\mathrm{d}x\
,$ (232)

and

$\tilde{M}^{(p)}=\int_{x_{0}}^{\infty}(x-x_{0})^{p}\,\mathrm{e}^{N\frac{g^{\prime\prime}(x_{0})}{2}(x-x_{0})^{2}}\mathrm{d}x=\begin{cases}\frac{1}{2}\sqrt{2\pi}(\frac{p-1}{2})!\left(-\frac{1}{Ng^{\prime\prime}(x_{0})}\right)^{\frac{p+1}{2}}&p\text{
odd,}\\\
\frac{1}{2}\sqrt{2\pi}(p-1)!!\left(-\frac{1}{Ng^{\prime\prime}(x_{0})}\right)^{\frac{p+1}{2}}&p\text{
even.}\end{cases}$ (233)

The leading order of the Laplace approximation is similar to the one found
earlier,

$\left(h(x_{0}^{-})+h(x_{0}^{+})\right)\tilde{M}^{(0)}=\frac{h(x_{0}^{-})+h(x_{0}^{+})}{2}\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\
.$ (234)

Here, we denote with $h(x_{0}^{\pm})$ the right/left limit of the function $h$
in $x_{0}$. If $h$ is continuous in $x_{0}$, the formula (234) reduces to
(231). The next-to-leading order term is

$\left(h^{\prime}(x_{0}^{-})-h^{\prime}(x_{0}^{+})\right)\tilde{M}^{(1)}=\frac{h^{\prime}(x_{0}^{-})-h^{\prime}(x_{0}^{+})}{2}\sqrt{2\pi}\frac{-1}{Ng^{\prime\prime}(x_{0})}\
,$ (235)

which is, in general, non-vanishing if the first derivative of $h$ is
discontinuous in $x_{0}$. The calculation for the next-to-next-leading order
term is similar. To summarize, if the function $h$ is discontinuous at the
maximum $x_{0}$ of $g$, the Laplace approximation for the integral (227) is

$I=\frac{h(x_{0}^{-})+h(x_{0}^{+})}{2}\sqrt{-\frac{2\pi}{Ng^{\prime\prime}(x_{0})}}\left(1+\frac{\tilde{C}_{1/2}}{\sqrt{N}}+\frac{\tilde{C}_{1}}{N}+o\left(\frac{1}{N}\right)\right)\
,$ (236)

where

$\tilde{C}_{1/2}=\frac{h^{\prime}(x_{0}^{-})-h^{\prime}(x_{0}^{+})}{h(x_{0}^{-})+h(x_{0}^{+})}\sqrt{-\frac{2}{N\pi
g^{\prime\prime}(x_{0})}}\ ,$ (237)

and

$\tilde{C}_{1}=\frac{h(x_{0}^{-}){C}_{1}(x_{0}^{-})+h(x_{0}^{+}){C}_{1}(x_{0}^{+})}{h(x_{0}^{-})+h(x_{0}^{+})}\
,$ (238)

where we denote by ${C}_{1}(x_{0}^{\pm})$ the expression (229) computed (as a
limit) in $x_{0}^{\pm}$. Therefore, a discontinuity results in a
$O(1/\sqrt{N})$ correction in the asymptotic expansion.

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# Towards 6G Hyper-Connectivity: Vision,
Challenges, and Key Enabling Technologies

Howon Lee, Byungju Lee, Heecheol Yang, Junghyun Kim, Seungnyun Kim,
Wonjae Shin, Byonghyo Shim, and H. Vincent Poor H. Lee is with Hankyong
National University. B. Lee is with Incheon National University. H. Yang is
with Chungnam National University. J. Kim is with Soonchunhyang University. W.
Shin is with Ajou University. S. Kim and B. Shim are with Seoul National
University. H. V. Poor is with Princeton University. (Co-first authors: H.
Lee, B. Lee, H. Yang, and J. Kim) (Corresponding authors: W. Shin and B. Shim)

###### Abstract

Technology forecasts anticipate a new era in which massive numbers of humans,
machines, and things are connected to wireless networks to sense, process,
act, and communicate with the surrounding environment in a real-time manner.
To make the visions come true, the sixth generation (6G) wireless networks
should be hyper-connected, implying that there are no constraints on the data
rate, coverage, and computing. In this article, we first identify the main
challenges for 6G hyper-connectivity, including terabits-per-second (Tbps)
data rates for immersive user experiences, zero coverage-hole networks, and
pervasive computing for connected intelligence. To overcome these challenges,
we highlight key enabling technologies for 6G such as distributed and
intelligence-aware cell-free massive multi-input multi-output (MIMO) networks,
boundless and fully integrated terrestrial and non-terrestrial networks, and
communication-aware distributed computing for computation-intensive
applications. We further illustrate and discuss the hyper-connected 6G network
architecture along with open issues and future research directions.

###### Index Terms:

6G hyper-connectivity, cell-free massive MIMO, terahertz communications, non-
terrestrial networks, over-the-air distributed computing, scalable deep
learning.

##  I Introduction

Moving toward the post-pandemic world, we are experiencing a new, previously
inconceivable change in our lives. A large part of our daily activities are
going online, and we are witnessing numerous use-cases in shopping, mobility,
healthcare, and manufacturing. As a platform to embrace this dramatic change,
the fifth generation (5G) wireless systems have been standardized and
commercialized in 2019, and as of now, 5G services have been rolled out in
more than 70 countries. Notwithstanding the many technological advances
brought by 5G, there are still many hurdles and challenges in realizing an
envisioned boundless and hyper-connected society where intelligent things such
as machines, vehicles, sensors, and robots are seamlessly connected both in
physical and virtual spaces without limitation on the data rates and/or
transmission delays [1]. To back up these disruptive changes and therefore
make the slogan a reality, we need to prepare the sixth generation (6G) of
wireless networks by identifying key enabling technologies that cannot be
achieved through a simple evolution of 5G.

One important expectation in the 6G era is that machines and things will be
the main consumers of mobile data traffic, and thus there will be more than 55
billion devices connected to the Internet by 2025. These devices will
continuously sense, process, act, and communicate with the surrounding
environment, generating more than 73 zettabytes of data per year [2]. Since 6G
wireless networks should support a variety of form factors with different
service requirements and hardware capabilities, enabling technologies should
be dearly distinct from those of 5G in various aspects.

The primary purpose of this article is to discuss key challenges arising from
these considerations. These include the provision of terabits-per-second
(Tbps) wireless connectivity, zero coverage-hole networks, and pervasive
computing for connected intelligence, along with enabling technologies that
can be used as a core for the 6G system. The key concepts of 6G hyper-
connectivity are illustrated in Fig. 1.

After the introduction, the rest of this article is organized as follows. In
Section II, we address the challenges facing 6G hyper-connectivity. In Section
III, we present the key enabling technologies for realizing this 6G vision. In
Section IV, we discuss future research issues and provide our concluding
remarks.

Figure 1: Key concepts of 6G hyper-connectivity.

##  II Challenges and Prospects Facing 6G Hyper-Connectivity

###  II-A Tbps for Immersive User Experience

Since the target of 5G is to support $20$ gigabits-per-second (Gbps) peak data
rate and $0.1$ Gbps user-experienced data rate, it is not too difficult to
support advanced multimedia services such as $8$K augmented reality (AR)
streaming, $16$K virtual reality (VR) streaming, and extended reality (XR)
requiring beyond-gigabit data rates. It is however not easy to support truly
immersive mobile services such as digital twin or metaverse since the virtual
model accurately reflecting the physical world using XR devices or high-
fidelity mobile holographic displays requires a rate of up to Tbps [1]. For
example, a full-view $24$K VR transmission with a $64$ pixels per degree (PPD)
minimum resolution, a $200\,\text{Hz}$ minimum refresh rate, and a
$300\\!\\!:\\!\\!1$ compression rate requires at least $6.37$ Gbps. Clearly,
this level of rate cannot be supported anytime anywhere in the current 5G
systems since the typical user-experienced data rate of 5G is in the range of
0.1$\sim$1 Gbps.

One way to support emerging services requiring tremendous data rates is to
exploit the terahertz (THz) frequency band. The THz band offers colossal
spectrum but the THz waves suffer high attenuation due to the attendant
significant path loss, molecular absorption, and atmospheric attenuation. One
straightforward option to overcome these problems is to densify the network,
meaning that we reduce the distance between the base station (BS) and mobile
device. In this ultra-dense scenarios, pencil beamforming via ultra-massive
multi-input multi-output (UM-MIMO) plasmonic nano-antenna arrays can be useful
in providing 10 to 100$\times$ improvement in data rate [3]. While the
densified small cells equipped with massive multi-input multi-output (MIMO)
systems are a good fit for THz communication systems, their deployment will
make the notions of cell and handover obsolete. Thus, we need to design the
cell-free networks from the scratch by considering various technical
challenges, such as beam tracking/management, user association and BS
coordination, and synchronization and initial access. Other key enabling
technologies to support Tbps data rates include orbital angular momentum (OAM)
multiplexing, full-duplex, and spatial modulation-MIMO (SM-MIMO).

###  II-B Zero Coverage-Hole Networks

Another more important goal of 5G systems is to provide advanced wireless
connectivity for a wide variety of vertical applications including self-
driving cars and drones, robot-aided smart factories and healthcare, and
remote control for disaster scenarios. To accommodate such applications,
wireless access should be available anywhere, meaning that there should be no
coverage holes. Although many network operators claim nationwide coverage for
5G, there are still many coverage holes in places such as tunnels, basements,
elevators, mountains, and oceans. Moreover, higher frequencies (3.5 GHz at FR1
and 28 GHz at FR2) used in 5G systems tend to be vulnerable to blockage owing
to their high directivity and severe path loss. For these reasons, more often
than not we experience link failure in our daily use of 5G services.

Recently, several new approaches to enhance the coverage and ensure universal
and seamless connectivity have been proposed. An intelligent reflecting
surface (IRS), an approach that provides an artificial path in the absence of
line-of-sight (LOS) links, is a viable candidate to eliminate coverage holes,
particularly in dense urban areas [4]. Wave-controlled holographic MIMO
surfaces that flexibly shape electromagnetic waves using sub-wavelength
metallic or dielectric scattering particle structures is another good option.
In remote areas such as deserts, oceans, and mountain areas, in which the
deployment of terrestrial networks (TNs) is almost impossible, non-terrestrial
networks (NTNs) with satellites, high-altitude platform stations (HAPSs), and
unmanned aerial vehicles (UAVs) can be an attractive solution because they
ensure coverage without relying on the ground infrastructure. By the
deliberate combination of TNs and NTNs [5], one can dramatically improve the
coverage and thus generate the network with virtually no coverage hole.

(a)

(b)

Figure 2: Left-hand side: Illustrations of 6G cell-free massive MIMO network;
right-hand side: CDFs of user SINR in cell-free massive MIMO networks, small
cell, and centralized massive MIMO networks.

###  II-C Pervasive Computing for Connected Intelligence

With the advent of truly immersive mobile services, such as XR and metaverse,
computational overhead has become a serious concern since these services
require high energy consumption, end-to-end latency, and processor and memory
overhead. In 5G, mobile edge computing (MEC) has been widely used to delegate
computation-intensive tasks to the edge (a local server near a BS). However,
MEC would not be enough to support future applications. In a VR service, for
example, a mobile VR headset must provide photorealistic and $6$ degrees of
freedom (6-DoF) rendered images within 20 ms of motion-to-render-to-photon
(M2R2P) latency. To control this overhead solely in a mobile device or offload
it to a nearby BS will be by no means easy so that new paradigm to deal with
the surge of computation is required.

Fortunately, as the density of wireless networks increases and beyond-gigabit
transmissions can be supported using mmWave and THz band transmissions, one
can exploit computational resources in nearby BSs and mobile devices that are
equipped with high performance computing processors. For instance, a vehicle
equipped with a high-performance battery ($1000\,\text{Wh/L}$) and powerful
graphics processing units (GPUs) can perform environment sensing, learning,
and computation-oriented tasks and send the results to the destination device.
Particularly, computation offloading is useful for the artificial intelligence
(AI) applications, such as object detection and classification, speech
recognition, and metaverse control, since these tasks usually require heavy
deep neural network (DNN) computation, but the inference results are often
very simple.

In order to facilitate the boundless use of computation resources over the
network, one should use advanced wireless distributed computing techniques.
Decentralized learning obtaining a global model from localized data via peer-
to-peer communication among mobile devices can be one good choice. When
security matters, federated learning where the mobile devices jointly train
the global model without exchanging the local dataset with the central server
would be a proper choice. Yet another option to support large-scale models is
split learning, in which large-scale DNNs are divided into several pieces and
then trained separately on mobile devices.

##  III Key Enabling Technologies for 6G Hyper-Connectivity

In this section, we discuss key enabling technologies to achieve beyond Gbps
for immersive user experience, zero coverage-hole networks, and pervasive
computing for connected intelligence.

###  III-A Distributed $\&$ Intelligence-Aware Cell-Free Massive MIMO Network

In the 6G era, a network will be heavily densified and thus, the areal density
of BSs will be comparable to, or even surpass, the density of mobile devices.
As the coverage of a cell gets smaller, the distance between BS and mobile
will also be shorter. To make the most of the proximity between BSs and mobile
devices, a network should be designed such that a group of small cells
coherently serve users in a user-centric manner (see in Fig. 2(a)) [6]. In
most urban scenarios, user-centric cell association will generate multiple LOS
links between BSs and mobile so that the mobile can easily achieve a high
degree of macro-diversity gain and spectral efficient improvement.

To evaluate the efficacy of the cell-free massive MIMO network, we plot
cumulative distribution functions (CDFs) of user signal-to-interference-and-
noise ratio (SINR) in mmWave ($100\,\text{GHz}$) and THz ($1\,\text{THz}$)
communication scenarios. One can see from Fig. 2(b) that the SINR variation of
the cell-free massive MIMO network is much smaller, meaning that the cell-free
massive MIMO network can well suppress the inter-cell interference through the
cooperative beamforming of BSs. For example, in the THz systems, the cell-free
massive MIMO network can provide at least $7\,$dB SINR to more than $95\%$ of
users while the conventional small cell network guarantees only -$2.5\,$dB. We
also observe that the SINR gain of the THz band is higher than that for the
mmWave band, meaning that it can effectively compensate for the huge path loss
of the THz systems. To fully enjoy the benefits of cell-free massive MIMO, we
need to address the following challenges.

Intelligence-aware network architecture:  In order to provide the user-centric
coverage through the joint beamforming of cooperating BSs, a cell control
information, channel state information (CSI), and transmit data of local BSs
should be shared with the central unit (CU) through the fronthaul link.
Sharing such a large information among all network entities will cause a
significant fronthaul overhead, not to mention the tremendous computational
overhead required for the channel estimation, data transmission/reception, and
resource allocation [7]. A promising approach to mitigate the overhead is the
decentralized processing. One such example is the multi-agent deep
reinforcement learning (MA-DRL) where the operations like beamforming,
resource allocation, and user association are performed locally and
individually by the DRL agent in each small cell.

Multi-sensory data fusion for channel prediction and beamforming: In order to
realize a truly zero coverage-hole cell-free massive MIMO network, the network
should have mobility-aware functions such as fast-fading channel estimation,
mobile trajectory tracking, blockage prediction, and dynamic BS clustering and
beamforming. One promising direction is to use the multi-modal sensory data
(e.g., RGB depth vision information of camera, range information of LiDAR,
SLAM-based user location) along with the computer vision techniques to
identify the 3D shape of nearby wireless environments (e.g., obstacle
locations, LOS and non-line-of-sight (NLOS) paths, and mobility patterns). In
this scheme, using a set of panoramic images obtained from the stereo camera,
the DL-based object detection can extract the location (3D Cartesian
coordinate $(x,y,z)$) of target objects in an image and then determine its
class. In acquiring the location information of a mobile device from input
images, a convolutional neural network (CNN) that extracts the spatially
correlated feature using the convolution filter can be employed. Once the
location of the mobile is identified, by measuring the angle and distance from
the BS to the target object (e.g., mobile phone, vehicle, or urban air
mobility (UAM)), the mmWave and THz beam can be generated without a
complicated beam management process.

Computation-oriented end-to-end cell-free massive MIMO network: In order to
support the computation-intensive and/or latency-critical applications, the
future 6G network should have the computation offloading functionality [8].
Computation offloading can be realized by utilizing the local computing power
at the BSs as well as cloud computing at the CU. For example, latency-
sensitive computation tasks will be processed at the BSs in proximity to the
user, and the computation-intensive tasks can be handled via the cooperation
of BSs and CU.

To ensure successful computation offloading, an integrated cell-free network
design exploring both communication and computational perspectives needs to be
suggested. In general, it should be guaranteed that computing operations can
be delivered by reliable and low-latency communication links through a user,
BSs, and CU. From a computation perspective, edge servers located at the BSs
may have limited computing capabilities in terms of computing power, memory
size, and energy consumption compared to the central server at the CU. In
addition, since the user needs to offload computation tasks to nearby BSs, the
randomness in computation latency at each BS should be taken into account to
guarantee low-latency and reliable computing services.

Figure 3: Satellite communication systems with different altitudes: Round-trip
delay and single satellite coverage as a fraction of the Earth’s surface.

Integrated RF and optical X-haul provisioning: In the cell-free massive MIMO
network, fronthaul and backhaul overhead is a serious yet underrepresented
problem. Recently, advanced fronthaul/backhaul techniques have been proposed.
Replacing the current CPRI interface with the optical fiber-based
fronthaul/backhaul is a simple option but will increase the CAPEX
significantly. A radio stripe network-based fronthaul and backhaul where the
BSs are serially connected using daisy-chain architecture to the same cables
would be a cost-effective option for the densely deployed BSs. Also, an
integrated access and backhaul (IAB) network where only a few BSs are
connected to traditional fiber infrastructures while other BSs wirelessly
relay the fronthaul/backhaul traffic would be a promising option. Yet another
simple solution is free-space optical (FSO) communication where the modulated
optical beams are transmitted through the free space. When the LOS link is
guaranteed, FSO-based backhaul can offer several Gbps data rates with a much
lower deployment cost than the optical fiber-based backhaul. When the LOS link
is absent, optical IRS, a planar metasurface that controls the optical
propagation environment by adjusting the phase shifts of reflecting elements
can be used to offer the virtual LOS link between the BSs and CU.

Rank-sufficient THz LOS communications using cell-free massive MIMO network:
In the THz band, due to the severe path loss and high directivity, the
conventional MIMO techniques exploiting the rich scattering of multipath
environments will not work. In the case where the LOS path is the dominant,
the cell-free massive MIMO systems can be an appealing solution to address the
rank-deficiency issue of THz communications. Specifically, through the joint
transmission of small BSs, the cell-free massive MIMO systems can generate a
virtual multipath propagation. Further, the shortened communication distance
along with the extremely large number of transmit antennas incur a spherical
wavefront which creates a rich non-linear pattern of phase variations and
therefore, increases the channel rank.

###  III-B Boundless and Fully Integrated Terrestrial $\&$ Non-Terrestrial
Networks

In remote and rural areas where the existing terrestrial cellular network is
unreachable [9], NTNs will be a useful means of providing cost-effective and
high-speed internet services. Depending on the flying altitude and coverage
area, NTN devices can be classified as UAV, HAPS, and satellite. Satellite
stations are further divided into geostationary orbit (GEO), medium earth
orbit (MEO), and low earth orbit (LEO) based on their orbital altitude.

In Fig. 3, we plot the round-trip delay (RTD) and single satellite coverage on
the Earth’s surface for various altitude levels. When a satellite revolves in
orbit close to the surface of the earth, a faster response time can be
guaranteed, but the satellite’s decreases owing to its proximity to the
ground. For example, only three equally spaced GEO satellites are needed to
can provide near-global coverage because each GEO satellite covers almost 35%
of the Earth’s area, but RTD in this case exceeds 0.5 second. Conversely, the
RTD of the LEO satellite is less than 0.1 second, but the coverage will be
just 0.2% at an altitude of 550 km. One can infer from this discussion that a
large number of LEO satellites, often called LEO mega-constellation, would be
a good option to guarantee the latency in the order of a few tens of
milliseconds and global coverage.

We note that there exists a trade-off between the ground observer’s minimum
elevation angle and link quality (see Fig. 3). If we increase the minimum
elevation angle, so is the LOS probability of the satellite link; however, the
beam coverage and visible time of the satellite will be reduced. To balance
this trade-off, the ground station’s minimum elevation angle should be
carefully determined in practice. Recently, SpaceX decided to change the
minimum elevation angle of Starlink LEO satellites from 40° to 25° (granted by
the FCC in April 2021) to improve coverage with a small number of LEO
satellites [10].

In analyzing the NTN communication of satellite and aerial devices, areal
capacity ($\mathrm{bits/s/m^{2}}$) is no longer useful; therefore, a new
metric called volumetric traffic capacity ($\mathrm{bits/s/m^{3}}$) that
measures the quality of user experience in 3D space is needed. To obtain a
sufficient volumetric traffic capacity to ensure universal global connectivity
for 6G systems, the following issues must be addressed.

Mobility-aware channel modeling and tracking in 3D space: Since the NTN
terminals are located at a high altitude and are moving fast with respect to
the ground stations, LOS propagation paths are guaranteed, but the wireless
signals suffer significant Doppler shift. The first step in addressing the
distortions caused by high mobility and a large coverage area is an accurate
3D channel model that accounts for the Doppler shift and propagation delay. In
addition, dynamic channel prediction based on the real-time location tracking
of NTN terminals is of great importance to achieve agile and effective NTN
management.

Boundless multi-layered TN/NTN architecture: To provide sufficient and
continuous coverage everywhere, the large LEO satellite constellations should
be combined with GEO/MEO using the coverage-aware protocol and NTN-backhaul
connection. In fact, a boundless multi-layered architecture fully integrating
TNs and NTNs is essential to guarantee the reliable connection in the ocean
and desert, as well as high-speed trains and airplanes [11]. UAV and HAPS play
a key role in improving the connection quality because they can be used as
relays between the LEO/MEO/GEO satellite and ground stations. In addition, an
integrated network protocol that includes multi-layer routing, network
mobility management, and resource allocation should be introduced.

Seamless inter-beam/satellite/orbit handover: Owing to the high-speed movement
of the LEO satellites, the maximum visible time of the satellites at the
ground station is at most 10$\sim$20 minutes, causing a frequent handover even
for a fixed ground station. To mitigate this problem, LEO satellites should be
connected to nearby satellites in the same or adjacent orbits through laser or
radio frequency (RF) links. This so-called inter-satellite link (ISL)
connectivity supports relatively low end-to-end latency even for the long
distance communication scenario. Candidate techniques to establish an
efficient ISL connectivity include spatial-mode diversity and multiplexing,
wavelength division multiplexing, and adaptive beam control using
pointing/acquisition/tracking (PAT) process. Another challenge of NTNs is that
the cell size varies according to the elevation angle of the NTN terminals;
therefore, conventional handover techniques might not be suitable. To address
this challenge, a forecast-based handover decision on the basis of satellite
position estimation is required.

Figure 4: Over-the-air distributed computing scenario where a VR headset
assigns linear computation tasks to four edges with uncoded/coded offloading.

3D cell planning and fast beam management: Since the cell size varies
according to the altitude and elevation angle of the satellite, we need to
introduce 3D cell planning based on the orbits of satellites. A satellite with
a higher altitude and lower elevation angle tends to form a larger cell size
owing to its wider beam footprint on the ground. Specifically, the GEO and LEO
satellites have beam widths of up to 3,500 km and 1,000 km at an altitude of
approximately 600 km, respectively. In addition, 3D cell types can be
classified into earth-moving and earth-fixed cells with respect to the Earth’s
surface, depending on whether the satellite beams can be fixed or steered. The
earth-fixed cell should be designed to be highly robust to satellite beam
pointing errors since even a very small error makes the satellite beam
footprint placed tens of kilometers away from the initially designated area on
the Earth; whereas, the earth-moving cell may require more frequent handovers
because of the dynamics of beam movement on the ground. As for the cell
planning in multi-layered TN/NTN architecture, a truncated octahedron-shaped
cell planning distinguished from the hexagon-shaped 2D cell layout is desired
when considering the volumetric quotient in 3D space.

Intelligent spectrum management and NTN system optimization: To achieve
seamless and broad coverage in integrated terrestrial and non-terrestrial
networks, powerful spectrum management scheme is needed. To this end,
intelligent and dynamic cooperation between primary and cognitive users is
preferred over the semi-static spectrum sharing. Note that because the S-band
(i.e., 2170$\sim$2200 MHz) is adjacent to the 3G/4G LTE band (i.e.,
2110$\sim$2170 MHz), the large Doppler shift of the high-speed satellites
causes severe interference in the adjacent band, deteriorating the service
quality of 3G/4G TN services. A possible option to mitigate the interference
is the on-board pre-compensation of the Doppler shift at the center of the
beam on the ground while the residual Doppler shift can be corrected at each
receiver side. To further improve spectral efficiency, we should have an
appropriate frequency reuse pattern among multiple beams, available channels,
and antenna gains.

Figure 5: Task completion times when a user offloads linear computation tasks
to four edges over two different channel conditions (Scenario 1: LOS channel
at a distance of 15m; Scenario 2: NLOS channel at a distance of 300m). Figure
6: Hyper-connected 6G network architecture

###  III-C Over-the-air Distributed Computing for Artificial Intelligence
Applications

A future trend in dealing with the relentless growth in computing-intensive
applications is pervasive computing, in which mobile devices offload
computational overhead to network edges or nearby devices. Notwithstanding the
rosy outlook, a number of issues have to be addressed since there is a
fundamental trade-off between computation and communication. In fact, when one
tries to offload computational overhead, one should investigate the overhead
of the communication link. Fig. 4 depicts computation offloading scenarios
where a VR headset assigns linear computations to four edges with various
offloading schemes [12]. Since the reduction in the communication load over
the network increases the computation overhead at the edges, we need to come
up with a proper offloading scheme optimized for the computation capability
and network conditions. One thing to note is that the wireless channels
experienced by devices and their computing capabilities are quite different
among devices so that there is a large variation in the task completion time.
Particularly, this problem is critical for the ultra-reliable and low-latency
communications (URLLC) services since the task completion time is determined
by the device with the worst channel condition or straggling/malfunctioning
devices (e.g., edge failure in Fig. 4). For the seamless integration of
communication and computing, the following approaches can be considered.

Communication-aware pervasive computation design:  To alleviate the
difficulties caused by the channel uncertainty, we need to exploit the mmWave
and THz frequency bands. If sufficient bandwidth is provided and LOS channel
conditions are guaranteed for every computing node, wireless links will not be
a serious bottleneck. Obviously, this ideal condition cannot be guaranteed
always, so we need to come up with an environment-aware and communication-
friendly computing mechanism. For example, the communication load can be
reduced by quantization, compression, and sparsification of the transmit
information as long as it does not severely degrade the task quality. For
special cases where the (weighted) average of the computation tasks is needed,
we can design the system such that the computation results are added during
the wireless transmission of the same time/frequency resources. This so-called
over-the-air computation can significantly reduce the communication and
computation overhead.

Collaborative and robust coded computation offloading: A naïve approach to
exploit multiple computing nodes is to split the whole computation task into
multiple pieces and then distribute them equally to multiple nodes. This
split-based offloading will not be effective in the presence of link failure
and straggling/malfunctioning devices, since the whole task can be completed
only when all subtasks are finished. One way to avoid this problem is to
replicate a part of computation tasks and assign them to the multiple devices.

One option to overcome difficulties caused by the channel heterogeneity, link
failure, and straggling/malfunctioning is to apply coding techniques to
distributed task allocation. By introducing redundancy in distributed tasks,
coded distributed computing can be more resilient to the link and edge
failures. For example, if computation tasks are designed using maximum
distance separable (MDS) codes, one can recover the straggling tasks from
other completed tasks [13]. As illustrated in Fig. 5, one can achieve the
better computation-communication trade-off by assigning coded computation sub-
tasks to computing nodes.

Scalable and over-the-air deep learning: To perform a large-scale deep
learning (DL) task over wireless networks, we must decide how to allocate the
tasks among the cloud, network edges, and the mobile devices [14]. For
instance, network edges perform DNN training, and the mobile device performs
the inference using the trained model from edges. To perform the flexible DL
using abundant network resources, we can consider the following: 1) federated
learning, which trains local models on each device and then passes updated
parameters to the central cloud for a global model; 2) distributed learning,
in which each device communicates directly with its neighbors; and 3) split
learning, in which devices or the central cloud trains each split model
individually.

These DL techniques can be combined with over-the-air techniques [15] to
effectively aggregate large-scale distributed data in ultra-low-latency
scenarios. In addition, a scalable approach can be applied to resource-
constrained mobile devices to reduce communication costs by leveraging
gradient clustering, correction, and quantization methods.

##  IV Summary and Outlook

In this article, we discussed the vision and challenges to achieve the hyper-
connected society in 6G. We have identified several key enabling technologies
and their roles in supporting ultra-broadband, ubiquitous 3D, and computation-
aware connectivity in 6G. With these disruptive technologies, 6G is envisioned
to collapse the existing boundaries of space/air/ground networks,
communication/computing/intelligence capabilities, as well as physical/virtual
worlds. We anticipate that this will bring the truly 6G hyper-connectivity
society to humans, machines, and sensors in ubiquitous 3D global coverage
(beyond ground-level 2D connectivity for 5G) even in space, oceans and
desserts. In Fig. 6, we summarized the hyper-connected 6G network architecture
by highlighting the key enabling technologies along with other remaining major
challenges.

Going forward, achieving Tbps data rates, zero coverage holes, and pervasive
computing for connected intelligence will contribute to reducing differences
in social and regional infrastructures and economic opportunities, thereby
ameliorating many social issues such as the digital divide, regional
polarization, and education inequality. Our hope is that this article will
spark further interest and expedite the research activities for 6G.

## References

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# Commuting Probability of Compact Groups

Alireza Abdollahi  Department of Pure Mathematics, Faculty of Mathematics and
Statistics, University of Isfahan, Isfahan 81746-73441, Iran.
<EMAIL_ADDRESS>and  Meisam Soleimani Malekan  Department of Pure
Mathematics, Faculty of Mathematics and Statistics, University of Isfahan,
Isfahan 81746-73441, Iran; Institute for Research in Fundamental Sciences,
School of Mathematics, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract.

For any (Hausdorff) compact group $G$ with the normalized Haar measure
${\mathbf{m}}_{G}$, denote by ${\rm cp}(G)$ the probability
${\mathbf{m}}_{G\times G}(\\{(x,y)\in G\times G\;|\;xy=yx\\})$ of commuting a
randomly chosen pair of elements of $G$. Here we prove that if ${\rm
cp}(G)>0$, then there exists a finite group $H$ such that ${\rm
cp}(G)=\frac{{\rm cp}(H)}{|G:F|^{2}}$, where $F$ is the FC-center of $G$ i.e.
the set of all elements of $G$ whose conjugacy classes are finite and $H$ is
isoclinic to $F$ with ${\rm cp}(F)={\rm cp}(H)$. The latter equality enables
one to transfer many existing results concerning commuting probability of
finite groups to one of compact groups. For example, here for a compact group
$G$ we prove that if ${\rm cp}(G)>\frac{3}{40}$ then either $G$ is solvable
or, else $G\cong A_{5}\times T$ for some abelian group $T$, in which case
${\rm cp}(G)=\frac{1}{12}$; where $A_{5}$ denotes the alternating group of
degree $5$.

###### Key words and phrases:

Commuting probability; Compact groups

###### 2010 Mathematics Subject Classification:

20E18; 20P05

##  1\. Introduction and Results

Let $G$ be a (Hausdorff) compact group. Then $G$ has a unique normalized Haar
measure denoted by ${\mathbf{m}}_{G}$. The set ${\mathcal{A}}(G):=\\{(x,y)\in
G\times G\;|\;xy=yx\\}$ is a measurable set (actually a closed set) of
$G\times G$ and its measure in $G\times G$ will be denoted by ${\rm cp}(G)$.
In the case $G$ is finite, ${\rm cp}(G)$ has been extensively studied for
example see [1] and [3] and references therein. It is famous (see e.g. [2])
that if $G$ is finite, ${\rm cp}(G)\leq\frac{5}{8}$ whenever $G$ is non-
abelian. The latter is generalized for any non-abelian compact group in [2].
The rationality of ${\rm cp}(G)$ is clear in the case of finite $G$ and for
compact groups it is proved in [5, Theorem 3.11]. It is also shown in [5,
Theorems 1.1 and 1.2] that if ${\rm cp}(G)>0$, then $Z(FC(G))$ is open in $G$,
where $Z(FC(G))$ is the center of the FC-center $FC(G)$ of $G$, i.e. $FC(G)$
consists of elements of $G$ whose conjugacy classes are finite. In the latter
case the derived subgroup of $FC(G)$ is also finite. We will use these results
in the sequel.
Here we prove a formula relating the commuting probability of compact groups
to one of finite groups.

######  Theorem 1.1.

For any compact group $G$ there exists a finite group $H$ such that ${\rm
cp}(G)=\frac{{\rm cp}(H)}{|G:FC(G)|^{2}}$. In particular, if ${\rm cp}(G)>0$,
the finite group $H$ can be chosen so that it is isoclinic to $FC(G)$ and
${\rm cp}(FC(G))={\rm cp}(H)$.

Here we are using the usual convention that if $|G:FC(G)|$ is infinite, then
$\frac{1}{|G:FC(G)|}=0$. Recall that, following [4], two groups $H$ and $K$
are called isoclinic if there exist group isomorphisms
$\alpha:H/Z(H)\rightarrow K/Z(K)$ and $\beta:H^{\prime}\rightarrow K^{\prime}$
such that for all $x,y\in H$ and all $x^{\prime}\in(xZ(H))^{\alpha}$ and
$y^{\prime}\in(yZ(H))^{\alpha}$, $[x^{\prime},y^{\prime}]=[x,y]^{\beta}$,
where for a group $T$, $Z(T)$ and $T^{\prime}$ are denoting the center and the
derived subgroups of $T$, respectively. The alternating group of degree $5$ is
denoted by $A_{5}$ as usual. In view of Theorem 1.1 it is now possible to
generalize some of results in finite case for commuting probability to compact
groups. For example we give the following.

######  Theorem 1.2.

Let $G$ be a compact group. Then
(1) if ${\rm cp}(G)>\frac{1}{4}$ then both $G^{\prime}$ and $G/Z(G)$ are
finite.
(2) if ${\rm cp}(G)>\frac{3}{40}$ then either $G$ is solvable or, else $G\cong
A_{5}\times T$ for some abelian group $T$, in which case ${\rm
cp}(G)=\frac{1}{12}$.

The second part of Theorem 1.1 has already been proved in [1, Theorem 12] for
finite groups and we will use it in our proof for the compact case.

##  2\. Proof of Theorem 1.1

Let $n=|G:Z|$ and $m=|F:Z|$, where $F$ is the FC-center of $G$ and $Z$ is its
center. By [5, Theorem 1.2] we may assume that ${\rm cp}(G)>0$ so that $n$ and
$m$ are both finite. Suppose that $F=\cup_{i=1}^{m}x_{i}Z$ and
$G=\cup_{j=1}^{n}y_{j}Z$, where $x_{i}\in F$ and $y_{j}\in G$. Let
$c_{\ell,k}=1$ if $x_{\ell}x_{k}=x_{k}x_{\ell}$ and $c_{\ell,k}=0$ otherwise.

We use the following in the sequel.

(1) If $y\in G\setminus F$, then ${\mathbf{m}}_{G}(C_{G}(y))=0$, where
$C_{G}(y)=\\{a\in G\;|\;ay=ya\\}$.
(2) By (1) if $y_{j}\in G\setminus F$, ${\mathbf{m}}_{G}(x_{i}Z\cap
C_{G}(y_{j}))=0$. If $y_{j}\in F$, we may assume without loss of generality
that $y_{j}=x_{\ell}$ for some $\ell$.
(3) $x_{i}Z\cap C_{G}(x_{\ell})=\begin{cases}x_{i}Z&{\rm if}\;c_{\ell,i}=1\\\
\varnothing&{\rm otherwise}\end{cases}$.
(4) For any $x\in F$ $C_{G}(x)\cap y_{j}Z=\begin{cases}y_{j}Z&{\rm if}\;x\in
C_{G}(y_{j})\\\ \varnothing&{\rm otherwise}\end{cases}$.

Let ${\mathbf{1}}_{{\mathcal{A}}(G)}$ be the characteristic function of
${\mathcal{A}}(G)=\\{(x,y)\in G\times G\;|\;xy=yx\\}$.
The following two first equlities are from [2].

$\displaystyle{\rm cp}(G)$ $\displaystyle=\int_{G\times
G}{\mathbf{1}}_{{\mathcal{A}}(G)}(x,y)d(x,y)=\int_{G}{\mathbf{m}}_{G}(C_{G}(x))dx\;\;\;{\rm
by\;Fubini\;theorem}$
$\displaystyle\overset{(1)}{=}\int_{F}{\mathbf{m}}_{G}(C_{G}(x))dx$
$\displaystyle=\sum_{i=1}^{m}\int_{x_{i}Z}{\mathbf{m}}_{G}(C_{G}(x))dx=\sum_{i=1}^{m}\sum_{j=1}^{n}\int_{x_{i}Z}{\mathbf{m}}_{G}(C_{G}(x)\cap
y_{j}Z)dx$
$\displaystyle\overset{(2)}{=}\sum_{i=1}^{m}\sum_{j=1}^{n}\int_{x_{i}Z\cap
C_{G}(y_{j})}{\mathbf{m}}_{G}(y_{j}Z)dx=\sum_{i=1}^{m}\sum_{j=1}^{n}\int_{x_{i}Z\cap
C_{G}(y_{j})}{\mathbf{m}}_{G}(Z)dx$
$\displaystyle={\mathbf{m}}_{G}(Z)\sum_{i=1}^{m}\sum_{j=1}^{n}\int_{x_{i}Z\cap
C_{G}(y_{j})}{\mathbf{1}}_{x_{i}Z\cap
C_{G}(y_{j})}(x)dx={\mathbf{m}}_{G}(Z)\sum_{i=1}^{m}\sum_{j=1}^{n}{\mathbf{m}}_{G}(x_{i}Z\cap
C_{G}(y_{j}))$
$\displaystyle\overset{(2)}{=}{\mathbf{m}}_{G}(Z)\sum_{i,j=1}^{m}{\mathbf{m}}_{G}(x_{i}Z\cap
C_{G}(x_{j}))\overset{(3)}{=}{\mathbf{m}}_{G}(Z)\sum_{i,j=1}^{m}{\mathbf{m}}_{G}(Z)c_{i,j}$
$\displaystyle={\mathbf{m}}_{G}(Z)^{2}\sum_{i,j=1}^{m}c_{i,j}=\frac{1}{|G:Z|^{2}}\sum_{i,j=1}^{m}c_{i,j}$

The latter, in particular, shows the rationality of ${\rm cp}(G)$. On the
other hand,

$\displaystyle{\rm cp}(F)$ $\displaystyle={\mathbf{m}}_{F\times
F}\big{(}\bigcup_{i,j=1}^{m}\\{x_{i}Z\times
x_{j}Z\;|\;x_{i}x_{j}=x_{j}x_{i}\\}\big{)}$
$\displaystyle=\sum_{i,j=1}^{m}{\mathbf{m}}_{F\times F}(x_{i}Z\times
x_{j}Z)c_{i,j}$
$\displaystyle=\sum_{i,j=1}^{m}{\mathbf{m}}_{F}(Z)^{2}c_{i,j}={\mathbf{m}}_{F}(Z)^{2}\sum_{i,j=1}^{m}c_{i,j}$

It follows that

(*) $\displaystyle\sum_{i,j=1}^{m}c_{i,j}=\frac{{\rm
cp}(F)}{{\mathbf{m}}_{F}(Z)^{2}}$

Thus we also find the following relation.

${\rm cp}(G)=\frac{|F:Z|^{2}}{|G:Z|^{2}}{\rm cp}(F)=\frac{{\rm
cp}(F)}{|G:F|^{2}}.$

By [4, p. 135, paragraph 4], there exists a group $H$ such that $Z(H)\leq
H^{\prime}$ and $H$ is isoclinic to $F$. Note that since $F/Z(F)$ and
$F^{\prime}$ are finite, $H$ is a finite group. If $d_{i,j}$ are defined
similar to $c_{i,j}$ for $H$, then the isoclinism of $H$ and $F$ implies that
$\sum_{i,j=1}^{m}c_{i,j}=\sum_{i,j=1}^{m}d_{i,j}$. Note that the same relation
as (*) holds for $H$ as well, thus

$\frac{{\rm cp}(F)}{{\mathbf{m}}_{F}(Z)^{2}}=\frac{{\rm
cp}(H)}{{\mathbf{m}}_{H}(Z(H))^{2}}.$

Since ${\mathbf{m}}_{F}(Z)=|F:Z|$ and ${\mathbf{m}}_{H}(Z(H))=|H:Z(H)|$, the
isoclinism again implies that ${\rm cp}(F)={\rm cp}(H)$. This completes the
proof of Theorem 1.1.

##  3\. Proof of Theorem 1.2

By Theorem 1.1 there exists a finite group $H$ isoclinic to $F:=FC(G)$ such
that ${\rm cp}(G)=\frac{{\rm cp}(H)}{|G:F|^{2}}$.

(1) Suppose that ${\rm cp}(G)>\frac{1}{4}$. If $G\neq F$, then
$\frac{1}{|G:F|^{2}}\leq\frac{1}{4}$ and so ${\rm cp}(H)>1$ which is
impossible. Thus $G=F$. Now [5, Theorem 3.11] follows that both $G^{\prime}$
and $G/Z(G)$ are finite.

(2) Suppose that ${\rm cp}(G)>\frac{3}{40}$. It follows that ${\rm
cp}(H)>\frac{3}{40}$ and $|G:F|\in\\{1,2,3\\}$ so that $G^{\prime}\leq F$. Now
[1, Theorem 12] implies that $H$ is solvable or $H\cong A_{5}\times T$ for
some abelian group $T$. If $H$ is solvable, by isoclinism, $H^{\prime}\cong
F^{\prime}$ and so $F$ is solvable and so is $G$, since $G/F$ is cyclic. Now
assume that $H\cong A_{5}\times T$ for some abelian group $T$ so that ${\rm
cp}(H)=\frac{1}{12}$. Now (1) implies that $G=F$. By isoclinism, $G/Z(G)\cong
A_{5}$ and $G^{\prime}\cong A_{5}$. Therefore $G=G^{\prime}Z(G)$ and so
$G\cong A_{5}\times Z(G)$.

## References

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