# Existence and emergent dynamics of quadratically separable states to the
Lohe tensor model

Seung-Yeal Ha
Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 08826 and
Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of
Korea<EMAIL_ADDRESS>,  Dohyun Kim
School of Mathematics, Statistics and Data Science,
Sungshin Women’s University, Seoul 02844, Republic of Korea
<EMAIL_ADDRESS>and  Hansol Park
Department of Mathematical Sciences,
Seoul National University, Seoul 08826, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor
model is an aggregation model on the space of tensors with the same rank and
size. It incorporates previously well-studied aggregation models on the space
of low-rank tensors such as the Kuramoto model, Lohe sphere and matrix models
as special cases. Due to its structural complexities in cubic interactions for
the Lohe tensor model, explicit construction of solutions with specific
structures looks daunting. Recently, we obtained completely separable states
by associating rank-1 tensors. In this paper, we further investigate another
type of solutions, namely “quadratically separable states” consisting of
tensor products of matrices and their component rank-2 tensors are solutions
to the double matrix model whose emergent dynamics can be studied using the
same methodology of the Lohe matrix model.

###### Key words and phrases:

Aggregation, double sphere model, gradient flow, Kuramoto model, Lohe matrix
model, Lohe tensor model, synchronization

###### 2010 Mathematics Subject Classification:

82C10, 82C22, 35B37

Acknowledgment. The work of S.-Y. Ha is supported by National Research
Foundation of Korea (NRF-2020R1A2C3A01003881).

##  1\. Introduction

Collective behaviors often appear in large population systems for weakly
coupled oscillators or interacting units [3, 6, 7] in diverse scientific
disciplines including biology, social sciences, engineering with various space
and time scales, for instance, colonies of bacteria [25], school of fish [4,
26], flock of starlings [29], pedestrian dynamics [18], opinion dynamics [17],
power grid networks [22], etc. For a brief introduction to collective
dynamics, we refer the reader to survey articles and book [1, 5, 10, 24].
Mathematical approach toward the understanding of collective motions has been
established in literature by Winfree [28] and Kuramoto [19] in the 1970s, and
by Vicsek [27] in the 1990s. After their remarkable works, the aforementioned
models have been extended in several directions, particularly in high-
dimensional extension to Riemannian manifolds including the hypersurfaces [2,
12] and the matrix Lie group [11] which have attracted lots of interest thanks
to its powerful application, for instance, nonconvex optimization. In this
work, among high-dimensional models, we are concerned with the Lohe tensor
model in [13].

Next, we briefly discuss tensors and an aggregation model on the space of
tensors, namely “the Lohe tensor model”. A rank-$m$ complex valued tensor can
be represented as a multi-dimensional array of complex numbers with multi-
indices. The rank of a tensor is the number of indices, say a rank-$m$ tensor
with size $d_{1}\times\cdots\times d_{m}$ is an element of
${\mathbb{C}}^{d_{1}\times\cdots\times d_{m}}$. For example, scalars, vectors
and matrices correspond to rank-0, 1 and 2 tensors, respectively. Let $T$ be a
rank-$m$ tensor with a size $d_{1}\times\cdots\times d_{m}$. Then, we denote
$(\alpha_{1},\cdots,\alpha_{m})$-th component of the tensor $T$ by
$[T]_{\alpha_{1}\cdots\alpha_{m}}$, and we set $\overline{T}$ by the rank-$m$
tensor whose components are the complex conjugate of the elements in $T$:

$[\overline{T}]_{\alpha_{1}\cdots\alpha_{m}}:=\overline{[T]_{\alpha_{1}\cdots\alpha_{m}}}.$

Let ${\mathcal{T}}_{m}(\mathbb{C};d_{1}\times\cdots\times d_{m})$ be the
collection of all rank-$m$ tensors with size $d_{1}\times\cdots\times d_{m}$.
Then, it is a complex vector space. Several well-known first-order aggregation
models, for instance, the Kuramoto model [19], the swarm sphere model [23] and
matrix models [8, 20] can be regarded as aggregation models on
${\mathcal{T}}_{0}(\mathbb{C};1),{\mathcal{T}}_{1}(\mathbb{R};d)$ and
${\mathcal{T}}_{2}(\mathbb{C};d\times d)$, respectively. Let $A_{j}$ be the
skew-hermitian rank-$2m$ tensor with size $(d_{1}\times\cdots\times
d_{m})\times(d_{1}\times\cdots\times d_{m})$. For simplicity, we introduce
handy notation as follows: for
$T\in{\mathcal{T}}_{m}(\mathbb{C};d_{1}\times\cdots\times d_{m})$ and
$A\in{\mathcal{T}}_{2m}(\mathbb{C};d_{1}\times\cdots\times d_{m}\times
d_{1}\times\cdots\times d_{m})$, we set

$\displaystyle\begin{aligned}
&[T]_{\alpha_{*}}:=[T]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}},\quad[T]_{\alpha_{*0}}:=[T]_{\alpha_{10}\alpha_{20}\cdots\alpha_{m0}},\quad[T]_{\alpha_{*1}}:=[T]_{\alpha_{11}\alpha_{21}\cdots\alpha_{m1}},\\\
&[T]_{\alpha_{*i_{*}}}:=[T]_{\alpha_{1i_{1}}\alpha_{2i_{2}}\cdots\alpha_{mi_{m}}},\quad[T]_{\alpha_{*(1-i_{*})}}:=[T]_{\alpha_{1(1-i_{1})}\alpha_{2(1-i_{2})}\cdots\alpha_{m(1-i_{m})}},\\\
&[A]_{\alpha_{*}\beta_{*}}:=[A]_{\alpha_{1}\alpha_{2}\cdots\alpha_{m}\beta_{1}\beta_{2}\cdots\beta_{m}}.\end{aligned}$

Then, the Lohe tensor model in a component form reads as follows:

$\begin{cases}\displaystyle\dot{[T_{j}]}_{\alpha_{*0}}=[A_{j}]_{\alpha_{*0}\alpha_{*1}}[T_{j}]_{\alpha_{*1}}\\\
\displaystyle\hskip
42.67912pt+\sum_{i_{*}\in\\{0,1\\}^{m}}\kappa_{i_{*}}\Big{(}[T_{c}]_{\alpha_{*i_{*}}}\bar{[T_{j}]}_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}\bar{[T_{c}]}_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\Big{)},\\\
\displaystyle\bar{[A_{j}]}_{\alpha_{*0}\alpha_{*1}}=-[A_{j}]_{\alpha_{*1}\alpha_{*0}},\end{cases}$
(1.1)

where $\kappa_{i_{*}}$’s are nonnegative coupling strengths.

Before we discuss our main issues, we introduce a concept of a “quadratically
separable state” for the Lohe tensor model (1.1).

######  Definition 1.1.

Let $\\{T_{i}\\}$ be a quadratically separable state to (1.1), if it is
decomposed as a tensor product of rank-2 tensors (or matrices):

$T_{i}=U_{i}^{1}\otimes U_{i}^{2}\otimes\cdots\otimes U_{i}^{m},\quad
U_{i}^{k}\in\mathbb{C}^{d_{1}^{k}\times
d_{2}^{k}},\quad\|U_{i}^{k}\|_{\textup{F}}=1,\quad 1\leq i\leq N,\quad 1\leq
k\leq m,$

where $\|\cdot\|_{\textup{F}}$ is the Frobenius norm induced by Frobenius
inner product: for matrices $A$ and $B$,

$\langle
A,B\rangle_{\textup{F}}:=\textup{tr}(A^{\dagger}B),\quad\|A\|_{\textup{F}}:=\sqrt{\textup{tr}(A^{\dagger}A)}.$

In this paper, we are interested in the following simple questions:

  * •

(Q1): Are there quadratically separable states for the Lohe tensor model?

  * •

(Q2): If so, do they exhibit collective behaviors under which circumstances?

Our main results deal with the raised two questions (Q1) and (Q2). More
precisely, our main results of this paper can be summarized as follows.

First, we introduce the double matrix model induced from the Lohe tensor model
whose elements have rank-4 with a specific condition on natural frequencies
$B_{j}$ and $C_{j}$ (see (3.9)):

$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
56.9055pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
56.9055pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
\end{cases}$ (1.2)

where $B_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{1}\times d_{2}}$ and
$C_{j}\in\mathbb{C}^{d_{3}\times d_{4}\times d_{3}\times d_{4}}$ are skew-
symmetric rank-4 tensors. For a solution $\\{(U_{i},V_{i})\\}$ to the double
matrix model (1.2), a special solution $T_{i}$ to (1.1) can be represented as
follows:

$T_{i}(t)\equiv U_{i}(t)\otimes V_{i}(t),\quad t>0.$

Precisely, if $T_{i}$ is initially decomposed into the tensor product of two
matrices $U_{i}$ and $V_{i}$, then its separability is propagated along the
flow for all time. For details, we refer the reader to Section 3.

Second, we study emergent dynamics of the double matrix model by investigating
several aggregation quantities:

$\displaystyle\mathcal{D}(\mathcal{U}(t)):=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t)):=\max_{1\leq
i,j\leq N}|n-\langle U_{i},U_{j}\rangle_{\textup{F}}(t)|,$
$\displaystyle\mathcal{D}(\mathcal{V}(t)):=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t)):=\max_{1\leq
i,j\leq N}|m-\langle V_{i},V_{j}\rangle_{\textup{F}}(t)|.$

For a homogeneous ensemble (i.e. $B_{i}=B_{j}$ and $C_{i}=C_{j}$ for all $i$
and $j$.) we show that system (1.2) exhibits complete aggregation in which all
relative distances for $\\{U_{i}\\}$ and $\\{V_{i}\\}$ tend to zero
respectively (see Theorem 4.1). On the other hand, for a heterogeneous
ensemble ($B_{i}\neq B_{j}$ and $C_{i}\neq C_{j}$ in general) complete
aggregation (one-point collapse) would not be expected. Instead, our concern
is dedicated to emergence of locked states in which relative distances
converge to positive definite values (see Theorem 4.2). For our analytical
results, we need to assume that the size of unitary matrices satisfy
$\min(n,m)>4\sqrt{\max(n,m)}$ that requires restriction on $n,m$. In fact,
this technical assumption on the sizes is mainly due to the fact that elements
are complex-valued. Thus, when the unitary groups $\mathbf{U}(n)$ and
$\mathbf{U}(m)$ are replaced by the special orthogonal groups $\textbf{SO}(n)$
and $\textbf{SO}(m)$, such restriction on $n,m$ would be removed (see Theorem
C.1 and Theorem C.2).

The rest of the paper is organized as follows. In Section 2, we begin with
previous results on the relation between the Lohe tensor model and the swarm
double sphere model presented in [9]. As a natural extension, we construct the
double matrix model in Section 3 and study existence and uniqueness of
quadratically separable states. In Section 4, we study emergent dynamics of
the double matrix model for both homogeneous and heterogeneous ensembles.
Next, the double matrix model is further generalized to the multi matrix model
in Section 5. Finally, Section 6 is devoted to a brief summary of the paper
and future work. In Appendix A and Appendix B, we provide proofs of Lemma 4.1
and Lemma 4.2, respectively. In Appendix C, emergent dynamics of the double
matrix model on $\textbf{SO}(n)\times\textbf{SO}(m)$ is provided.

For simplicity of presentation, we use the following abbreviated jargons:

  * •

LT model: Lohe tensor model, LM model: Lohe matrix model,

  * •

SDS model: swarm double sphere model, SMS model: swarm multi-sphere model,

  * •

DM model: double matrix model, DUM model: double unitary matrix model,

  * •

DSOM model: double special orthogonal matrix model, QSS: quadratically
separable state,

  * •

MM model: multiple matrix model, MUM model: multiple unitary matrix model.

##  2\. Preliminaries

In this section, we review how the SDS model [9, 21] can be related to the LT
model via completely separable states, and discuss extension of the SDS model
to the SMS model leading to the DM model.

###  2.1. From the LT model to the SDS model

In this subsection, we briefly recall the relation between the LT model and
the SDS model which was first observed in [9]. In [21], Lohe introduced a
first-order aggregation model on the product of two unit spheres
$(u_{i},v_{i})\in\mathbb{S}^{d_{1}-1}\times\mathbb{S}^{d_{2}-1}$:

$\displaystyle\begin{cases}\dot{u}_{i}=\Omega_{i}u_{i}+\displaystyle\frac{\kappa}{N}\sum_{j=1}^{N}\langle
v_{i},v_{j}\rangle(u_{j}-\langle u_{i},u_{j}\rangle u_{i}),\quad t>0,\\\
\dot{v}_{i}=\Lambda_{i}v_{i}+\displaystyle\frac{\kappa}{N}\sum_{j=1}^{N}\langle
u_{i},u_{j}\rangle(v_{j}-\langle v_{i},v_{j}\rangle v_{i}),\\\
(u_{i},v_{i})(0)=(u_{i}^{0},v_{i}^{0})\in\mathbb{S}^{d_{1}-1}\times\mathbb{S}^{d_{2}-1},\quad
1\leq i\leq N,\end{cases}$ (2.1)

where $\Omega_{i}$ and $\Lambda_{i}$ are skew-symmetric matrices of sizes
$d_{1}\times d_{1}$ and $d_{2}\times d_{2}$, respectively:

$\Omega_{i}^{\top}=-\Omega_{i},\quad\Lambda_{i}^{\top}=-\Lambda_{i},\quad
1\leq i\leq N,$

and ${\kappa}$ denotes the (uniform) coupling strength.

On the other hand, if we choose the following parameters:

$m=2,\quad{\kappa}_{00}={\kappa}_{11}=0,\quad{\kappa}_{01}={\kappa}_{10}={\kappa},$

system (1.1) reduces to the generalized Lohe matrix model in [15]:

$\begin{cases}\displaystyle{\dot{T}}_{i}=A_{i}T_{i}+\kappa(T_{c}T_{i}^{\dagger}T_{i}-T_{i}T_{c}^{\dagger}T_{i})+\kappa(T_{i}T_{i}^{\dagger}T_{c}-T_{i}T_{c}^{\dagger}T_{i}),\quad
t>0,\\\ \displaystyle
T_{i}(0)=T_{i}^{0},\quad\|T_{i}^{0}\|_{\textup{F}}=1,\quad
T_{c}:=\frac{1}{N}\sum_{k=1}^{N}T_{k},\quad i=1,\cdots,N.\end{cases}$ (2.2)

Next, we present how models (2.1) and (2.2) can be viewed as equivalent
systems under well-prepared natural frequency tensors and initial data in the
following proposition.

######  Proposition 2.1.

[9] Systems (2.1) and (2.2) are equivalent in the following sense.

  1. (1)

Suppose $\\{(u_{i},v_{i})\\}$ is a solution to (2.1). Then, rank-2 real
tensors $T_{i}$ defined by $T_{i}:=u_{i}\otimes v_{i}$ is a solution to (2.2)
with initial data $T_{i}^{0}=u_{i}^{0}\otimes v_{i}^{0}$ and well-prepared
free flow tensors $A_{i}$:

$A_{i}T_{i}:=\Omega_{i}T_{i}+T_{i}\Lambda_{i}^{\top}.$ (2.3)

  2. (2)

Suppose $T_{i}$ is a solution to (2.2)-(2.3) with completely factorized
initial data:

$T_{i}^{0}=:u_{i}^{0}\otimes v_{i}^{0},\quad 1\leq i,j\leq N,$

for rank-1 real tensors $u_{i}^{0}\in\mathbb{S}^{d_{1}1}$ and
$v_{i}^{0}\in\mathbb{S}^{d_{2}-1}$. Then, there exists a pair of unit vectors
$(u_{i}(t),v_{i}(t))$ such that

$T_{i}(t)=u_{i}(t)\otimes v_{i}(t),\quad t>0,$

where $(u_{i},v_{i})$ is a solution to (2.1) with initial data
$(u_{i},v_{i})(0)=(u_{i}^{0},v_{i}^{0})$.

By applying the completely separability stated in Proposition 2.1, emergent
behaviors for (2.1) and those for (2.2) are exactly the same. Thus, it
suffices to investigate the SDS model (2.1).

######  Proposition 2.2.

_[9]_ Suppose the initial data $\\{(u_{i}^{0},v_{i}^{0})\\}$ satisfy the
following conditions:

$\min_{1\leq i,j\leq N}\langle u_{i}^{0},u_{j}^{0}\rangle>0,\quad\min_{1\leq
i,j\leq N}\langle v_{i}^{0},v_{j}^{0}\rangle>0,$

and let $\\{(U,V)\\}$ be a solution to system (2.1). Then, we have

$\lim_{t\to\infty}\max_{1\leq i,j\leq
N}|u_{i}(t)-u_{j}(t)|=0\quad\textup{and}\quad\lim_{t\to\infty}\max_{1\leq
i,j\leq N}|v_{i}(t)-v_{j}(t)|=0.$

Now, it is worthwhile mentioning that system (2.1) can be represented as a
coupled gradient flow:

$\begin{cases}\displaystyle{\dot{u}}_{i}=-\frac{N\kappa}{2}{\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}\Big{(}\nabla_{u_{i}}\mathcal{E}(U,V)\Big{)},\vspace{0.2cm}\\\
\displaystyle{\dot{v}}_{i}=-\frac{N\kappa}{2}{\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}\Big{(}\nabla_{v_{i}}\mathcal{E}(U,V)\Big{)},\end{cases}$
(2.4)

where the projection operators ${\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}$
and ${\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}$ onto the tangent spaces of
$\mathbb{S}^{d_{1}-1}$ and $\mathbb{S}^{d_{2}-1}$ at $u_{i}$ and $v_{i}$ are
defined by the formulae, respectively: for $w_{1}\in\mathbb{R}^{d_{1}}$ and
$w_{2}\in\mathbb{R}^{d_{2}}$,

$\begin{cases}\displaystyle{\mathbb{P}}_{T_{u_{i}}\mathbb{S}^{d_{1}-1}}(w_{1}):=w_{1}-\langle
w_{1},u_{i}\rangle u_{i},\\\
\displaystyle{\mathbb{P}}_{T_{v_{i}}\mathbb{S}^{d_{2}-1}}(w_{2}):=w_{2}-\langle
w_{2},v_{i}\rangle v_{i},\end{cases}$

and the potential function $\mathcal{E}(U,V)$ is defined as

$\mathcal{E}(U,V):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
u_{i},u_{j}\rangle\langle v_{i},v_{j}\rangle.$ (2.5)

Thanks to the gradient flow formulation (2.4), any solution to system (2.1)
converges to an equilibrium as $t\to\infty$.

###  2.2. From the SDS model to the SMS model

In this subsection, we extend the SDS model (2.1) on the product of two unit
spheres to an aggregation model on the product of multiple unit spheres,
namely, the SMS model. Note that the SDS model can be represented as a
gradient flow with a potential function as can be seen in (2.4)–(2.5). Thus,
we first generalize the potential function (2.5) as follows: for
$u_{i}^{k}\in\mathbb{S}^{d_{k}-1},\quad i=1,\cdots,N,\quad k=1,\cdots,m,$ we
set

$\mathcal{E}(U^{1},U^{2},\cdots,U^{m}):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\prod_{k=1}^{m}\langle
u_{i}^{k},u_{j}^{k}\rangle,\quad
U^{k}:=\\{u_{1}^{k},u_{2}^{k},\cdots,u_{N}^{k}\\}.$ (2.6)

Using the same spirit for a gradient flow with the potential function (2.6),
we propose the SMS model as follows.

$\displaystyle\begin{cases}\displaystyle{\dot{u}}_{i}^{k}=\frac{\kappa}{N}\sum_{j=1}^{N}\left(\prod_{l\neq
k}\langle u_{i}^{l},u_{j}^{l}\rangle\right)\Big{(}u_{j}^{k}-\langle
u_{i}^{k},u_{j}^{k}\rangle u_{i}^{k}\Big{)},\quad t>0,\\\
u_{i}^{k}(0)=u_{i}^{k,0}\in\mathbb{S}^{d_{k}-1}\qquad
i\in\\{1,2,\cdots,N\\},\quad k\in\\{1,2,\cdots,m\\}.\end{cases}$ (2.7)

As in Section 2.1, we set rank-$m$ real tensor $T_{i}$:

$T_{i}:=u_{i}^{1}\otimes\cdots\otimes u_{i}^{m},\quad i=1,\cdots,N.$

Then, it is easy to check that $T_{i}$ satisfies

$[\dot{T}_{p}]_{\alpha_{*0}}=\frac{\kappa}{N}\sum_{k=1}^{m}\sum_{\ell=1}^{N}\left([T_{\ell}]_{\alpha_{*i_{*}^{k}}}[\bar{T}_{p}]_{\alpha_{*1}}[T_{p}]_{\alpha_{*(1-i_{*}^{k})}}-[T_{p}]_{\alpha_{*i_{*}^{k}}}[\bar{T}_{\ell}]_{\alpha_{*1}}[T_{p}]_{\alpha_{*(1-i_{*}^{k})}}\right).$
(2.8)

It should be noted that (2.8) can be derivable from the Lohe tensor model
(1.1) with the following conditions:

$\displaystyle\kappa_{i_{*}}=\begin{cases}\kappa\quad&\text{when
}i_{*}=i_{*}^{k},\quad 1\leq k\leq m,\\\
0&\text{otherwise},\end{cases}\quad\text{where}\quad
i_{*}^{k}:=\underbrace{(0,\cdots,0,1,0,\cdots,0)}_{\text{only $k^{th}$ index
is $1$}}.$

Hence, systems (2.7) and (2.8) can be related in view of a completely
separable state, and since the emergent dynamics of the LT model has been
discussed in literature [13, 14, 15], we conclude that system (2.7) exhibits
complete aggregation under suitable circumstances.

######  Proposition 2.3.

_[9]_ Suppose that initial data $T^{0}=\\{T_{i}^{0}\\}$ are completely
factorized as a tensor product of rank-1 real tensors:

$\displaystyle\begin{aligned} &T_{i}^{0}=u_{i}^{1,0}\otimes
u_{i}^{2,0}\otimes\cdots\otimes u_{i}^{m,0},\quad i=1,\cdots,N,\\\
&{\mathcal{A}}(U^{k,0}):=\min_{1\leq i,j\leq N}\langle
u^{k,0}_{i},u^{k,0}_{j}\rangle>0,\quad k=1,\cdots,m,\end{aligned}$

and let $T=\\{T_{i}\\}$ be a solution to system (2.8) and
$\\{U^{1},\cdots,U^{d}\\}$ be a solution to system (2.7). Then, the following
assertions hold.

  1. (1)

$T_{i}=T_{i}(t)$ is completely separable in the sense that

$T_{i}(t)=u_{i}^{1}(t)\otimes u_{i}^{2}(t)\otimes\cdots\otimes
u_{i}^{m}(t),\quad t>0,\quad i=1,\cdots,N.$

  2. (2)

The solution exhibits the complete aggregation:

$\lim_{t\to\infty}\max_{1\leq i,j\leq N}\|T_{i}(t)-T_{j}(t)\|_{\textup{F}}=0.$

###### Proof.

For a proof, we refer the reader to Theorem 6.2 and Proposition 7.1 in [9]. ∎

##  3\. Existence of rank-4 quadratically separable states

In this section, we present existence of the QSS for the LT model with rank-4
tensors and introduce the DM model than can be induced from the LT model.

###  3.1. The DM model

In this subsection, we propose the DM model consisting of two generalized Lohe
matrix model on the rectangular matrices with possibly different sizes:

$U_{j}\in\mathbb{C}^{d_{1}\times d_{2}}\quad\mbox{and}\quad
V_{j}\in\mathbb{C}^{d_{3}\times d_{4}},\quad j=1,\cdots,N.$

Below, we sketch our strategy how to derive the DM model from the LT model:

  * •

Step A (A homogeneous ensemble): we present a DM model for a homogeneous
ensemble (Section 3.1.1).

  * •

Step B (A heterogeneous ensemble): by adding natural frequency tensors with
suitable structure conditions, we derive the DM model from the LT model
(Section 3.1.2).

####  3.1.1. A homogeneous ensemble

Let $T_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{3}\times d_{4}}$ be a
rank-4 tensor that is a solution to (1.1) with zero natural frequency tensors
$A_{j}\equiv O$:

$[\dot{T}_{j}]_{\alpha_{*0}}=\displaystyle\sum_{i_{*}\in\\{0,1\\}^{4}}\left[\frac{\kappa_{i_{*}}}{N}\sum_{k=1}^{N}\left([T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\right)\right].$
(3.1)

For a given solution $T_{j}$ to (3.1), we assume that there exist two matrices
$U_{j}\in\mathbb{C}^{d_{1}\times d_{2}}$ and $V_{j}\in\mathbb{C}^{d_{3}\times
d_{4}}$ such that

$T_{j}=U_{j}\otimes
V_{j},\quad[T_{j}(t)]_{\alpha\beta\gamma\delta}=[U_{j}(t)]_{\alpha\beta}[V_{j}(t)]_{\gamma\delta}\quad\mbox{in
a component form}.$

Next, we rewrite cubic interaction terms in (3.1) in terms of $U_{j}$ and
$V_{j}$. For this, we decompose the index vectors $i_{*}$ and
$\alpha_{*i_{*}}$ as

$i_{*}:=(i_{1},i_{2},i_{3},i_{4}),\quad\alpha_{*i_{*}}:=(\beta_{*j_{*}},\gamma_{*k_{*}}),\quad
j_{*}:=(i_{1},i_{2}),\quad k_{*}:=(i_{3},i_{4}),$

where $j_{*}$ and $k_{*}$ correspond to the index vectors for $U_{j}$ and
$V_{j}$, respectively. We now observe

$\displaystyle\begin{aligned}
&[T_{i}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{k}]_{\alpha_{*(1-i_{*})}}\\\
&\hskip
14.22636pt=[T_{i}]_{(\beta_{*j_{*}},\gamma_{*k_{*}}}[\bar{T}_{j}]_{(\beta_{*1},\gamma_{*1})}[T_{k}]_{(\beta_{*(1-j_{*})},\gamma_{*(1-k_{*})})}\\\
&\hskip
14.22636pt=[U_{i}]_{\beta_{*j_{*}}}[V_{i}]_{\gamma_{*k_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[\bar{V}_{j}]_{\gamma_{*1}}[U_{k}]_{\beta_{*(1-j_{*})}}[V_{k}]_{\gamma_{*(1-k_{*})}}\\\
&\hskip
14.22636pt=\left([U_{i}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{k}]_{\beta_{*(1-j_{*})}}\right)\left([V_{i}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{k}]_{\gamma_{*(1-k_{*})}}\right).\end{aligned}$

By interchanging the roles of $j\leftrightarrow k$, the term inside of the
summation in the right-hand side of (3.1) becomes

$\displaystyle[T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}$
$\displaystyle\hskip
28.45274pt=\left([U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\right)\left([V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\right)$
$\displaystyle\hskip
42.67912pt-\left([U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\right)\left([V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\right).$

Since the left-hand side of (3.1) has the form of

$\dot{T}_{j}=\dot{U}_{j}\otimes V_{j}+U_{j}\otimes\dot{V}_{j},$ (3.2)

one should impose either ${\kappa}_{*}=(1,1)$ or $j_{*}=(1,1)$ to derive the
restriction on ${\kappa}_{i_{*}}$:

$\kappa_{i_{*}}=0\quad\textup{for all $i_{*}\in\\{0,1\\}^{4}$ with
$(i_{1},i_{2})\neq(1,1)$ and $(i_{3},i_{4})\neq(1,1)$.}$

Then, the right-hand side of (3.1) further reduces to

$\displaystyle\begin{aligned}
&\displaystyle\sum_{i_{*}\in\\{0,1\\}^{4}}\left[\frac{\kappa_{i_{*}}}{N}\sum_{k=1}^{N}\left([T_{k}]_{\alpha_{*i_{*}}}[\bar{T}_{j}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}-[T_{j}]_{\alpha_{*i_{*}}}[\bar{T}_{k}]_{\alpha_{*1}}[T_{j}]_{\alpha_{*(1-i_{*})}}\right)\right]\\\
&\hskip
28.45274pt=\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}[T_{k}]_{\beta_{*j_{*}}\gamma_{*1}}[\bar{T}_{j}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*(1-j_{*})}\gamma_{*0}}\\\
&\hskip
142.26378pt-[T_{j}]_{\beta_{*j_{*}}\gamma_{*1}}[\bar{T}_{k}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*(1-j_{*})}\gamma_{*0}}\Big{)}\biggr{]}\\\
&\hskip
42.67912pt+\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}[T_{k}]_{\beta_{*1}\gamma_{*k_{*}}}[\bar{T}_{j}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*0}\gamma_{*(1-k_{*})}}\\\
&\hskip
142.26378pt-[T_{j}]_{\beta_{*1}\gamma_{*k_{*}}}[\bar{T}_{k}]_{\beta_{*1}\gamma_{*1}}[T_{j}]_{\beta_{*0}\gamma_{*(1-k_{*})}}\Big{)}\biggr{]}\\\
&\hskip
28.45274pt=[V_{j}]_{\gamma_{*0}}\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}[U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\\\
&\hskip 142.26378pt-\langle
V_{k},V_{j}\rangle_{\textup{F}}[U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\Big{)}\biggr{]}\\\
&\hskip
42.67912pt+[U_{j}]_{\beta_{*0}}\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{[}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{j},U_{k}\rangle_{\textup{F}}[V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\\\
&\hskip 142.26378pt-\langle
U_{k},U_{j}\rangle_{\textup{F}}[V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\Big{)}\biggr{]}.\end{aligned}$

By comparing $\cdot\otimes V_{j}$ and $U_{j}\otimes\cdot$ in (3.2), one has

$\begin{cases}[\dot{U}_{j}]_{\beta_{*0}}=\displaystyle\sum_{j_{*}\in\\{0,1\\}^{2}}\biggl{(}\frac{\kappa_{(j_{*},1,1)}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}[U_{k}]_{\beta_{*j_{*}}}[\bar{U}_{j}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\\\
\displaystyle\hskip 170.71652pt-\langle
V_{k},V_{j}\rangle_{\textup{F}}[U_{j}]_{\beta_{*j_{*}}}[\bar{U}_{k}]_{\beta_{*1}}[U_{j}]_{\beta_{*(1-j_{*})}}\Big{)}\biggr{)},\\\
[\dot{V}_{j}]_{\gamma_{*0}}=\displaystyle\sum_{k_{*}\in\\{0,1\\}^{2}}\biggl{(}\frac{\kappa_{(1,1,k_{*})}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{j},U_{k}\rangle_{\textup{F}}[V_{k}]_{\gamma_{*k_{*}}}[\bar{V}_{j}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\\\
\displaystyle\hskip 170.71652pt-\langle
U_{k},U_{j}\rangle_{\textup{F}}[V_{j}]_{\gamma_{*k_{*}}}[\bar{V}_{k}]_{\gamma_{*1}}[V_{j}]_{\gamma_{*(1-k_{*})}}\Big{)}\biggr{)}.\\\
\end{cases}$

If we choose the coupling strengths as for remaining $i_{*}$:

$\displaystyle\kappa_{(0,1,1,1)}=\kappa_{(1,1,0,1)}=\kappa_{1},\quad\kappa_{(1,0,1,1)}=\kappa_{(1,1,1,0)}=\kappa_{2},\quad\kappa_{i_{*}}=0,$
(3.3)

we obtain the desired system for $(U_{j},V_{j})$:

$\displaystyle\begin{cases}\dot{U}_{j}=\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right).\\\
\end{cases}$ (3.4)

####  3.1.2. A heterogeneous ensemble

Similar to several aggregation models such as the Lohe matrix model [20] and
the Lohe tensor model [13], a natural candidate for heterogeneous (or non-
identical) extension of (3.4) would be the model (3.4) together with natural
frequency tensors $B_{j}$ and $C_{j}$ whose ranks are 4. Thus, the DM model
for a heterogeneous ensemble reads as

$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}U_{j}^{\dagger}U_{j}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{j}U_{j}^{\dagger}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}V_{j}^{\dagger}V_{j}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right)\\\ \hskip
28.45274pt+\displaystyle\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{j}V_{j}^{\dagger}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
\end{cases}$ (3.5)

where $B_{j}\in\mathbb{C}^{d_{1}\times d_{2}\times d_{1}\times d_{2}}$ and
$C_{j}\in\mathbb{C}^{d_{3}\times d_{4}\times d_{3}\times d_{4}}$ are rank-4
tensors satisfying skew-symmetric properties: for suitable indices,

$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=-[\bar{B}_{j}]_{\alpha_{2}\beta_{2}\alpha_{1}\beta_{1}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=-[\bar{C}_{j}]_{\gamma_{2}\delta_{2}\gamma_{1}\delta_{1}}.$
(3.6)

Moreover, $U_{j}B_{j}$ and $C_{j}V_{j}$ can be understood as tensor
contractions between rank-4 and rank-2 tensors:

$[B_{j}U_{j}]_{\alpha\beta}=[B_{j}]_{\alpha\beta\gamma\delta}[U_{j}]_{\gamma\delta},\quad[C_{j}V_{j}]_{\alpha\beta}=[C_{j}]_{\alpha\beta\gamma\delta}[V_{j}]_{\gamma\delta}.$

Next, we find a condition for $A_{j}$ in (1.1) in terms of $B_{j}$ and $C_{j}$
in (3.5) so that models (1.1) and (3.5) are equivalent. For this, it suffices
to focus on the free flows by setting ${\kappa}_{1}={\kappa}_{2}=0$, i.e.,

$\dot{U}_{j}=B_{j}U_{j},\quad\dot{V}_{j}=C_{j}V_{j},\quad j=1,\cdots,N.$

If we use the relations above and

$(U_{j}\otimes V_{j})^{\prime}=(B_{j}U_{j})\otimes
V_{j}+U_{j}\otimes(C_{j}V_{j}),$

then we can find

$A_{j}(U_{j}\otimes V_{j})=(B_{j}U_{j})\otimes
V_{j}+U_{j}\otimes(C_{j}V_{j}).$ (3.7)

In addition, if we rewrite (3.7) as a component form, then it becomes

$\displaystyle\begin{aligned}
0&=[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}-[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{1}\delta_{1}}\\\
&\hskip
14.22636pt-[U_{j}]_{\alpha_{1}\beta_{1}}[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}\\\
&=\Big{(}[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}-[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}\delta_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}-[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}\Big{)}\\\
&\hskip
14.22636pt\times[U_{j}]_{\alpha_{2}\beta_{2}}[V_{j}]_{\gamma_{2}\delta_{2}}.\end{aligned}$
(3.8)

Since (3.8) holds for arbitrary $U_{j}$ and $V_{j}$, we can find an explicit
relation between $A_{j}$ and $B_{j}$, $C_{j}$:

$\displaystyle[A_{j}]_{\alpha_{1}\beta_{1}\gamma_{1}\delta_{1}\alpha_{2}\beta_{2}\gamma_{2}\delta_{2}}=[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}\delta_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}+[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}.$
(3.9)

So far, we have derived the DM model (3.5) with (3.9) from the LT model (1.1).
However, if we use the same argument reversely and recall that system (3.5)
admits a unique solution, then one can identify the LT model from the DM
model. Thus, we can say that system (3.5)–(3.9) and system (1.1) are
equivalent in some sense. Below, we summarize the argument above in the
following proposition.

######  Proposition 3.1.

  1. (1)

Suppose $\\{(U_{i},V_{i})\\}$ is a solution to (3.5). Then, a rank-4 tensor
$T_{i}$ defined by $T_{i}:=U_{i}\otimes V_{i}$ is a QSS to (1.1) with a well-
prepared free flow tensor $A_{i}$ satisfying (3.9).

  2. (2)

Suppose a rank-4 tensor $T_{i}$ is a solution to (1.1) with (3.9) and
quadratically separable initial data:

$T_{i}^{0}=:U_{i}^{0}\otimes V_{i}^{0},\quad 1\leq i\leq N,$

for rank-2 tensors $U_{i}^{0}\in\mathbb{C}^{d_{1}\times d_{2}}$ and
$V_{i}^{0}\in\mathbb{C}^{d_{3}\times d_{4}}$ with unit norms. Then, there
exist two matrices with unit norms $U_{i}=U_{i}(t)$ and $V=V_{i}(t)$ such that

$T_{i}(t)=U_{i}(t)\otimes V_{i}(t),\quad t>0,$

where $(U_{i},V_{i})$ is a solution to (3.5) with
$(U_{i},V_{i})(0)=(U_{i}^{0},V_{i}^{0})$.

###  3.2. Gradient flow formulation of the DM model

In this subsection, we show that system (3.5) can be formulated as a gradient
flow with a suitable analytical potential induced from the Lohe tensor model.
Recall from [14] that the following functional can be associated with the LT
model:

$\mathcal{V}(T)=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
T_{i},T_{j}\rangle_{\textup{F}}.$ (3.10)

If we use the decomposition $T_{i}=U_{i}\otimes V_{i}$, then we find the
corresponding functional for (3.5):

$\mathcal{E}(U,V):=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle U_{i}\otimes
V_{i},U_{j}\otimes
V_{j}\rangle_{\textup{F}}=1-\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle
U_{i},U_{j}\rangle_{\textup{F}}\langle V_{i},V_{j}\rangle_{\textup{F}}.$
(3.11)

In the following lemma, we show that $\mathcal{E}(U,V)$ is non-increasing
along the flow (3.5).

######  Lemma 3.1.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (3.5) with $(B_{j},C_{j})=(O,O)$.
Then, the functional $\mathcal{E}(U,V)$ is non-increasing in time:

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{E}(U,V)=-\frac{\kappa_{1}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\Big{(}\langle
V_{j},V_{i}\rangle_{\textup{F}}U_{i}U_{j}^{\dagger}-\langle
V_{i},V_{j}\rangle_{\textup{F}}U_{j}U_{i}^{\dagger}\Big{)}\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{1}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\left(\langle
U_{j},U_{i}\rangle_{\textup{F}}V_{i}V_{j}^{\dagger}-\langle
U_{i},U_{j}\rangle_{\textup{F}}V_{j}V_{i}^{\dagger}\right)\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{2}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\Big{(}\langle
V_{j},V_{i}\rangle_{\textup{F}}U_{j}^{\dagger}U_{i}-\langle
V_{i},V_{j}\rangle_{\textup{F}}U_{i}^{\dagger}U_{j}\Big{)}\right\|_{\textup{F}}^{2}$
$\displaystyle\hskip
19.91684pt-\frac{\kappa_{2}}{N}\sum_{j=1}^{N}\left\|\frac{1}{N}\sum_{i=1}^{N}\left(\langle
U_{j},U_{i}\rangle_{\textup{F}}V_{j}^{\dagger}V_{i}-\langle
U_{i},U_{j}\rangle_{\textup{F}}V_{i}^{\dagger}V_{j}\right)\right\|_{\textup{F}}^{2}.$

###### Proof.

It follows from Proposition 4.1 in [14] that $\mathcal{V}(T)$ in (3.10) for
the LT model (1.1) satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{V}(T)=-\frac{1}{N}\sum_{j=1}^{N}\sum_{i_{*}\in\\{0,1\\}^{4}}\kappa_{i_{*}}\|M^{i_{*}}(T_{c})M^{i_{*}}(T_{j})^{\dagger}-M^{i_{*}}(T_{j})M^{i_{*}}(T_{c})^{\dagger}\|_{\textup{F}}^{2}.$
(3.12)

Here, we recall from [14, Definition 3.1] that for a rank-4 tensor $T$,
$\mathcal{M}^{i_{*}}(T)$ is defined as a rank-2 tensor reshaped from $T$ or it
can be understood as a bijective linear map which conserves the Frobenius
norm. Since we only consider four types among $i_{*}\in\\{0,1\\}^{4}$ as in
(3.3), we set

$\kappa_{(0,1,1,1)}=\kappa_{(1,1,0,1)}=\kappa_{1},\quad\kappa_{(1,0,1,1)}=\kappa_{(1,1,1,0)}=\kappa_{2},\quad\kappa_{i_{*}}=0\quad\text{for
other }i_{*}.$ (3.13)

On the other hand, in (3.12), we are concerned with the term
$M^{i_{*}}(T_{i})M^{i_{*}}(T_{j})^{\dagger}$ for $T_{i}=U_{i}\otimes V_{i}$
and $i_{*}$ in (3.13), for instance, if $i_{*}=(0,1,1,1)$, then we have

$\displaystyle[M^{i_{*}}(T_{i})M^{i_{*}}(T_{j})^{\dagger}]_{\alpha_{20}\alpha_{21}}$
$\displaystyle=[T_{i}]_{\alpha_{11}\alpha_{20}\alpha_{31}\alpha_{41}}[\bar{T}_{j}]_{\alpha_{11}\alpha_{21}\alpha_{31}\alpha_{41}}$
$\displaystyle=[U_{i}]_{\alpha_{11}\alpha_{20}}[V_{i}]_{\alpha_{31}\alpha_{41}}[\bar{U}_{j}]_{\alpha_{11}\alpha_{21}}[\bar{V}_{j}]_{\alpha_{31}\alpha_{41}}=\langle
V_{j},V_{i}\rangle_{\textup{F}}[U_{j}^{\dagger}U_{i}]_{\alpha_{21}\alpha_{20}}.$

If we repeat similar calculations as above, we obtain the desired dissipative
estimate. ∎

###  3.3. The DM model

In this subsection, we further reduce the DM model (3.5) to the model defined
on the product of two unitary matrices $\mathbf{U}(n)\times\mathbf{U}(m)$
which we call as the DUM model. By the construction of model (3.5), we know
that the Frobenius norms of $U_{j}$ and $V_{j}$ are conserved. In addition to
the conservation of the Frobenius norms, we show that unitarity is also
preserved, when rectangular matrices are replaced by square matrices.

######  Lemma 3.2.

Suppose that the system parameters and initial data satisfy

$d_{1}=d_{2}=n,\quad d_{3}=d_{4}=m,\quad U_{j}^{0}\in\mathbf{U}(n),\quad
V_{j}^{0}\in\mathbf{U}(m),$

and let $\\{(U_{j},V_{j})\\}$ be a solution to (3.5). Then, one has

$\displaystyle U_{j}(t)\in\mathbf{U}(n),\quad V_{j}(t)\in\mathbf{U}(m),\quad
t>0,\quad j=1,\cdots,N.$

###### Proof.

Since $U_{j}$ and $V_{j}$ have the same structure, we focus only on $U_{j}$.
We rewrite the dynamics of $U_{j}$ as a simpler form:

$\dot{U}_{j}=B_{j}U_{j}+(P_{j}-P_{j}^{\dagger})U_{j}+U_{j}(Q_{j}-Q_{j}^{\dagger}),$

where $P_{j}$ and $Q_{j}$ are defined as

$\displaystyle P_{j}:=\frac{{\kappa}_{1}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger},\quad
Q_{j}:=\frac{{\kappa}_{2}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{j}^{\dagger}U_{k}.$

Then, one has

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(U_{j}U_{j}^{\dagger})$
$\displaystyle=B_{j}U_{j}U_{j}^{\dagger}-U_{j}U_{j}^{\dagger}B_{j}+(P_{j}-P_{j}^{\dagger})U_{j}U_{j}^{\dagger}+U_{j}(Q_{j}-Q_{j}^{\dagger})U_{j}^{\dagger}$
$\displaystyle\hskip 14.22636pt-
U_{j}U_{j}^{\dagger}(P_{j}-P_{j}^{\dagger})-U_{j}(Q_{j}-Q_{j}^{\dagger})U_{j}^{\dagger}$
$\displaystyle=(B_{j}+P_{j}-P_{j}^{\dagger})U_{j}U_{j}^{\dagger}-U_{j}U_{j}^{\dagger}(B_{j}+P_{j}-P_{j}^{\dagger}).$

By straightforward calculation, we find

$\frac{{\textup{d}}}{{\textup{d}t}}\|I_{n}-U_{j}U_{j}^{\dagger}\|_{\textup{F}}=0.$
(3.14)

Since we assume $U_{j}^{0}\in\mathbf{U}(n)$, i.e.,
$I_{n}-U_{j}U_{j}^{\dagger}=O$, the relation (3.14) yields the desired result.
∎

Due to the unitarity, system (3.5) can further be reduced to the model on
$\mathbf{U}(n)\times\mathbf{U}(m)$ by using the relations:

$U_{j}U_{j}^{\dagger}=I_{n}=U_{j}^{\dagger}U_{j},\quad
V_{j}V_{j}^{\dagger}=I_{m}=V_{j}^{\dagger}V_{j},\quad j=1,\cdots,N.$

Thus, the DM model reads as

$\displaystyle\begin{cases}\dot{U}_{j}=B_{j}U_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=C_{j}V_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\end{cases}$

where $\kappa=\kappa_{1}+\kappa_{2}$.

Note that natural frequency tensors $B_{j}$ and $C_{j}$ have rank-4 satisfying
skew-symmetric properties as in (3.6). In order to give a meaning of
Hamiltonian, we associate two Hermitian matrices, namely,
$H_{j}\in\mathbb{C}^{n\times n}$ and $G_{j}\in\mathbb{C}^{m\times m}$:

$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=:[-\mathrm{i}H_{j}]_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=:[-\mathrm{i}G_{j}]_{\gamma_{1}\gamma_{2}}\delta_{\delta_{1}\delta_{2}}.$

Then, system (3.10) reduces to the model on
$\mathbf{U}(n)\times\mathbf{U}(m)$:

$\displaystyle\begin{cases}\dot{U}_{j}=-\mathrm{i}H_{j}U_{j}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\\\
\dot{V}_{j}=-\mathrm{i}G_{j}V_{j}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\end{cases}$
(3.15)

where $H_{j}U_{j}$ and $G_{j}V_{j}$ are now usual matrix products. Then as in
Lemma 3.1, system (3.15) also satisfies the dissipative energy estimate. Since
the proof can be directly obtained from Lemma 3.1, we omit it.

######  Corollary 3.1.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (3.15) with $H_{j}=G_{j}\equiv O$.
Then the Lyapunov functional (5.1) satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{E}(U,V)=-\frac{\kappa}{N}\sum_{j=1}^{N}\left(\|\dot{U}_{j}\|_{\textup{F}}^{2}+\|\dot{V}_{j}\|_{\textup{F}}^{2}\right),\quad
t>0.$

##  4\. Emergent dynamics of rank-4 quadratically separable states

In this section, we study the emergent behavior of the QSS for the Lohe tensor
model by investigating the dynamics of the DUM model which reads as follows:

$\begin{cases}\dot{U}_{j}=\displaystyle-\mathrm{i}H_{j}U_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\quad
t>0,\\\
\dot{V}_{j}=\displaystyle-\mathrm{i}G_{j}V_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{U}(n)\times\mathbf{U}(m),\quad
1\leq j\leq N,\end{cases}$ (4.1)

where $H_{j}\in\mathbb{C}^{n\times n}$ and $G_{j}\in\mathbb{C}^{m\times m}$
are Hermitian matrices which play roles of natural frequencies for each
oscillator.

First, we recall several definitions of emergent behaviors for (4.1).

######  Definition 4.1.

_[16]_ Let $(\mathcal{U},\mathcal{V}):=\\{U_{j},V_{j}\\}_{j=1}^{N}$ be a
solution to (4.1).

  1. (1)

System (4.1) exhibits complete aggregation if the following estimate holds:

$\lim_{t\to\infty}\max_{1\leq i,j\leq
N}\Big{(}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}}+\|V_{i}(t)-V_{j}(t)\|_{\textup{F}}\Big{)}=0.$
(4.2)

  2. (2)

The state $(\mathcal{U},\mathcal{V})$ tends to a phase-locked state if the
following relation holds:

$\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}(t)^{\dagger}\quad\textup{and}\quad\exists~{}\lim_{t\to\infty}V_{i}(t)V_{j}(t)^{\dagger}.$

In order to investigate emergent behaviors for (4.1), we denote the following
quantities for notational simplicity:

$\displaystyle\begin{aligned} &X_{ij}:=U_{i}U_{j}^{\dagger},\quad
S_{ij}:=I_{n}-U_{i}U_{j}^{\dagger},\quad d_{ij}:=\langle
U_{i},U_{j}\rangle_{\textup{F}},\\\ &Y_{ij}:=V_{i}V_{j}^{\dagger},\quad
T_{ij}:=I_{m}-V_{i}V_{j}^{\dagger}\quad c_{ij}:=\langle
V_{i},V_{j}\rangle_{\textup{F}}.\end{aligned}$ (4.3)

It follows from simple observations that

$\displaystyle\|U_{j}\|_{\textup{F}}=n,\quad\|V_{j}\|_{\textup{F}}=m,\quad|d_{ij}|\leq
n,\quad|c_{ij}|\leq m,$
$\displaystyle\|S_{ij}\|^{2}_{\textup{F}}=\|U_{i}-U_{j}\|_{\textup{F}}^{2}=2\textup{Re}(n-d_{ij}),\quad\|T_{ij}\|_{\textup{F}}^{2}=\|V_{i}-V_{j}\|_{\textup{F}}^{2}=2\textup{Re}(m-c_{ij}).$

Then, it is easy to see that

$\|U_{i}-U_{j}\|_{\textup{F}}\to 0\Longleftrightarrow|n-d_{ij}|\to
0,\quad\|V_{i}-V_{j}\|_{\textup{F}}\to 0\Longleftrightarrow|m-c_{ij}|\to 0.$

Thus, the complete aggregation in (4.2) can be represented in terms of the
quantities in (4.3):

$\displaystyle\lim_{t\to\infty}\max_{1\leq i,j\leq
N}\Big{(}\|S_{ij}(t)\|_{\textup{F}}+\|T_{ij}(t)\|_{\textup{F}}\Big{)}=0\quad\textup{or}\quad\lim_{t\to\infty}\max_{1\leq
i,j\leq N}\Big{(}|n-d_{ij}(t)|+|m-c_{ij}(t)|\Big{)}=0.$

In this regard, we define aggregation quantities: for $t>0$,

$\displaystyle\mathcal{D}(\mathcal{U}(t)):=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t)):=\max_{1\leq
i,j\leq N}|n-d_{ij}(t)|,$
$\displaystyle\mathcal{D}(\mathcal{V}(t)):=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t)):=\max_{1\leq
i,j\leq N}|m-c_{ij}(t)|.$

Note that $\mathcal{S}(\mathcal{U})$ has a second-order with respect to the
state $\mathcal{U}$, whereas $\mathcal{D}(\mathcal{U})$ has a first-order.

In the following two subsections, we present emergent dynamics for homogeneous
and heterogeneous ensembles respectively.

###  4.1. Homogeneous ensemble

In this subsection, we deal with the emergent dynamics of (4.1) with a
homogeneous ensemble:

$H_{j}\equiv H,\quad G_{j}\equiv G,\quad j=1,\cdots,N,$

and by the solution splitting property, we may assume

$H=O,\quad G=O.$

In this setting, system (3.5) becomes

$\displaystyle\begin{cases}\dot{U}_{j}=\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
V_{j},V_{k}\rangle_{\textup{F}}~{}U_{k}-\langle
V_{k},V_{j}\rangle_{\textup{F}}~{}U_{j}U_{k}^{\dagger}U_{j}\right),\quad
t>0,\\\ \dot{V}_{j}=\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\langle
U_{j},U_{k}\rangle_{\textup{F}}~{}V_{k}-\langle
U_{k},U_{j}\rangle_{\textup{F}}~{}V_{j}V_{k}^{\dagger}V_{j}\right),\\\
(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{U}(n)\times\mathbf{U}(m).\end{cases}$
(4.4)

Without loss of generality, we may assume

$n\geq m.$ (4.5)

Our goal of this subsection is to find a sufficient condition under which

$\lim_{t\to\infty}\mathcal{L}(t)=0,\quad\mathcal{L}(t):=\mathcal{D}(\mathcal{U}(t))+\mathcal{D}(\mathcal{V}(t))+\mathcal{S}(\mathcal{U}(t))+\mathcal{S}(\mathcal{V}(t)),$

where $\mathcal{L}=\mathcal{L}(t)$ is called as a total aggregation
functional.

In [16], it suffices to study the temporal evolutions of
$\mathcal{D}(\mathcal{U})$. However in our case, time evolutions of
$\mathcal{S}(\mathcal{U})$ as well as $\mathcal{S}(\mathcal{V})$ are needed to
achieve complete aggregation estimates. Below, we derive a differential
inequality for $\mathcal{L}$.

######  Lemma 4.1.

Let $\\{(U_{j},V_{j})\\}$ be a solution to (4.4) with (4.5). Then, the total
aggregation functional $\mathcal{L}$ satisfies

$\displaystyle\begin{aligned}
\dot{\mathcal{L}}\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3},\quad
t>0.\end{aligned}$ (4.6)

###### Proof.

Since a proof is lengthy, we provide it in Appendix A. ∎

We are now ready to provide a sufficient condition leading to the complete
aggregation for (4.4).

######  Theorem 4.1.

Suppose that the system parameters and initial data satisfy

$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq m>4\sqrt{n}.\\\
&\textup{(ii)}~{}~{}\mathcal{L}^{0}<\alpha_{n,m}:=\frac{-(12n+27)+\sqrt{(12n+27)^{2}+48(m-4\sqrt{n})(3n+4)}}{4(3n+4)},\end{aligned}$
(4.7)

and let $\\{(U_{j},V_{j})\\}$ be a solution to (4.4). Then, we have

$\lim_{t\to\infty}\mathcal{L}(t)=0.$

Moreover, the convergence rate is at least exponential. In other words, system
(4.4) exhibits complete aggregation with an exponential convergence.

###### Proof.

Consider an auxiliary quadratic polynomial:

$f(s):=\left(2n+\frac{8}{3}\right)s^{2}+(4n+9)s-2(m-4\sqrt{n}).$

Since $m>4\sqrt{n}$, the algebraic relation $f=0$ admits a unique positive
root $\alpha_{n,m}$ defined in (4.7)(ii). Then, the relation (4.6) is
rewritten in terms of $f$:

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq{\kappa}\mathcal{L}f(\mathcal{L}),\quad
t>0.$

Since initial data satisfy (4.7)(ii), the desired result follows from
dynamical systems theory. ∎

######  Remark 4.1.

(i) In (4.7)(i), we have assumed that $n\geq m>4\sqrt{n}$ which imposes a
restriction on the size of $U_{i}$ such as

$n>16.$

It also should be mentioned that such restriction arises, for instance, when
we estimate the terms $\mathcal{I}_{15}$ in (A.7) and $\mathcal{I}_{25}$ in
(A.10). We indeed show in Appendix C that this technical assumption would be
removed.

(ii) Since $m\leq n$, we have

$\limsup_{n\to\infty}\alpha_{n,m}\leq\lim_{n\to\infty}\frac{-(12n+27)+\sqrt{(12n+27)^{2}+48(n-4\sqrt{n})(3n+4)}}{4(3n+4)}=\frac{1}{2}.$

###  4.2. Heterogeneous ensemble

In this subsection, we study the case of heterogeneous Hamiltonians in which
$H_{j}$ and $G_{j}$ in (4.1) are given to be different in general. In order to
establish the emergence of the phase-locked state, we will follow a strategy
developed in [16]. For any two solutions $\\{U_{j},V_{j}\\}$ and
$\\{\tilde{U}_{j},\tilde{V}_{j}\\}$ to (4.1), we define the diameters
measuring the dissimilarity of two configurations:

$\displaystyle\begin{aligned} &d(U,\tilde{U})(t):=\max_{1\leq i,j\leq
N}\|U_{i}(t)U_{j}^{\dagger}(t)-\tilde{U}_{i}(t)\tilde{U}_{j}^{\dagger}(t)\|_{\textup{F}},\\\
&d(V,\tilde{V})(t):=\max_{1\leq i,j\leq
N}\|V_{i}(t)V_{j}^{\dagger}(t)-\tilde{V}_{i}(t)\tilde{V}_{j}^{\dagger}(t)\|_{\textup{F}}.\end{aligned}$
(4.8)

Then, we will show that the diameters above converge to zero:

$\lim_{t\to\infty}\Big{(}d(U,\tilde{U})(t)+d(V,\tilde{V})(t)\Big{)}=0.$ (4.9)

As a next step, since our system is autonomous, for any $T>0$, we choose
$\tilde{U}_{j}$ and $\tilde{V}_{j}$ as

$\tilde{U}_{j}(t)=U_{j}(t+T),\quad\tilde{V}_{j}(t)=V_{j}(t+T).$

By discretizing the time $t\in\mathbb{R}_{+}$ as $n\in\mathbb{Z}_{+}$ and
setting $T=m\in\mathbb{Z}_{+}$, we conclude that
$\\{U_{i}(n)U_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ and
$\\{V_{i}(n)V_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ are indeed Cauchy
sequences in the complete spaces $\mathbf{U}(n)$ and $\mathbf{U}(m)$,
respectively. Hence, there exist two constant unitary matrices
$U_{ij}^{\infty}\in\mathbf{U}(n)$ and $V_{ij}^{\infty}\in\mathbf{U}(m)$ such
that

$\lim_{t\to\infty}\|U_{i}(t)U_{j}^{\dagger}(t)-U_{ij}^{\infty}\|_{\textup{F}}=0,\quad\lim_{t\to\infty}\|V_{i}(t)V_{j}^{\dagger}(t)-V_{ij}^{\infty}\|_{\textup{F}}=0.$

Hence, we aim to find a sufficient condition under which (4.9) holds. To show
(4.9), we associate another diameters measuring the difference between two
solution configurations $\\{U_{j},V_{j}\\}$ and
$\\{\tilde{U}_{j},\tilde{V}_{j}\\}$:

$\displaystyle\begin{aligned} &\mathcal{S}(U,\tilde{U})(t):=\max_{1\leq
i,j\leq N}|\langle
U_{i},U_{j}\rangle(t)-\langle\tilde{U}_{i},\tilde{U}_{j}\rangle(t)|,\\\
&\mathcal{S}(V,\tilde{V})(t):=\max_{1\leq i,j\leq N}|\langle
V_{i},V_{j}\rangle(t)-\langle\tilde{V}_{i},\tilde{V}_{j}\rangle(t)|.\end{aligned}$
(4.10)

Note that

$\mathcal{S}(U,\tilde{U})\leq\sqrt{n}d(U,\tilde{U}),\quad\mathcal{S}(V,\tilde{V})\leq\sqrt{m}d(V,\tilde{V}).$

To this end, our goal of this subsection is to find a sufficient framework
leading to

$\lim_{t\to\infty}\mathcal{F}(t)=0,\quad\mathcal{F}(t):=d(U,\tilde{U})(t)+d(V,\tilde{V})(t)+\mathcal{S}(U,\tilde{U})(t)+\mathcal{S}(V,\tilde{V})(t).$
(4.11)

Below, we derive a differential inequality for $\mathcal{F}$ in (4.11).

######  Lemma 4.2.

Let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any two
solutions to (4.1) with (4.5), respectively. Then, the aggregation functional
$\mathcal{F}$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}{\mathcal{F}}\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F},$
(4.12)

where $\mathcal{D}(\mathcal{H})$ and $\mathcal{D}(\mathcal{G})$ are defined as

$\mathcal{D}(\mathcal{H}):=\max_{1\leq i,j\leq
N}\|H_{i}-H_{j}\|_{\infty},\quad\mathcal{D}(\mathcal{G}):=\max_{1\leq i,j\leq
N}\|G_{i}-G_{j}\|_{\infty}.$

###### Proof.

We postpone its proof in Appendix B. ∎

In what follows, using the differential inequality (4.12), we provide a
sufficient condition leading to the phase-locked state. First, in order to
make a leading coefficient of (4.12) negative, we assume that a coupling
strength ${\kappa}$ is sufficiently large:

${\kappa}>\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{2(m-4\sqrt{n})}.$

For a handy notation, we denote

$\Lambda:=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$

Next, we show that we can make the total aggregation functional $\mathcal{L}$
small as we wish by controlling the coupling strength ${\kappa}$.

######  Proposition 4.1.

Suppose system parameters and initial data satisfy

$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq
m,\quad\mathcal{D}(\mathcal{H})\geq\mathcal{D}(\mathcal{G}),\qquad\textup{(ii)}~{}~{}{\kappa}\geq{\kappa}_{\textup{c}},\quad\mathcal{L}^{0}\ll\nu_{2},\end{aligned}$
(4.13)

where ${\kappa}_{\textup{c}}$ and $\nu_{2}$ are specified later in (4.18) and
(4.19), respectively, and let $\\{(U_{j},V_{j})\\}$ be a solution to (4.1).
Then, we have

$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$

###### Proof.

Since we assumed (4.13)(i), it follows from (4.6) that

$\dot{\mathcal{L}}\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}+2(1+3\sqrt{n})\mathcal{D}(\mathcal{H}).$
(4.14)

Now, we introduce an auxiliary cubic polynomial:

$g(s):=2(m-4\sqrt{n})s-(4n+9)s^{2}-\left(2n+\frac{8}{3}\right)s^{3}.$ (4.15)

Then, (4.14) can be rewritten as

$\dot{\mathcal{L}}\leq{\kappa}\left(\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}-g(\mathcal{L})\right).$

By investigating roots of the polynomial $g$ in (4.15), we deduce that for a
sufficient large ${\kappa}$, the polynomial $g$ admits one negative root, say,
$\nu_{0}<0$, and two positive roots, say, $0<\nu_{1}<\nu_{2}$ with continuous
dependence of ${\kappa}$, i.e.,

$\lim_{{\kappa}\to\infty}\nu_{1}({\kappa})=0,\quad\lim_{{\kappa}\to\infty}\nu_{2}({\kappa})=\alpha_{*},\quad\textup{$\alpha_{*}$:
a unique positive root of $g$}.$ (4.16)

Since we assume (4.13)(ii), there exists a finite entrance time $T_{*}>0$ such
that

$\mathcal{L}(t)<\nu_{1},\quad t>T_{*}.$ (4.17)

Finally, we combine (4.16) and (4.17) to obtain the desired estimate:

$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$

∎

######  Remark 4.2.

For an explicit value for ${\kappa}_{\textup{c}}$ in (4.13)(ii), by simple
calculus, we know that the polynomial $g=g(s)$ in (4.15) admits a global
maximum in $\mathbb{R}_{+}$ at $s=s_{*}$:

$s_{*}:=\frac{-(4n+9)+\sqrt{(4n+9)^{2}+2(6n+8)(m-4\sqrt{n})}}{6n+8}.$

Thus, in order to guarantee the existence of a positive root for $g$, one
should impose

$g(s_{*})>\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}},\quad\textup{i.e.,}\quad{\kappa}>\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{g(s_{*})}=:{\kappa}_{\textup{c}}.$
(4.18)

For this ${\kappa}_{\textup{c}}$, $\nu_{2}$ in (4.13)(ii) can be explicitly
determined as the largest positive root of

$g(s)=\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}\quad\textup{when
${\kappa}>{\kappa}_{\textup{c}}$}.$ (4.19)

It follows from Proposition 4.1 that under the assumption on the smallness of
the initial data $\mathcal{L}^{0}$ and largeness of the coupling strength
${\kappa}$, we have

$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$

Hence, there exists a (large) coupling strength ${\kappa}_{\textup{p}}$ larger
than ${\kappa}_{\textup{c}}$ such that for $t\gg 1$,

$(4n+22)\mathcal{L}(t)+20\mathcal{L}(t)^{2}<\frac{\Lambda}{2},\quad\Lambda=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$

Then, (4.12) becomes

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{F}$
$\displaystyle\leq-\Lambda{\kappa}\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}\leq-\Lambda{\kappa}\mathcal{F}+\frac{\Lambda{\kappa}}{2}\mathcal{F}=-\frac{\Lambda{\kappa}}{2}\mathcal{F}.$

This yields a desired exponential decay for $\mathcal{F}$. The argument above
can be stated and shown as follows.

######  Theorem 4.2.

Suppose system parameters and initial data satisfy

$\displaystyle\begin{aligned} &\textup{(i)}~{}~{}n\geq
m,\quad\mathcal{D}(\mathcal{H})\geq\mathcal{D}(\mathcal{G}),\qquad&\textup{(ii)}~{}~{}{\kappa}>{\kappa}_{\textup{p}}>{\kappa}_{\textup{c}},\quad\max\\{\mathcal{L}^{0},\tilde{\mathcal{L}}^{0}\\}\leq\nu_{2},\end{aligned}$

and let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any
two solutions to (4.1), respectively. Then, the following assertions hold.

  1. (1)

The functional $\mathcal{F}$ converges to zero with an exponential rate.

  2. (2)

The normalized velocities $\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}$ and
$\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}$ synchronize:

$\lim_{t\to\infty}\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}=0,\quad\lim_{t\to\infty}\|\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}-\mathrm{i}\dot{\tilde{V}}_{j}\tilde{V}_{j}^{\dagger}\|_{\textup{F}}=0.$

  3. (3)

There exist unitary matrices $X_{\infty}\in\mathbf{U}(n)$ and
$Y_{\infty}\in\mathbf{U}(m)$ such that

$\displaystyle\lim_{t\to\infty}U_{i}^{\dagger}(t)\tilde{U}_{i}(t)=X_{\infty},\quad\lim_{t\to\infty}\|\tilde{U}_{i}(t)-U_{i}(t)X_{\infty}\|_{\textup{F}}=0,$
$\displaystyle\lim_{t\to\infty}V_{i}^{\dagger}(t)\tilde{V}_{i}(t)=Y_{\infty},\quad\lim_{t\to\infty}\|\tilde{V}_{i}(t)-V_{i}(t)Y_{\infty}\|_{\textup{F}}=0.$

  4. (4)

Asymptotic phase-locking emerge: for any indices $i$ and $j$,

$\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t)\quad\textup{and}\quad\exists~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t).$

Moreover, there exist phase-locked state
$\mathcal{X}^{\infty}:=\\{X_{i}^{\infty}\\}_{i=1}^{N}$ and
$\mathcal{Y}^{\infty}:=\\{Y_{i}^{\infty}\\}_{i=1}^{N}$, and unitary matrices
$P\in\mathbf{U}(n)$ and $Q\in\mathbf{U}(m)$ such that

$\displaystyle\lim_{t\to\infty}\|U_{i}(t)-X_{i}^{\infty}P\|_{\textup{F}}=0=\lim_{t\to\infty}\|V_{i}(t)-Y_{i}^{\infty}Q\|_{\textup{F}}=0,\quad\max\\{\mathcal{D}(\mathcal{X}^{\infty}),\mathcal{D}(\mathcal{Y}^{\infty})\\}<\nu_{2}.$

###### Proof.

(i) It follows from (4.12) in Proposition 4.1 that $\mathcal{F}$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}{\mathcal{F}}\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.$
(4.20)

Since $\mathcal{L}$ can be sufficiently small under (4.13), i.e.,

$\mathcal{L}(t)<\nu_{1},\quad t>T_{*},$

we choose ${\kappa}_{\textup{p}}>{\kappa}_{\textup{c}}$ sufficiently large
such that

$(4n+22)\mathcal{L}+20\mathcal{L}^{2}<\frac{\Lambda}{2},\quad\Lambda=2(m-4\sqrt{n})-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}.$

Thus, (4.20) becomes

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{F}\leq-\frac{\Lambda{\kappa}}{2}\mathcal{F},\quad
t>T_{*},$ (4.21)

and the relation (4.21) yields the desired exponential convergence of
$\mathcal{F}$ toward zero.

(ii) For the second assertion, we claim:

$\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F},\qquad\|\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}-\mathrm{i}\dot{\tilde{V}}_{j}\tilde{V}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$ (4.22)

Once the relations (4.22) hold, it follows from the first assertion to derive
the second assertion. Note that

$\displaystyle\begin{aligned}
&\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}=\left\|\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle
V_{k},V_{j}\rangle_{\textup{F}}U_{j}U_{k}^{\dagger}\Big{)}\right.\\\ &\hskip
142.26378pt\left.-\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}}\tilde{U}_{k}\tilde{U}_{j}^{\dagger}-\langle\tilde{V}_{k},\tilde{V}_{j}\rangle_{\textup{F}}\tilde{U}_{j}\tilde{U}_{k}^{\dagger}\Big{)}\right\|_{\textup{F}}.\end{aligned}$
(4.23)

By algebraic manipulations, one has

$\displaystyle\|\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}}\tilde{U}_{k}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}$
$\displaystyle\hskip 14.22636pt\leq\|\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}U_{j}^{\dagger}-\tilde{U}_{k}\tilde{U}_{j}^{\dagger})\|_{\textup{F}}+\|(\langle
V_{j},V_{k}\rangle_{\textup{F}}-\langle\tilde{V}_{j},\tilde{V}_{k}\rangle_{\textup{F}})\tilde{U}_{k}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq(m+n)\mathcal{F}.$

Thus, the relation (4.23) yields

$\|\mathrm{i}\dot{U}_{j}U_{j}^{\dagger}-\mathrm{i}\dot{\tilde{U}}_{j}\tilde{U}_{j}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$

This shows the desired synchronization of the normalized velocities and the
exactly same argument is applied to $\mathrm{i}\dot{V}_{j}V_{j}^{\dagger}$.

(iii) Since system (4.1) is autonomous, we directly use Theorem 2(3) of [16].
Hence, we briefly sketch a proof. We observe

$\displaystyle\left\|\frac{{\textup{d}}}{{\textup{d}}s}(U_{i}^{\dagger}\tilde{U}_{i})\right\|_{\textup{F}}=\|\dot{U}_{i}U_{i}^{\dagger}-\dot{\tilde{U}}_{i}\tilde{U}_{i}^{\dagger}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$

Since $\mathcal{F}$ tends to zero exponentially,

$\lim_{t\to\infty}(U_{i}^{\dagger}(t)\tilde{U}_{i}(t))=U_{i}^{0,{\dagger}}\tilde{U}_{i}^{0}+\int_{0}^{\infty}\frac{{\textup{d}}}{{\textup{d}}s}(U_{i}^{\dagger}(s)\tilde{U}_{i}(s)){\textup{d}}s.$
(4.24)

In addition, since $d(U,\tilde{U})$ converges to zero, one deduces that the
right-hand side of (4.24) does not depend on the index $i$. This establishes
the desired assertion.

(iv) We first show the existence of the asymptotic limit of
$U_{i}U_{j}^{\dagger}$ and $V_{i}V_{j}^{\dagger}$. For any $T>0$, since system
(4.1) is autonomous, we choose $\tilde{U}_{i}$ and $\tilde{V}_{i}$ as

$\tilde{U}_{i}(t)=U_{i}(t+T),\quad V_{i}(t)=V_{i}(t+T).$

If we discretize the time $t\in\mathbb{R}_{+}$ and $n\in\mathbb{Z}_{+}$ and
choose $T=m\in\mathbb{Z}_{+}$, we use the convergence of $d(U,\tilde{U})$ and
$d(V,\tilde{V})$ to conclude that
$\\{U_{i}(n)U_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ and
$\\{V_{i}(n)V_{j}^{\dagger}(n)\\}_{n\in\mathbb{Z}_{+}}$ become Cauchy
sequences in the complete spaces $\mathbf{U}(n)$ and $\mathbf{U}(m)$,
respectively. Thus, the limits of $U_{i}U_{j}^{\dagger}$ and
$V_{i}V_{j}^{\dagger}$ exist. In particular, if we denote

$X_{i}^{\infty}:=\lim_{t\to\infty}U_{i}(t)U_{1}^{\dagger}(t),$

then one has

$\lim_{t\to\infty}U_{i}(t)U_{j}^{\dagger}(t)=\lim_{t\to\infty}U_{i}U_{1}^{\dagger}(t)(U_{j}(t)U_{1}^{\dagger}(t))^{\dagger}=X_{i}^{\infty}X_{j}^{\infty,{\dagger}},\quad\lim_{t\to\infty}\langle
U_{i},U_{j}\rangle_{\textup{F}}=\langle
X_{i}^{\infty},X_{j}^{\infty}\rangle_{\textup{F}}.$

On the other hand, we recall (A.2)

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(U_{i}U_{j}^{\dagger})=\mathrm{i}(U_{i}U_{j}^{\dagger}H_{j}-H_{i}U_{i}U_{j}^{\dagger})\\\
&+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{i},V_{k}\rangle_{\textup{F}}U_{k}U_{j}^{\dagger}-\langle
V_{k},V_{i}\rangle_{\textup{F}}U_{i}U_{k}^{\dagger}U_{i}U_{j}^{\dagger}+\langle
V_{k},V_{j}\rangle_{\textup{F}}U_{i}U_{k}^{\dagger}-\langle
V_{j},V_{k}\rangle_{\textup{F}}U_{i}U_{j}^{\dagger}U_{k}U_{j}^{\dagger}\Big{)}.\end{aligned}$
(4.25)

In (4.25), if we let $t\to\infty$ and apply Barbalat’s lemma, the left-hand
side of (4.25) vanishes and consequently, the relation (4.25) becomes

$\displaystyle\begin{aligned}
O&=\mathrm{i}(X_{i}^{\infty}X_{j}^{\infty,{\dagger}}H_{j}-H_{i}X_{i}^{\infty}X_{j}^{\infty,{\dagger}})\\\
&\hskip 14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{i}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty}X_{j}^{\infty,{\dagger}}-\langle
Y_{k}^{\infty},Y_{i}^{\infty,{\dagger}}\rangle_{\textup{F}}X_{i}^{\infty}X_{k}^{\infty,{\dagger}}X_{i}^{\infty}X_{j}^{\infty,{\dagger}}\\\
&\hskip 14.22636pt+\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{i}^{\infty}X_{k}^{\infty,{\dagger}}-\langle
Y_{j}^{\infty},Y_{k}^{\infty,{\dagger}}\rangle_{\textup{F}}X_{i}^{\infty}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}X_{j}^{\infty,{\dagger}}\Big{)}.\end{aligned}$
(4.26)

By performing left-multiplication $X_{i}^{\infty,{\dagger}}$ and right-
multiplication $X_{j}^{\infty}$ with (4.26), we obtain

$\displaystyle\begin{aligned}
&-\mathrm{i}X_{j}^{\infty,{\dagger}}H_{j}X_{j}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}\Big{)}\\\
&\hskip
14.22636pt=-\mathrm{i}X_{i}^{\infty,{\dagger}}H_{i}X_{i}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{i}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{i}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{i}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{i}^{\infty}\Big{)}.\end{aligned}$
(4.27)

Since the relation(4.27) does not depend on the index, we can set

$-\mathrm{i}\Theta:=-\mathrm{i}X_{j}^{\infty,{\dagger}}H_{j}X_{j}^{\infty}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}-\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}\Big{)}.$

This yields

$X_{j}^{\infty}\Theta
X_{j}^{\infty,{\dagger}}=H_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
Y_{k}^{\infty},Y_{j}^{\infty}\rangle_{\textup{F}}X_{k}^{\infty,{\dagger}}X_{j}^{\infty}-\langle
Y_{j}^{\infty},Y_{k}^{\infty}\rangle_{\textup{F}}X_{j}^{\infty,{\dagger}}X_{k}^{\infty}\Big{)}.$

Similarly for $\\{V_{i}\\}$, we can define a matrix $\Gamma$ independent of
the index such that

$Y_{j}^{\infty}\Gamma
Y_{j}^{\infty,{\dagger}}=G_{j}++\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
X_{k}^{\infty},X_{j}^{\infty}\rangle_{\textup{F}}Y_{k}^{\infty,{\dagger}}Y_{j}^{\infty}-\langle
X_{j}^{\infty},X_{k}^{\infty}\rangle_{\textup{F}}Y_{j}^{\infty,{\dagger}}Y_{k}^{\infty}\Big{)}.$

Then, $\\{\\{X_{i}^{\infty}\\},\Theta\\}$ and
$\\{\\{Y_{i}^{\infty}\\},\Gamma\\}$ indeed consist of the phase-locked state.
Moreover, one has

$\mathcal{D}(\mathcal{X}^{\infty})=\max_{1\leq i,j\leq
N}\|X_{i}^{\infty}X_{j}^{\infty,{\dagger}}-I_{n}\|=\lim_{t\to\infty}\max_{1\leq
i,j\leq
N}\|U_{i}(t)U_{j}^{\dagger}(t)-I_{n}\|_{\textup{F}}=\lim_{t\to\infty}\mathcal{D}(\mathcal{U})<\nu_{2}.$

Exactly the same estimate holds for $\mathcal{D}(\mathcal{Y}^{\infty})$ as
well. ∎

######  Remark 4.3.

In [20], the phase-locked state of (4.1) are defined to be of the following
form:

$U_{i}(t)=U_{i}^{\infty}e^{-\mathrm{i}\Gamma_{U}t},\quad
V_{i}(t)=V_{i}^{\infty}e^{-\mathrm{i}\Gamma_{V}t},$

where $U_{i}^{\infty}\in\mathbf{U}(n)$ and $V_{i}^{\infty}\in\mathbf{U}(m)$
are unitary matrices, and $\Lambda_{U}$ and $\Lambda_{V}$ satisfy

$\displaystyle\begin{aligned}
&U_{i}^{\infty}\Gamma_{U}U_{i}^{\infty,{\dagger}}=H_{i}+\frac{\mathrm{i}{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
V_{k}^{\infty},V_{j}^{\infty}\rangle_{\textup{F}}U_{k}^{\infty}U_{i}^{\infty,{\dagger}}-\langle
V_{j}^{\infty},V_{k}^{\infty}\rangle_{\textup{F}}U_{i}^{\infty}U_{k}^{\infty,{\dagger}}\Big{)},\\\
&V_{i}^{\infty}\Gamma_{V}V_{i}^{\infty,{\dagger}}=G_{i}+\frac{\mathrm{i}{\kappa}}{N}\sum_{k=1}^{N}\Big{(}\langle
U_{k}^{\infty},U_{j}^{\infty}\rangle_{\textup{F}}V_{k}^{\infty}V_{i}^{\infty,{\dagger}}-\langle
U_{j}^{\infty},U_{k}^{\infty}\rangle_{\textup{F}}V_{i}^{\infty}V_{k}^{\infty,{\dagger}}\Big{)}.\end{aligned}$

##  5\. Rank-$2m$ quadratically separable state

In this section, we study a quadratically separable state of the LT model for
rank-$2m$ tensors by introducing the MM model whose solution configuration is
given as follows.

$\\{\mathcal{U}^{1},\mathcal{U}^{2},\cdots,\mathcal{U}^{m}\\},\quad\mathcal{U}^{p}:=(U_{1}^{p},\cdots,U_{N}^{p}),\quad
p=1,\cdots,m.$

In the following two subsections, we introduce extended models for the DM
model (3.5) and the DUM model (3.15). Since the procedures are similar as
those in Section 3, we omit details.

###  5.1. The MM model

In this subsection, we introduce the MM model:

$\displaystyle\begin{cases}\displaystyle\dot{U}_{j}^{p}=B_{j}^{p}U_{j}^{p}+\displaystyle\frac{\kappa_{1}}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{\ell},U_{k}^{\ell}\rangle_{\textup{F}}U_{k}^{p}\big{(}U_{j}^{p}\big{)}^{\dagger}U_{j}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{l},U_{j}^{l}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right)\\\
\hskip
28.45274pt\displaystyle+\frac{\kappa_{2}}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{\ell},U_{k}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{j}^{p}\big{)}^{\dagger}U_{k}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{\ell},U_{j}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right),\end{cases}$
(5.1)

where $B_{j}^{p}$ is a rank-$4m$ tensor satisfying the skew-symmetric
property.

For a solution $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ to (5.1), we set a rank-$2m$
tensor denoted by $T_{i}$:

$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{m}(t),\quad U_{i}^{p}\in\mathbb{C}^{d_{1}^{p}\times d_{2}^{p}},\quad
i=1,\cdots,N,\quad p=1,\cdots,m,$ (5.2)

which can be written in a component form:

$[T_{i}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}\cdots\alpha_{m}\beta_{m}}=[U_{i}^{1}]_{\alpha_{1}\beta_{1}}[U_{i}^{2}]_{\alpha_{2}\beta_{2}}\cdots[U_{i}^{m}]_{\alpha_{m}\beta_{m}}.$

In order to relate the model (5.1) with the LT model (1.1), we consider the
index vector $i_{*}$ in ${\kappa}_{i_{*}}$. Then, we introduce two subsets of
$\\{0,1\\}^{2d}$: for $q=1,\cdots,m$,

$\displaystyle\Lambda_{1}$ $\displaystyle=\\{i_{*}\in\\{0,1\\}^{2d}:\text{0
appears once at $2q-1$-th coordinate}\\},$ $\displaystyle\Lambda_{2}$
$\displaystyle=\\{i_{*}\in\\{0,1\\}^{2d}:\text{0 appears once at $2q$-th
coordinate}\\}.$

Note that $|\Lambda_{1}|=|\Lambda_{2}|=m$ and for instance with $m=2$,

$(0,1,1,1),(1,1,0,1)\in\Lambda_{1},\quad(1,0,1,1),(1,1,1,0)\in\Lambda_{2}.$

In this regard, we choose the index vector as

$\displaystyle\kappa_{i_{*}}=\begin{cases}\kappa_{1}\quad&\text{if
}i_{*}\in\Lambda_{1},\\\ \kappa_{2}\quad&\text{if }i_{*}\in\Lambda_{2},\\\
0&\text{otherwise},\end{cases}$ (5.3)

which generalizes (3.3) for the DM model.

Next, for the natural frequency tensors, we use $\\{B_{j}^{p}\\}_{p=1}^{m}$ to
associate a rank-$4m$ tensor $A_{j}$ as

$\displaystyle[A_{j}]_{\alpha_{1}\alpha_{2}\cdots\alpha_{2m}\beta_{1}\beta_{2}\cdots\beta_{2m}}=\sum_{k=1}^{m}\left([B_{j}^{k}]_{\alpha_{2k-1}\alpha_{2k}\beta_{2k-1}\beta_{2k}}\prod^{d}_{\begin{subarray}{c}\ell=1\\\
\ell\neq
k\end{subarray}}\left(\delta_{\alpha_{2\ell-1}\beta_{2\ell-1}}\delta_{\alpha_{2\ell}\beta_{2\ell}}\right)\right),$
(5.4)

which corresponds to (3.9) for the DM model. Then, it follows from
straightforward calculation that the Lohe tensor model with (5.3) and (5.4)
reduces to (5.1) whose solutions can be related by (5.2). The argument above
is summarized in the following proposition analogous to Proposition 3.1.

######  Proposition 5.1.

The following assertions hold.

  1. (1)

Suppose $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ is a solution to (5.1). Then, a
rank-$2m$ tensor defined by $T_{i}:=U_{i}^{1}\otimes
U_{i}^{2}\otimes\cdots\otimes U_{i}^{m}$ is the QSS to (1.1) with well-
prepared initial data and free flow tensors $A_{i}$ satisfying (5.4).

  2. (2)

Suppose a rank-$2m$ tensor $T_{i}$ is a solution to (1.1) with (5.4) and
quadratically separable initial data:

$T_{i}^{0}:=U_{i}^{1,0}\otimes U_{i}^{2,0}\otimes\cdots\otimes
U_{i}^{p,0},\quad 1\leq i\leq N,$

for rank-2 tensors $U_{i}^{p,0}\times\mathbb{C}^{d_{1}^{p}\times d_{2}^{p}}$
with unit norms. Then, there exist matrices $\\{U_{i}^{p}\\}$ with unit norms
such that

$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{p}(t),\quad t>0,$

where $\\{U_{i}^{p}\\}$ is a solution to (5.1) with
$U_{i}^{p,0}=U_{i}^{p}(0)$.

###  5.2. The MUM model

In this subsection, we further reduce to the MM model to the model on the
product of the unitary groups. Since the unitary group is concerned, we set

$d_{1}^{p}=d_{2}^{p}=:d_{p},\quad 1\leq p\leq m.$ (5.5)

For the modeling of natural frequencies, we also define Hermitian matrices
with a size $d_{p}\times d_{p}$:

$[-\mathrm{i}H_{j}^{p}]_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}:=[B_{j}^{p}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}.$

In addition, $\\{H_{j}^{p}\\}$ satisfy

$\displaystyle[A_{j}]_{\alpha_{1}\alpha_{2}\cdots\alpha_{2m}\beta_{1}\beta_{2}\cdots\beta_{2m}}$
$\displaystyle=\sum_{k=1}^{m}\left([-\mathrm{i}H_{j}^{k}]_{\alpha_{2k-1}\beta_{2k-1}}\delta_{\alpha_{2k}\beta_{2k}}\prod^{m}_{\begin{subarray}{c}\ell=1\\\
\ell\neq
k\end{subarray}}\left(\delta_{\alpha_{2\ell-1}\beta_{2\ell-1}}\delta_{\alpha_{2\ell}\beta_{2\ell}}\right)\right).$
(5.6)

By Lemma 3.2, one can verify that system (5.1) with (5.5) conserves the
unitarity of $U_{j}^{p}$. Thus, system (5.1) reduces to the following model on
the unitary group:

$\displaystyle\begin{cases}\displaystyle\dot{U}_{j}^{p}=-\mathrm{i}H_{j}^{p}U_{j}^{p}+\displaystyle\frac{\kappa}{N}\sum_{k=1}^{N}\left(\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{j}^{l},U_{k}^{l}\rangle_{\textup{F}}U_{k}^{p}-\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq p\end{subarray}}^{m}\langle
U_{k}^{\ell},U_{j}^{\ell}\rangle_{\textup{F}}U_{j}^{p}\big{(}U_{k}^{p}\big{)}^{\dagger}U_{j}^{p}\right),\\\
U_{j}^{p}(0)=U_{j}^{p,0}\in\mathbf{U}(d_{p}),\end{cases}$ (5.7)

where $\kappa=\kappa_{1}+\kappa_{2}$ and $H_{j}^{p}U_{j}^{p}$ is a usual
matrix product. As in Proposition 5.1, existence and uniqueness of the QSS for
(5.7) can be stated as follows.

######  Proposition 5.2.

The following assertions hold.

  1. (1)

Suppose $\\{\mathcal{U}^{p}\\}_{p=1}^{m}$ is a solution to (5.7). Then, a
rank-$2m$ tensor defined by $T_{i}:=U_{i}^{1}\otimes
U_{i}^{2}\otimes\cdots\otimes U_{i}^{m}$ is a quadratically separable state to
(1.1) with a well-prepared free flow tensor $A_{i}$ satisfying (5.6).

  2. (2)

Suppose a rank-$2m$ tensor $T_{i}$ is a solution to (1.1) with (5.6) and
quadratically separable initial data:

$T_{i}^{0}:=U_{i}^{1,0}\otimes U_{i}^{2,0}\otimes\cdots\otimes
U_{i}^{p,0},\quad 1\leq i\leq N,$

for rank-2 tensors $U_{i}^{p,0}\times\mathbb{C}^{n_{1}^{p}\times n_{2}^{p}}$
with unit norms. Then, there exist matrices $\\{U_{i}^{p}\\}$ with unit norms
such that

$T_{i}(t)=U_{i}^{1}(t)\otimes U_{i}^{2}(t)\otimes\cdots\otimes
U_{i}^{p}(t),\quad t>0,$

where $\\{U_{i}^{p}\\}$ is a solution to (5.7) with
$U_{i}^{p,0}=U_{i}^{p}(0)$.

######  Remark 5.1.

In Section 4, we provided the emergent dynamics of the double unitary matrix
model (4.1). However for its generalized model (5.7), emergent dynamics will
not be studied, since it can be straightforwardly obtained from the results in
Section 4.

##  6\. Conclusion

In this paper, we have studied the existence and emergent dynamics of
quadratically separable states for the Lohe tensor model which incorporates
several well-known low-rank aggregation models such as the Kuramoto model, the
Lohe sphere model and the Lohe matrix model, etc. In our previous work [9], we
obtained completely separable states as special solutions to the Lohe tensor
model defined as tensor products of rank-1 real tensors (or vectors). In
analogy with the previous work, we consider the states in which a solution can
be decomposed as a tensor product of rank-2 tensors (or matrices), namely, a
quadratically separable state. Precisely, if initial data are quadratically
separable, then such separability is preserved along the Lohe tensor flow.
Moreover, by introducing and analyzing double matrix and unitary matrix
models, we are able to study asymptotic behavior of the quadratically
separable states to the Lohe tensor model. Of course, there are several issues
that are not discussed in this work. For example, one can naturally consider
the state consisting of tensors with possibly different ranks and sizes. We
explore this issue in a future work.

##  Appendix A Proof of Lemma 4.1

In this appendix, we provide a proof of Lemma 4.1 in which a differential
inequality for the aggregation functional
$\mathcal{L}=\mathcal{D}(\mathcal{U})+\mathcal{D}(\mathcal{V})+\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V})$
is derived. We divide a proof into two steps:

  * •

Step A: we derive differential inequalities for $\mathcal{D}(\mathcal{U})$ and
$\mathcal{D}(\mathcal{V})$ (see Lemma A.1).

  * •

Step B: we derive differential inequalities for $\mathcal{S}(\mathcal{U})$ and
$\mathcal{S}(\mathcal{V})$ (see Lemma A.2).

######  Lemma A.1.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.4). Then,
$\mathcal{D}(\mathcal{U})$ and $\mathcal{D}(\mathcal{V})$ satisfy

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}S(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}S(\mathcal{U}).\end{aligned}$
(A.1)

###### Proof.

By straightforward calculations, one finds a differential equation for
$G_{ij}=U_{i}U_{j}^{\dagger}$:

$\frac{{\textup{d}}}{{\textup{d}t}}G_{ij}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}G_{kj}-c_{ki}G_{ik}G_{ij}+c_{kj}G_{ik}-c_{jk}G_{ij}G_{kj}).$
(A.2)

By using the relation $G_{ij}=I_{n}-S_{ij}$, we see that $S_{ij}$ satisfies

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}S_{ij}&=\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}c_{ik}S_{kj}-c_{ki}S_{ik}-c_{ki}S_{ij}+c_{i}S_{k}S_{ij}+c_{kj}S_{ik}-c_{jk}S_{kj}-c_{jk}S_{ij}+c_{jk}S_{ij}S_{kj}\\\
&\hskip 71.13188pt+(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}\Big{]}.\end{aligned}$
(A.3)

After algebraic manipulation, we rewrite (A.3) in terms of $S_{ij}$ and
$c_{ij}-m$ that are expected to converge to zero:

$\displaystyle\begin{aligned}
\frac{{\textup{d}}S_{ij}}{dt}&=-2m{\kappa}S_{ij}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\Big{(}S_{ik}S_{ij}+S_{ij}S_{kj}\Big{)}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij}\Big{]}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}\Big{]}\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\Big{[}(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}\Big{]}.\end{aligned}$
(A.4)

On the other hand for an $n\times n$ matrix $A$, one has

$\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|A\|_{\textup{F}}^{2}=\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\textup{tr}(AA^{\dagger})=\frac{1}{2}\textup{tr}(\dot{A}A^{\dagger}+A\dot{A}^{\dagger})=\textup{Re}\textup{tr}(\dot{A}A^{\dagger}).$

We multiply (A.4) with $S_{ij}^{\dagger}$ to find

$\displaystyle\begin{aligned}
&\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|S_{ij}\|_{\textup{F}}^{2}=-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger}+S_{ij}S_{kj}S_{ij}^{\dagger})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ik}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}(c_{ik}-m)\|S_{ij}\|_{\textup{F}}^{2}\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{kj}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}[(c_{jk}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}(c_{jk}-m)\|S_{ij}\|_{\textup{F}}^{2}\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger})]+\textup{Re}[(c_{jk}-m)\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})]\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-c_{ik}+c_{jk}-c_{kj})\textup{tr}(S_{ij}^{\dagger})]\\\
&=:-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+\mathcal{I}_{11}+\mathcal{I}_{12}+\mathcal{I}_{13}+\mathcal{I}_{14}+\mathcal{I}_{15}.\end{aligned}$
(A.5)

Below, we present estimates for $\mathcal{I}_{1k},~{}k=1,\cdots,5$,
respectively.

$\bullet$ (Estimate of $\mathcal{I}_{11}$): We use (A.6) and

$S_{ij}+S_{ji}=S_{ij}S_{ji},\quad S_{ij}^{\dagger}=S_{ji}$ (A.6)

to derive

$\displaystyle\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ji})$
$\displaystyle=\frac{1}{2}\textup{tr}(S_{ik}S_{ij}S_{ji}+S_{ij}S_{ji}S_{ki})=\frac{1}{2}\textup{tr}(S_{ij}S_{ji}(S_{ki}+S_{ik}))$
$\displaystyle=\frac{1}{2}\textup{tr}(S_{ij}S_{ji}S_{ki}S_{ik})=\frac{1}{2}\|S_{ki}S_{ij}\|_{\textup{F}}^{2}.$

Similarly, one has

$\textup{Re}\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})=\frac{1}{2}\|S_{ij}S_{jk}\|_{\textup{F}}^{2}.$

Hence, $\mathcal{I}_{11}$ satisfies

$\mathcal{I}_{11}=\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger}+S_{ij}S_{kj}S_{ij}^{\dagger})=\frac{m{\kappa}}{2N}\sum_{k=1}^{N}\big{(}\|S_{ki}S_{ij}\|_{\textup{F}}^{2}+\|S_{ij}S_{jk}\|_{\textup{F}}^{2}\big{)}\leq
m{\kappa}\mathcal{D}(\mathcal{U})^{4}.$

$\bullet$ (Estimates of $\mathcal{I}_{12}$ and $\mathcal{I}_{13}$): By the
maximality of $\mathcal{D}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$, we
have

$\displaystyle\mathcal{I}_{12}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ik}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}(c_{ik}-m)\|S_{ij}\|_{\textup{F}}^{2}$
$\displaystyle\leq
3{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},$
$\displaystyle\mathcal{I}_{13}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{kj}-m)\textup{tr}(S_{ik}S_{ij}^{\dagger})]-\textup{Re}[(c_{jk}-m)\textup{tr}(S_{kj}S_{ij}^{\dagger})]-\textup{Re}(c_{jk}-m)\|S_{ij}\|_{\textup{F}}^{2}$
$\displaystyle\leq
3{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}.$

$\bullet$ (Estimate of $\mathcal{I}_{14}$): By straightforward calculation,
one has

$\displaystyle\mathcal{I}_{14}$
$\displaystyle=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-m)\textup{tr}(S_{ik}S_{ij}S_{ij}^{\dagger})]+\textup{Re}[(c_{jk}-m)\textup{tr}(S_{ij}S_{kj}S_{ij}^{\dagger})]\leq
2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{3}.$

$\bullet$ (Estimate of $\mathcal{I}_{15}$): Note that for an $n\times n$
matrix $A$,

$\textup{tr}(A)=\textup{tr}(I_{n}A)\leq\|I_{n}\|_{\textup{F}}\|A\|_{\textup{F}}=\sqrt{n}\|A\|_{\textup{F}}.$

This yields

$\mathcal{I}_{15}=\frac{{\kappa}}{N}\sum_{k=1}^{N}\textup{Re}[(c_{ki}-c_{ik}+c_{jk}-c_{kj})\textup{tr}(S_{ij}^{\dagger})]\leq
4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U}).$ (A.7)

In (A.5), we collect all the estimates for $\mathcal{I}_{1k},~{}k=1,\cdots,5$
to obtain

$\frac{1}{2}\frac{{\textup{d}}}{{\textup{d}t}}\|S_{ij}\|_{\textup{F}}^{2}\leq-2m{\kappa}\|S_{ij}\|_{\textup{F}}^{2}+m{\kappa}\mathcal{D}(\mathcal{U})^{4}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{3}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U}).$

Hence, $\mathcal{D}(\mathcal{U})$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$

Similarly, one can find a differential inequality for
$\mathcal{D}(\mathcal{V})$ by exchanging the roles of $\mathcal{U}$ and
$\mathcal{V}$:

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).$

∎

In (A.1), note that $\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$
appear in the differential inequalities for $\mathcal{D}(\mathcal{U})$ and
$\mathcal{D}(\mathcal{V})$. Hence, we derive the differential inequalities for
$\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$ below.

######  Lemma A.2.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.4). Then,
$\mathcal{S}(\mathcal{U})$ and $\mathcal{S}(\mathcal{V})$ satisfy

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).\end{aligned}$
(A.8)

###### Proof.

In (A.4), we take trace to obtain

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(n-d_{ij})=-2m{\kappa}(n-d_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}\\\
&\hskip
28.45274pt=:-2m{\kappa}(n-d_{ij})+\mathcal{I}_{21}+\mathcal{I}_{22}+\mathcal{I}_{23}+\mathcal{I}_{24}+\mathcal{I}_{25}.\end{aligned}$
(A.9)

Below, we present estimates of $\mathcal{I}_{2k},~{}k=1,\cdots,5$, separately.

$\bullet$ (Estimate of $\mathcal{I}_{21}$): We use the definition of
$\mathcal{D}(\mathcal{U})$ to find

$\displaystyle\mathcal{I}_{21}=\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\leq
2m{\kappa}\mathcal{D}(\mathcal{U})^{2}.$

$\bullet$ (Estimates of $\mathcal{I}_{22}$ and $\mathcal{I}_{23}$): It is easy
to see that

$\displaystyle\mathcal{I}_{22}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\leq
3{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}),$
$\displaystyle\mathcal{I}_{23}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\leq
3{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}).$

$\bullet$ (Estimate of $\mathcal{I}_{24}$): Similar to $\mathcal{I}_{21}$, one
has

$\displaystyle\mathcal{I}_{24}=\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\leq
2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}.$

$\bullet$ (Estimate of $\mathcal{I}_{25}$): We find

$\mathcal{I}_{25}=\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}\leq
4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$ (A.10)

In (A.9), we combine all estimates to obtain

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}).$

By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we derive a
differential inequality for $\mathcal{S}(\mathcal{V})$:

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).$

∎

######  Remark A.1.

For homogeneous Hamiltonians, we add $\eqref{AA-12}_{1}$ and
$\eqref{AA-12}_{2}$ to find

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}(\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}))$
$\displaystyle\leq-2{\kappa}(m-2\sqrt{m})\mathcal{S}(\mathcal{U})-2{\kappa}(n-2\sqrt{n})\mathcal{S}(\mathcal{V})+\mathcal{O}((\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}))^{2}).$

Hence, in order to obtain the desired convergence, we assume

$n>2\sqrt{n},\quad m>2\sqrt{m},\quad\textup{i.e.,}\quad n,m>4,$

which requires the restriction on the size of $U_{i}$ and $V_{j}$. This
technical assumption arises from the estimate of $\mathcal{I}_{25}$ in (A.10).

Now, we are ready to present a proof of Lemma 4.1 using Lemma A.1 and Lemma
A.2.

Proof of Lemma 4.1: Recall the inequalities in (A.1) and (A.8):

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2}+4{\kappa}\sqrt{n}\mathcal{S}(\mathcal{V}),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}+4{\kappa}\sqrt{m}\mathcal{S}(\mathcal{U}).\end{aligned}$
(A.11)

Without loss of generality, we may assume $n\geq m$, and add all the
inequalities in (A.11) to find

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}&\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+n{\kappa}(\mathcal{D}(\mathcal{U})^{3}+2\mathcal{D}(\mathcal{U})^{2}+\mathcal{D}(\mathcal{V})^{3}+2\mathcal{D}(\mathcal{V})^{2})\\\
&\hskip
14.22636pt+6{\kappa}(\mathcal{D}(\mathcal{U})\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})\mathcal{S}(\mathcal{U})+2\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V}))+4{\kappa}(\mathcal{D}(\mathcal{U})^{2}\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})^{2}\mathcal{S}(\mathcal{U}))\\\
&=:-2{\kappa}(m-4\sqrt{n})\mathcal{L}+n{\kappa}\mathcal{I}_{31}+6{\kappa}\mathcal{I}_{32}+4{\kappa}\mathcal{I}_{33}.\end{aligned}$
(A.12)

In the sequel, we provide estimates for $\mathcal{I}_{3k},~{}k=1,2,3$,
respectively.

$\bullet$ (Estimate of $\mathcal{I}_{31}$): We use a rough estimate to find

$\displaystyle\mathcal{I}_{31}=\mathcal{D}(\mathcal{U})^{3}+2\mathcal{D}(\mathcal{U})^{2}+\mathcal{D}(\mathcal{V})^{3}+2\mathcal{D}(\mathcal{V})^{2}\leq\mathcal{L}^{3}+2\mathcal{L}^{2}.$

$\bullet$ (Estimate of $\mathcal{I}_{32}$): By straightforward calculation,
one has

$\displaystyle\mathcal{I}_{32}$
$\displaystyle=\mathcal{D}(\mathcal{U})\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})\mathcal{S}(\mathcal{U})+2\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})$
$\displaystyle\leq\frac{1}{2}\mathcal{D}(\mathcal{U})^{2}+\frac{1}{2}\mathcal{D}(\mathcal{V})^{2}+\frac{3}{2}\mathcal{S}(\mathcal{U})^{2}+\frac{3}{2}\mathcal{S}(\mathcal{V})^{2}\leq\frac{3}{2}\mathcal{L}^{2}.$

$\bullet$ (Estimate of $\mathcal{I}_{33}$): We use Young’s inequality that for
$a,b>0$,

$a^{2}b\leq\frac{2}{3}a^{3}+\frac{1}{3}b^{3}$

to find

$\displaystyle\mathcal{I}_{33}=\mathcal{D}(\mathcal{U})^{2}\mathcal{S}(\mathcal{V})+\mathcal{D}(\mathcal{V})^{2}\mathcal{S}(\mathcal{U})\leq\frac{2}{3}\mathcal{D}(\mathcal{U})^{3}+\frac{1}{3}\mathcal{S}(\mathcal{V})^{3}+\frac{2}{3}\mathcal{D}(\mathcal{V})^{3}+\frac{1}{3}\mathcal{S}(\mathcal{U})^{3}\leq\frac{2}{3}\mathcal{L}^{3}.$

In (A.12), we combine all the estimates to obtain the desired inequality for
$\mathcal{L}$:

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}$
$\displaystyle\leq-2{\kappa}(m-4\sqrt{n})\mathcal{L}+2{\kappa}n(\mathcal{L}^{3}+2\mathcal{L}^{2})+9{\kappa}\mathcal{L}^{2}+\frac{8{\kappa}}{3}\mathcal{L}^{3}$
$\displaystyle=-2{\kappa}(m-4\sqrt{n})\mathcal{L}++{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}.$

##  Appendix B Proof of Lemma 4.2

In this appendix, we present a proof of Lemma 4.2 in which the differential
inequality for the aggregation functional $\mathcal{F}$ in (4.11) will be
derived.

######  Lemma B.1.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.1). Then, $d(U,\tilde{U})$ and
$d(V,\tilde{V})$ satisfy

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})&\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}).\\\
\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})&\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.1)

###### Proof.

First, we recall (A.4):

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}S_{ij}$
$\displaystyle=-2m{\kappa}S_{ij}+\frac{m{\kappa}}{N}\sum_{k=1}^{N}S_{ik}S_{ij}+S_{ij}S_{kj}+\mathrm{i}(H_{i}-H_{j})+\mathrm{i}(S_{ij}H_{j}-H_{i}S_{ij})$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij}$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}$
$\displaystyle\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-c_{ik}+c_{jk}-c_{kj})I_{n}.$

Thus, $S_{ij}-\tilde{S}_{ij}$ satisfies

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(S_{ij}-\tilde{S}_{ij})=-2m{\kappa}(S_{ij}-\tilde{S}_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\underbrace{(S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj})}_{=:\mathcal{I}_{41}}\\\
&\hskip
28.45274pt+\underbrace{\mathrm{i}((S_{ij}-\tilde{S}_{ij})H_{j}-H_{i}(S_{ij}-\tilde{S}_{ij}))}_{=:\mathcal{I}_{42}}+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{43}-\tilde{\mathcal{I}}_{43})+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{44}-\tilde{\mathcal{I}}_{44})\\\
&\hskip
28.45274pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}\underbrace{((c_{ki}-c_{ik}+c_{jk}-c_{kj})-(\tilde{c}_{ki}-\tilde{c}_{ik}+\tilde{c}_{jk}-\tilde{c}_{kj}))I_{n}}_{=:\mathcal{I}_{45}},\end{aligned}$
(B.2)

where $\mathcal{I}_{43}$ and $\mathcal{I}_{44}$ are defined as

$\displaystyle\mathcal{I}_{43}:=(c_{ik}-m)S_{kj}-(c_{ki}-m)S_{ik}-(c_{ik}-m)S_{ij}+(c_{ki}-m)S_{ik}S_{ij},$
$\displaystyle\mathcal{I}_{44}:=(c_{kj}-m)S_{ik}-(c_{jk}-m)S_{kj}-(c_{jk}-m)S_{ij}+(c_{jk}-m)S_{ij}S_{kj}.$

Below, we provide the estimates for $\mathcal{I}_{4k},~{}k=1,\cdots,5$,
respectively.

$\bullet$ (Estimate of $\mathcal{I}_{41}$): Note that

$\displaystyle\mathcal{I}_{41}$
$\displaystyle=S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj}$
$\displaystyle=S_{ik}(S_{ij}-\tilde{S}_{ij})+(S_{ik}-\tilde{S}_{ik})\tilde{S}_{ij}+S_{ij}(S_{kj}-\tilde{S}_{kj})+(S_{ij}-\tilde{S}_{ij})\tilde{S}_{kj}.$

This yields

$\|\mathcal{I}_{41}\|_{\textup{F}}\leq 4\mathcal{L}d(U,\tilde{U}).$ (B.3)

$\bullet$ (Estimate of $\mathcal{I}_{42}$): Note that for a skew-Hermitian
matrix $\Omega$ and a matrix $A$, one has

$\displaystyle\textup{tr}(\Omega AA^{\dagger})=0.$

This implies

$\displaystyle\textup{Re}\textup{tr}(\mathcal{I}_{42}(S_{ij}-\tilde{S}_{ij})^{\dagger})=\textup{Re}\textup{tr}(\mathrm{i}H_{j}(S_{ij}-\tilde{S}_{ij})(S_{ij}-\tilde{S}_{ij})^{\dagger})=0.$

$\bullet$ (Estimate of $\mathcal{I}_{43}$): We observe

$\displaystyle\begin{aligned}
&\|(c_{ik}-m)S_{kj}-(\tilde{c}_{ik}-m)\tilde{S}_{kj}\|_{\textup{F}}\\\ &\hskip
28.45274pt=\|(c_{ik}-m)(S_{kj}-\tilde{S}_{kj})+(c_{ik}-\tilde{c}_{ik})\tilde{S}_{kj}\|_{\textup{F}}\\\
&\hskip
28.45274pt\leq\mathcal{L}d(U,\tilde{U})+\mathcal{L}\mathcal{S}(V,\tilde{V})=\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).\end{aligned}$

Moreover, we use (B.3)

$\displaystyle\|(c_{ki}-m)S_{ik}S_{ij}-(\tilde{c}_{ki}-m)\tilde{S}_{ik}\tilde{S}_{ij}\|_{\textup{F}}$
$\displaystyle\hskip
28.45274pt=\|(c_{ki}-m)(S_{ik}S_{ij}-\tilde{S}_{ik}\tilde{S}_{ij})+(c_{ik}-\tilde{c}_{ik})\tilde{S}_{ik}\tilde{S}_{ij}\|_{\textup{F}}$
$\displaystyle\hskip 28.45274pt\leq\mathcal{L}\cdot
4\mathcal{L}d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})\mathcal{L}^{2}=\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$

to obtain

$\displaystyle\|\mathcal{I}_{43}-\tilde{\mathcal{I}}_{43}\|_{\textup{F}}\leq
3\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$

$\bullet$ (Estimate of $\mathcal{I}_{44}$): Similar to $\mathcal{I}_{43}$, one
finds

$\displaystyle\|\mathcal{I}_{44}-\tilde{\mathcal{I}}_{44}\|_{\textup{F}}\leq
3\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$

$\bullet$ (Estimate of $\mathcal{I}_{45}$): We directly find

$\displaystyle\|\mathcal{I}_{45}\|_{\textup{F}}\leq
4\sqrt{n}\mathcal{S}(V,\tilde{V}).$

In (B.2), we combine all the estimates to obtain

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})$
$\displaystyle\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}).$

By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we find the
differential inequality for $d(V,\tilde{V})$:

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})$
$\displaystyle\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))$
$\displaystyle\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}).$

∎

Below, we derive differential inequalities for $\mathcal{S}(U,\tilde{U})$ and
$\mathcal{S}(V,\tilde{V})$.

######  Lemma B.2.

Let $\\{(U_{i},V_{i})\\}$ be a solution to (4.1). Then,
$\mathcal{S}(U,\tilde{U})$ and $\mathcal{S}(V,\tilde{V})$ satisfy

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})&\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}).\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})&\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{H})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.4)

###### Proof.

We recall (A.9):

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}(n-d_{ij})&=-2m{\kappa}(n-d_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj})\\\
&\hskip
14.22636pt+\mathrm{i}\textup{tr}(H_{i}-H_{j})+\mathrm{i}\textup{tr}(S_{ij}H_{j}-H_{i}S_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ik}-m)(n-d_{kj})-(c_{ki}-m)(n-d_{ik})-(c_{ik}-m)(n-d_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{kj}-m)(n-d_{ik})-(c_{jk}-m)(n-d_{kj})-(c_{jk}-m)(n-d_{ij})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(c_{ki}-m)\textup{tr}(S_{ik}S_{ij})+(c_{jk}-m)\textup{tr}(S_{ij}S_{kj})\\\
&\hskip
14.22636pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}[(c_{ki}-c_{ik})+(c_{jk}-c_{kj})]\sqrt{n}.\end{aligned}$

We denote

$p_{ij}:=n-d_{ij},\quad q_{ij}:=m-c_{ij}.$

Then, $p_{ij}-\tilde{p}_{ij}$ satisfies

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}(p_{ij}-\tilde{p}_{ij})=-2m{\kappa}(p_{ij}-\tilde{p}_{ij})+\frac{m{\kappa}}{N}\sum_{k=1}^{N}\underbrace{\textup{tr}(S_{ik}S_{ij}+S_{ij}S_{kj}-\tilde{S}_{ik}\tilde{S}_{ij}-\tilde{S}_{ij}\tilde{S}_{kj})}_{=:\mathcal{I}_{51}}\\\
&\hskip
5.69046pt+\underbrace{\mathrm{i}\textup{tr}((S_{ij}-\tilde{S}_{ij})(H_{j}-H_{i}))}_{=:\mathcal{I}_{52}}+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{53}-\tilde{\mathcal{I}}_{53})+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{54}-\tilde{\mathcal{I}}_{54})\\\
&\hskip
5.69046pt+\frac{{\kappa}}{N}\sum_{k=1}^{N}(\mathcal{I}_{55}-\tilde{\mathcal{I}}_{55})+\frac{{\kappa}}{N}\sum_{k=1}^{N}\underbrace{(p_{ik}-\tilde{p}_{ik}+p_{kj}-\tilde{p}_{kj})+(\tilde{p}_{ki}-\tilde{p}_{ki})+(\tilde{p}_{jk}-p_{jk})}_{=:\mathcal{I}_{56}}\sqrt{n}.\end{aligned}$
(B.5)

Below, we present the estimates for $\mathcal{I}_{5k},~{}k=1,\cdots,6$,
separately.

$\bullet$ (Estimate of $\mathcal{I}_{51}$): We recall (B.3) to find

$|\mathcal{I}_{51}|\leq 4\mathcal{L}d(U,\tilde{U}).$

$\bullet$ (Estimate of $\mathcal{I}_{52}$): we set

$\mathcal{D}(\mathcal{H}):=\max_{1\leq i,j\leq N}\|H_{i}-H_{j}\|_{\infty},$

Then, one has

$|\mathcal{I}_{52}|\leq\mathcal{D}(\mathcal{H})d(U,\tilde{U}).$

$\bullet$ (Estimate of $\mathcal{I}_{53}$): Note that

$\displaystyle|(c_{ik}-m)(n-d_{kj})-(\tilde{c}_{ik}-m)(n-\tilde{d}_{kj})|$
$\displaystyle\hskip
28.45274pt\leq|p_{ik}-\tilde{p}_{ik}||q_{ik}|+|\tilde{p}_{ik}||q_{kj}-\tilde{q}_{kj}|\leq\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$

Thus, we find

$\displaystyle\|\mathcal{I}_{53}-\tilde{\mathcal{I}}_{53}\|_{\textup{F}}\leq
3\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$

$\bullet$ (Estimate of $\mathcal{I}_{54}$): We observe

$\displaystyle\|\mathcal{I}_{54}-\tilde{\mathcal{I}}_{54}\|_{\textup{F}}\leq
3\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V})).$

$\bullet$ (Estimate of $\mathcal{I}_{55}$): Note that

$\displaystyle
q_{ki}\textup{tr}(S_{ik}S_{ij}-\tilde{q}_{ki}\textup{tr}(\tilde{S}_{ik}\tilde{S}_{ij})$
$\displaystyle\hskip
14.22636pt=q_{ij}\textup{tr}(S_{ik}S_{ij}-\tilde{S}_{ik}\tilde{S}_{ij})+(q_{ij}-\tilde{q}_{ij})\textup{tr}(\tilde{S}_{ik}\tilde{S}_{ij})\leq
4\mathcal{L}^{2}d(U,\tilde{U})+\mathcal{L}^{2}\mathcal{S}(V,\tilde{V}).$

Thus, we have

$\displaystyle\|\mathcal{I}_{55}-\tilde{\mathcal{I}}_{55}\|_{\textup{F}}\leq
8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V}).$

$\bullet$ (Estimate of $\mathcal{I}_{56}$): We directly find

$\displaystyle|\mathcal{I}_{56}|\leq 4\sqrt{n}\mathcal{S}(V,\tilde{V}).$

In (B.5), we collect all the estimates to find

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})$
$\displaystyle\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}).$

By exchanging the roles of $\mathcal{U}$ and $\mathcal{V}$, we can derive a
differential inequality for $\mathcal{S}(V,\tilde{V})$:

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})$
$\displaystyle\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{H})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))$
$\displaystyle\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).$

∎

Now, we are ready to provide a proof of Lemma 4.2 with the help of Lemma B.1
and Lemma B.2.

Proof of Lemma 4.2: recall the inequalities in (B.1) and (B.4):

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}d(U,\tilde{U})&\leq-2m{\kappa}d(U,\tilde{U})+4m{\kappa}\mathcal{L}d(U,\tilde{U})+6\mathcal{L}(d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+4\sqrt{n}\mathcal{S}(V,\tilde{V}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}d(V,\tilde{V})&\leq-2n{\kappa}d(V,\tilde{V})+4n{\kappa}\mathcal{L}d(V,\tilde{V})+6\mathcal{L}(d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+2\mathcal{L}^{2}(4d(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))+4\sqrt{m}\mathcal{S}(U,\tilde{U}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(U,\tilde{U})&\leq-2m{\kappa}\mathcal{S}(U,\tilde{U})+4\mathcal{L}d(U,\tilde{U})+\mathcal{D}(\mathcal{H})d(U,\tilde{U})+6\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(U,\tilde{U})+2\mathcal{L}^{2}\mathcal{S}(V,\tilde{V})+4\sqrt{n}\mathcal{S}(V,\tilde{V}),\\\
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(V,\tilde{V})&\leq-2n{\kappa}\mathcal{S}(V,\tilde{V})+4\mathcal{L}d(V,\tilde{V})+\mathcal{D}(\mathcal{G})d(V,\tilde{V})+6\mathcal{L}(\mathcal{S}(V,\tilde{V})+\mathcal{S}(U,\tilde{U}))\\\
&\hskip
14.22636pt+8\mathcal{L}^{2}d(V,\tilde{V})+2\mathcal{L}^{2}\mathcal{S}(U,\tilde{U})+4\sqrt{m}\mathcal{S}(U,\tilde{U}).\end{aligned}$
(B.6)

We add all the inequalities in (B.6) to obtain the desired inequality for
$\mathcal{F}$:

$\displaystyle\frac{{\textup{d}}\mathcal{F}}{dt}$
$\displaystyle\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\mathcal{D}(\mathcal{H})}{{\kappa}}\right)\mathcal{F}+4(n+1){\kappa}\mathcal{L}(d(U,\tilde{U})+d(V,\tilde{V}))+6{\kappa}\mathcal{L}\mathcal{F}$
$\displaystyle\hskip
14.22636pt+12{\kappa}\mathcal{L}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))+16{\kappa}\mathcal{L}^{2}(d(U,\tilde{U})+d(V,\tilde{V}))+4{\kappa}\mathcal{L}^{2}(\mathcal{S}(U,\tilde{U})+\mathcal{S}(V,\tilde{V}))$
$\displaystyle\leq-{\kappa}\left(2m-8\sqrt{n}-\frac{\max\\{\mathcal{D}(\mathcal{H}),\mathcal{D}(\mathcal{G})\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.$

##  Appendix C Emergent dynamics of the DSOM model

In this section, we replace the product of two unitary groups
$\mathbf{U}(n)\times\mathbf{U}(m)$ by the product of two special orthogonal
groups $\mathbf{SO}(n)\times\mathbf{SO}(m)$. Since all elements of special
orthogonal matrices are real-valued, one has for any (real-valued) square
matrices $A$ and $B$,

$\langle
A,B\rangle_{\textup{F}}=\textup{tr}(AB^{\top})=\textup{tr}(BA^{\top})=\langle
B,A\rangle_{\textup{F}}.$ (C.1)

Thus, model (3.5) reduces to

$\begin{cases}\displaystyle\dot{U}_{j}=B_{j}U_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}-U_{j}U_{k}^{\top}U_{j}),\quad t>0,\\\
\displaystyle\dot{V}_{j}=C_{j}V_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
U_{j},U_{k}\rangle_{\textup{F}}(V_{k}-V_{j}V_{k}^{\top}V_{j}),\\\
\displaystyle(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{SO}(n)\times\mathbf{SO}(m),\quad
1\leq j\leq N,\end{cases}$ (C.2)

where $B_{j}\in\mathbb{R}^{n\times n\times n\times n}$ and
$C_{j}\in\mathbb{R}^{m\times m\times m\times m}$ are given rank-4 tensors
satisfying skew-symmetric properties:

$[B_{j}]_{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}=-[B_{j}]_{\alpha_{2}\beta_{2}\alpha_{1}\beta_{1}},\quad[C_{j}]_{\gamma_{1}\delta_{1}\gamma_{2}\delta_{2}}=-[{C}_{j}]_{\gamma_{2}\delta_{2}\gamma_{1}\delta_{1}}.$

######  Remark C.1.

Due to the symmetric property of the Frobenius inner product on real-valued
matrices, one has

$d_{ij}=\langle U_{i},U_{j}\rangle_{\textup{F}}=\langle
U_{j},U_{i}\rangle_{\textup{F}}=d_{ji},\quad c_{ij}=\langle
V_{i},V_{j}\rangle_{\textup{F}}=\langle
V_{j},V_{i}\rangle_{\textup{F}}=c_{ji}.$ (C.3)

Due to the relation (C.3), for example, the last term
$c_{ki}-c_{ik}+c_{jk}-c_{kj}$ in (A.3) vanishes. Thus, it would be expected
that the relation (C.3) relaxes the condition such as the dimension condition
$n\geq m>4\sqrt{n}$ in (4.7)(i) (see also Remark A.1).

The contents of this appendix are exactly the same as those of Section 4, and
we only provide a sketch of the proof. In Section C.1, we are concerned with
the identical (or homogeneous) Hamiltonian, and in Section C.2, the non-
identical (or heterogeneous) Hamiltonian is considered to exhibit the phase-
locked state.

###  C.1. Homogeneous ensemble

In this subsection, thanks to the solution splitting property, we assume that

$B_{j}\equiv O,\quad C_{j}\equiv O,\quad j=1,\cdots,N.$ (C.4)

Below, we state the main theorem for an identical ensemble under which the
complete aggregation occurs. For this, we recall diameters and aggregation
functional:

$\displaystyle\begin{aligned} &\mathcal{D}(\mathcal{U}(t))=\max_{1\leq i,j\leq
N}\|U_{i}(t)-U_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{U}(t))=\max_{1\leq
i,j\leq N}|n-d_{ij}(t)|,\\\ &\mathcal{D}(\mathcal{V}(t))=\max_{1\leq i,j\leq
N}\|V_{i}(t)-V_{j}(t)\|_{\textup{F}},\quad\mathcal{S}(\mathcal{V}(t))=\max_{1\leq
i,j\leq N}|m-c_{ij}(t)|,\\\
&\mathcal{L}=\mathcal{D}(\mathcal{U})+\mathcal{D}(\mathcal{V})+\mathcal{S}(\mathcal{U})+\mathcal{S}(\mathcal{V}).\end{aligned}$

######  Theorem C.1.

Suppose initial data satisfy

$\mathcal{L}^{0}<\alpha_{m,n}=\frac{-(12n+27)+\sqrt{(12n+27)^{2}+24m(3n+4)}}{4(4n+3)},$
(C.5)

and let $\\{(U_{i},V_{i})\\}$ be a solution to (C.2) with (C.4). Then, we have

$\lim_{t\to\infty}\mathcal{L}(t)=0.$

###### Proof.

By recalling the inequalities in (A.1) with the relation (C.1), we see

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})\leq-2m{\kappa}\mathcal{D}(\mathcal{U})+m{\kappa}\mathcal{D}(\mathcal{U})^{3}+6{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{V})\leq-2n{\kappa}\mathcal{D}(\mathcal{V})+n{\kappa}\mathcal{D}(\mathcal{V})^{3}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}.\end{aligned}$
(C.6)

Similarly, we use (A.8) to find

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{U})\leq-2m{\kappa}\mathcal{S}(\mathcal{U})+2m{\kappa}\mathcal{D}(\mathcal{U})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{V})\mathcal{D}(\mathcal{U})^{2},\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{S}(\mathcal{V})\leq-2n{\kappa}\mathcal{S}(\mathcal{V})+2n{\kappa}\mathcal{D}(\mathcal{V})^{2}+6{\kappa}\mathcal{S}(\mathcal{U})\mathcal{S}(\mathcal{V})+2{\kappa}\mathcal{S}(\mathcal{U})\mathcal{D}(\mathcal{V})^{2}.\end{aligned}$
(C.7)

Now, we add (C.6) and (C.7) to obtain a differential inequality for
$\mathcal{L}$:

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq-2{\kappa}m\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3}.$
(C.8)

If we introduce an auxiliary polynomial:

$f(s):=\left(2n+\frac{8}{3}\right)s^{2}+(4n+9)s-2m,$ (C.9)

then (C.8) can be written as

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{L}\leq{\kappa}\mathcal{L}f(\mathcal{L}).$

Furthermore, we notice that $\alpha_{m,n}$ in (C.5) becomes a unique positive
root of the quadratic polynomial $f$ in (C.9). Hence, the desired zero
convergence of $\mathcal{L}$ directly follows from the dynamical systems
theory. ∎

###  C.2. Heterogeneous ensemble

In this subsection, we are concerned with the heterogeneous Hamiltonians where
the natural frequency tensors are different in general. Similar to Section
4.2, there exist two skew-symmetric matrices $\Omega_{j}\in\mathbb{R}^{n\times
n}$ and $\Psi_{j}\in\mathbb{R}^{m\times m}$ such that

$B_{j}U_{j}=\Omega_{j}U_{j},\quad C_{j}V_{j}=\Psi_{j}V_{j},\quad 1\leq j\leq
N.$

Thus, our model reads as

$\begin{cases}\displaystyle\dot{U}_{j}=\Omega_{j}U_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
V_{j},V_{k}\rangle_{\textup{F}}(U_{k}-U_{j}U_{k}^{\top}U_{j}),\\\
\displaystyle\dot{V}_{j}=\Psi_{j}V_{j}+\frac{{\kappa}}{N}\sum_{k=1}^{N}\langle
U_{j},U_{k}\rangle_{\textup{F}}(V_{k}-V_{j}V_{k}^{\top}V_{j}),\\\
\displaystyle(U_{j},V_{j})(0)=(U_{j}^{0},V_{j}^{0})\in\mathbf{SO}(n)\times\mathbf{SO}(m),\quad
1\leq j\leq N,\end{cases}$ (C.10)

A crucial estimate used for the emergence of the phase-locked state is (4.12)
stated in Lemma 4.2 where the aggregation functional $\mathcal{F}$ tends to
zero. Here, $\mathcal{F}$ measures the inter-distance between any two solution
configurations $\\{(U_{i},V_{i})\\}$ and
$\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$:

$\mathcal{F}(t)=d(U,\tilde{U})(t)+d(V,\tilde{V})(t)+\mathcal{S}(U,\tilde{U})(t)+\mathcal{S}(V,\tilde{V})(t),$

where the diameters are defined in (4.8) and (4.10). We also denote the
diameters for natural frequencies:

$\mathcal{D}(\Omega):=\max_{1\leq i,j\leq
N}\|\Omega_{i}-\Omega_{j}\|_{\infty},\quad\mathcal{D}(\Psi):=\max_{1\leq
i,j\leq N}\|\Psi_{i}-\Psi_{j}\|_{\infty}.$

Without loss of generality, we may assume

$\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi).$

In the following lemma, we provide a temporal evolution of $\mathcal{F}$.

######  Lemma C.1.

Let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any two
solutions to (C.10). Then, the aggregation functionals $\mathcal{L}$ and
$\mathcal{F}$ satisfy

$\displaystyle\begin{aligned} &\frac{{\textup{d}}\mathcal{L}}{dt}\leq
2(1+3\sqrt{n})\mathcal{D}(\Omega)-2{\kappa}m\mathcal{L}+{\kappa}(4n+9)\mathcal{L}^{2}+{\kappa}\left(2n+\frac{8}{3}\right)\mathcal{L}^{3},\quad
t>0,\\\
&\frac{{\textup{d}}\mathcal{F}}{dt}\leq-{\kappa}\left(2m-\frac{\max\\{\mathcal{D}(\Omega),\mathcal{D}(\Psi)\\}}{{\kappa}}\right)\mathcal{F}+{\kappa}(4n+22)\mathcal{L}\mathcal{F}+20{\kappa}\mathcal{L}^{2}\mathcal{F}.\end{aligned}$
(C.11)

###### Proof.

For the inequality for $\mathcal{L}$, it directly follows from (C.8) in
Theorem C.1. Similarly, if we closely follow a proof of Lemma 4.2 presented in
Appendix B, then we find the desired inequality for $\mathcal{F}$. ∎

By applying $\eqref{Y-15}_{2}$, we establish the desired practical aggregation
estimate for (C.10).

######  Proposition C.1.

Suppose that the system parameters and initial data satisfy

$\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi),\quad{\kappa}>{\kappa}_{\textup{c}},\quad\mathcal{L}^{0}<\nu_{2},$
(C.12)

where ${\kappa}_{\textup{c}}$ and $\nu_{2}$ are specified in (C.14) and
(C.15), respectively, and let $\\{(U_{i},V_{i})\\}$ be a solution to (C.10).
Then, one has

$\lim_{{\kappa}\to\infty}\limsup_{t\to\infty}\mathcal{L}(t)=0.$

###### Proof.

We introduce an auxiliary polynomial:

$g(s):=2ms-(4n+9)s^{2}-\left(2n+\frac{8}{3}\right)s^{3}.$ (C.13)

Then, we notice that $g$ has three roots, say, $\alpha_{0}<0<\alpha_{1}$.
Since $\eqref{Y-15}_{2}$ is rewritten as

$\dot{\mathcal{L}}\leq{\kappa}\left(\frac{2(1+3\sqrt{n})\mathcal{D}(\mathcal{H})}{{\kappa}}-g(\mathcal{L})\right)=:{\kappa}p(s),$

for a sufficiently large ${\kappa}>0$, $p$ admits one negative root, say,
$\nu_{0}<0$ and two positive roots, say, $0<\nu_{1}<\nu_{2}$ with continuous
dependence on ${\kappa}$: for $\nu_{1}=\nu_{1}({\kappa})$ and
$\nu_{2}=\nu_{2}({\kappa})$,

$\lim_{{\kappa}\to\infty}\nu_{1}({\kappa})=0,\quad\lim_{{\kappa}\to\infty}\nu_{2}({\kappa})=\alpha_{1}.$

Since we assume $\eqref{Y-18}_{3}$, it follows from dynamical systems theory
that there exists a finite entrance time $T_{*}>0$ such that

$\mathcal{L}(t)<\nu_{1},\quad t>T_{*}.$

Hence, this yields the desired result. ∎

######  Remark C.2.

For the explicit value of ${\kappa}_{\textup{c}}$, we see that the cubic
polynomial $g$ in (C.13) admits the local maximum at $s=s_{*}$:

$s_{*}=\frac{-(4n+9)+\sqrt{(4n+9)^{2}+4m(3n+4)}}{6n+8}.$

Hence, ${\kappa}_{\textup{c}}$ is chosen to be

${\kappa}_{\textup{c}}:=\frac{2(1+3\sqrt{n})\mathcal{D}(\Omega)}{g(s_{*})}\quad\textup{so
that}\quad g(s_{*})>\frac{2(1+3\sqrt{n})}{{\kappa}_{\textup{c}}}.$ (C.14)

For this ${\kappa}_{\textup{c}}$, $\nu_{2}=\nu_{2}({\kappa})$ is completely
determined as the largest positive root of

$p(s)=0\quad\textup{for ${\kappa}>{\kappa}_{\textup{c}}$}.$ (C.15)

It follows from Proposition 5.1 that we can make $\mathcal{L}$ sufficiently
small by increasing the value of ${\kappa}$ for $t>T_{*}$. Thus, there exists
${\kappa}_{\textup{p}}>{\kappa}_{\textup{c}}$ such that for
${\kappa}>{\kappa}_{\textup{p}}$,

$(4n+22)\mathcal{L}+20\mathcal{L}^{2}<\frac{1}{2}\left(2m-\frac{\max\\{\mathcal{D}(\Omega),\mathcal{D}(\Psi)\\}}{{\kappa}}\right)=:\frac{1}{2}\Lambda.$

Hence, $\eqref{Y-15}_{2}$ yields

$\frac{d\mathcal{F}}{dt}\leq-\frac{1}{2}\Lambda\mathcal{F},\quad t>T_{*}.$

Since we have established the zero convergence of $\mathcal{F}$, we can obtain
the same result in Theorem 4.2. Thanks to exactly the same proof, we only
state the results.

######  Theorem C.2.

Suppose that the system parameters and initial data satisfy

$n\geq
m,\quad\mathcal{D}(\Omega)\geq\mathcal{D}(\Psi),\quad{\kappa}>{\kappa}_{\textup{p}},\quad\max\\{\mathcal{L}^{0},\tilde{\mathcal{L}}^{0}\\}\leq\nu_{2},$

and let $\\{(U_{i},V_{i})\\}$ and $\\{(\tilde{U}_{i},\tilde{V}_{i})\\}$ be any
two solutions to (C.10). Then, the following assertions hold.

  1. (1)

The aggregation functional $\mathcal{F}$ converges to zero exponentially.

  2. (2)

The normalized velocities $\dot{U}_{i}U_{i}^{\top}$ and
$\tilde{\dot{U}}_{i}\tilde{U}_{i}^{\top}$ synchronize:

$\|\dot{U}_{i}U_{i}^{\top}-\tilde{\dot{U}}_{i}\tilde{U}_{i}^{\top}\|_{\textup{F}}\leq
2{\kappa}(m+n)\mathcal{F}.$

  3. (3)

There exist special orthogonal matrices $X_{\infty}\in\mathbf{SO}(n)$ and
$Y_{\infty}\in\mathbf{SO}(m)$ such that

$\displaystyle\lim_{t\to\infty}U_{i}^{\top}(t)\tilde{U}_{i}(t)=X_{\infty},\quad\lim_{t\to\infty}\|\tilde{U}_{i}(t)-U_{i}(t)X_{\infty}\|_{\textup{F}}=0,$
$\displaystyle\lim_{t\to\infty}V_{i}^{\top}(t)\tilde{V}_{i}(t)=Y_{\infty},\quad\lim_{t\to\infty}\|\tilde{V}_{i}(t)-V_{i}(t)Y_{\infty}\|_{\textup{F}}=0.$

  4. (4)

System (C.10) exhibits asymptotic phase-locking: for any indices $i$ and $j$,

$\exists~{}~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\top}(t)\quad\textup{and}\quad\exists~{}~{}\lim_{t\to\infty}V_{i}(t)V_{j}^{\top}(t).$

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