# Pretty good fractional revival via magnetic fields:
theory and examples

Whitney Drazen, Mark Kempton111Department of Mathematics, Brigham Young
University, Provo, UT<EMAIL_ADDRESS>Gabor
Lippner222Department of Mathematics, Northeastern University, Boston, MA,
<EMAIL_ADDRESS>
###### Abstract

We develop the theory of pretty good quantum fractional revival in arbitrary
sized subsets of a graph, including the theory for fractional cospectrality of
subsets of arbitrary size. We use this theory to give conditions under which a
magnetic field can induce pretty good fractional revival, and give several
examples.

##  1 Introduction

“Fractional revival is a quantum transport phenomenon important for
entanglement generation in spin networks” [2]. In a continuous time quantum
walk fractional revival (FR) refers to the situation when the walk “preserves”
a subset at a certain moment in time. That is, there is a subset $K$ of the
nodes and a time $t$ such that if the initial state of the walk is supported
on $K$ then it is also supported on $K$ at time $t$. For entanglement
generation one would, typically, be interested in starting the walk from a
single vertex $v\in K$ and obtaining a superposition of the nodes in $K$ at
time $t$. For more background and a comprehensive characterization of FR see
[1].

Fractional revival is, in a sense, a relaxation of perfect state transfer
(PST). Nevertheless, finding examples of FR turned out to be nearly as
difficult as for PST. This naturally led to the study of further relaxations
of the phenomenon. Just as with PST, one can introduce an asymptotic variant
that has now been routinely dubbed “pretty good PST” in the literature [10,
5]. Informally, _pretty good fractional revival_ (PGFR) requires a sequence of
times at which the walk is closer and closer to actually preserving the subset
$K$. A complete characterization of PGFR between a pair of nodes on paths and
cycles was given in [3].

In a series of papers [7, 8, 4] (some of) the present authors have developed
methods to construct examples of pretty good (or even perfect) state transfer
using a diagonal perturbation of the matrix - sometimes referred to as a
magnetic field in the context of quantum spin networks. The goal of the
current paper is to extend these ideas to the case of PGFR. In particular, we
further develop the theory of PGFR to obtain a practical, verifiable condition
that guarantees that a subset of nodes will exhibit pretty good fractional
revival after adding a “generic” constant diagonal perturbation to the matrix.

In order to achieve this goal, we

  * •

devise a way to split the characterization of PGFR into somewhat separate
“eigenvector” and “eigenvalue” conditions,

  * •

introduce the notion of fractional cospectrality and provide a comprehensive
characterization of it in order to construct families of graphs that satisfy
the eigenvector part of the condition,

  * •

prove a suitable generalization of the well-known Kronecker condition for
pretty good state transfer to the case of pretty good fractional revival,

  * •

extend the field-trace method of [8] into a tool that allows one to verify
this new Kronecker-type condition if certain factors of the characteristic
polynomial are irreducible

  * •

prove that under a generic diagonal perturbation the relevant factors are
indeed irreducible.

The paper is structured as follows: in Section 2 we introduce pretty good
fractional revival (PGFR) and provide a spectral characterization. Then we
generalize Kronecker’s criterion to our setting and, using the field-trace
method developed in [8], we derive a sufficient condition for PGFR based on
the irreducibility of certain factors of the characteristic polynomial
together with a trace and degree condition on these factors.

In Section 3 we explain our generalization of the idea of cospectrality to the
fractional setting, and prove a theory analogous to the characterizations of
the original notion. This allows us to prove that for fractionally cospectral
subsets, under suitable diagonal perturbations of the adjacency matrix, the
factors of the characteristic polynomial relevant to PGFR are indeed
irreducible.

Finally, in Section 4 we construct examples where we can prove fractional
cospectrality of certain subsets and also verify the trace and degree
condition of Theorem 2.11, thereby guaranteeing PGFR in these graphs.

##  2 Pretty good fractional revival

We work in the following general setting: fix an index set $X$ and consider a
real symmetric matrix $M\in\mathbb{R}^{X\times X}$.

######  Definition 2.1.

Let $K\subset X$ be a subset of indices. We say that the $X\times X$ matrix
$M$ exhibits pretty good fractional revival with respect to $K$ if

$\operatorname{cl}\\{\exp(itM)_{K\times K}:t\geq 0\\}\cap
U(K)\nsubseteq\\{\rho\operatorname{Id}_{K\times K}:\rho\in\mathbb{C}\\},$

that is, if we take the family of $K\times K$ submatrices of $\exp(itM)$ for
all $t\geq 0$, then the closure of this family contains a unitary matrix with
at least two distinct eigenvalues.

Equivalently, there is a $K\times K$ unitary matrix $H$ with at least two
distinct eigenvalues, and a sequence $0<t_{1}\leq t_{2}\leq\dots$ such that
$\lim_{k\to\infty}\exp(it_{k}M)_{K\times K}=H$. This includes fractional
revival if $t=t_{1}=t_{2}=\dots$. The convergence can be understood entry wise
or, equivalently, with respect to any standard matrix norm.

Since the $K\times K$ submatrix of a unitary matrix $A$ is unitary if and only
if $A$ is block-diagonal relative to $K$, a further equivalent way to describe
pretty good fractional revival is to require that there is a sequence
$0<t_{1}\leq t_{2}\leq\dots$ and a matrix $A$ that is block-diagonal relative
to $K$ such that $\lim_{k\to\infty}\exp(it_{k}M)=A$, and $A_{K\times K}$ has
at least two distinct eigenvalues.

######  Remark 2.2.

This generalizes the concept of pretty good fractional revival on 2 nodes of a
graph from [3]: Two vertices $u$ and $v$ of a graph $G$ with adjacency matrix
$A$ exhibit pretty good fractional revival if, for all $\epsilon>0$ there is
some time $t>0$ such that

$|e^{itA}(u,u)|^{2}+|e^{itA}(u,v)|^{2}>1-\epsilon.$

######  Remark 2.3.

It would be perhaps more natural to require a non-diagonal $K\times K$ unitary
matrix in the closure of $\exp(itM)_{K\times K}$ instead of simply one that’s
not the multiple of the identity. However, it turns out that for primitive
matrices (eg for adjacency matrices of connected graphs) this stronger
requirement is equivalent to the one in Definition 2.1. See the second part of
Theorem 2.7 for an explanation of this.

Notice that $\\{\exp(itM):t\geq 0\\}\subset\langle M\rangle$ is a bounded
subset of the (finite dimensional) polynomial algebra generated by $M$. Hence
if $M$ exhibits pretty good fractional revival with respect to $K$ and
$\lim_{k\to\infty}\exp(it_{k}M)_{K\times K}=H$ then there is a unitary matrix
$\hat{H}\in\langle M\rangle$ that is block diagonal relative to $K$, and
$\operatorname{Id}_{K\times K}\neq H=\hat{H}_{K\times K}$.

###  2.1 Non-degenerate partition of the spectrum

Here we provide a spectral characterization of pretty good fractional revival
that can be summarized as follows: the subset $K$ induces a natural partition
$\mathcal{P}_{K}$ of the eigenvalues of $M$, and pretty good fractional
revival is exhibited with respect to $K$ if and only if a certain simultaneous
approximation problem is solvable in this partition.

Let us start by recalling some important notions and facts from [1].

######  Definition 2.4.

Let $M=\sum_{i=1}^{d}\theta_{i}E_{i}$ be the spectral decomposition of $M$,
and $K\subset X$ a non-empty set of indices.

  1. 1.

We denote by $D_{K}$ the $X\times X$ diagonal matrix whose entries on the
diagonal are 1 in $K$ and 0 outside of $K$.

  2. 2.

The _eigenvalue support_ of $K$ is the binary relation

$\Phi_{K}=\\{(\theta_{r},\theta_{s}):E_{r}D_{K}E_{s}\neq 0\\}.$

  3. 3.

Define $\mathcal{P}_{K}=(\Pi_{0},\Pi_{1},\dots,\Pi_{s})$ to be the partition
of $\\{1,2,\dots,d\\}$ where

$\Pi_{0}=\\{i:(E_{i})_{K\times K}=0\\},$

and $\Pi_{1},\dots,\Pi_{s}$ are the remaining equivalence classes of the
transitive closure of $\Phi_{K}$.

######  Lemma 2.5 (Lemma 2.5 and Theorem 2.10 from [1]).

$A=\sum_{j}c_{j}E_{j}$ is block-diagonal relative to $K$ if and only if the
$c_{j}$s are equal to each other within each part $\Pi_{r}:1\leq r\leq s$.

######  Definition 2.6.

The partition $\mathcal{P}_{K}$ is _non-degenerate_ if there is a _mod 1 non-
constant_ vector $(\rho_{1},\dots,\rho_{s})\in\mathbb{R}^{s}$ such that for
all $\varepsilon>0$ there is a $t>0$ so that for any $1\leq r\leq s$

$\forall j\in\Pi_{r}:\lVert t\cdot\theta_{j}-\rho_{r}\rVert<\varepsilon.$ (1)

where $\lVert x\rVert=\min\\{\lvert x-n\rvert:n\in\mathbb{Z}\\}$ is the
distance of $x$ to the nearest integer. In particular, $s\geq 2$ is required.

######  Theorem 2.7.

The matrix $M$ exhibits pretty good fractional revival with respect to $K$ if
and only if the partition $\mathcal{P}_{K}$ is non-degenerate. Furthermore, if
$M$ is primitive then pretty good fractional revival also implies that
$\operatorname{cl}\\{\exp(itM)_{K\times K}:t\geq 0\\}\cap U(K)$ contains a
non-diagonal matrix.

###### Proof.

First assume $M$ exhibits pretty good fractional revival with respect to $K$,
that is, there is a block-diagonal matrix $A$ such that $H=A_{K\times K}$ has
at least two distinct eigenvalues, and a sequence $0<t_{1}\leq t_{2}\leq\dots$
such that $\lim_{k\to\infty}\exp(2\pi it_{k}M)=A$. Since $\exp(2\pi
itM)=\sum_{j}\exp(2\pi it\theta_{j})E_{j}$, it follows that
$\mu_{j}=\lim_{k\to\infty}exp(2\pi it_{k}\theta_{j})$ exists for all $j$, and
that $A=\sum\mu_{j}E_{j}$. By Lemma 2.5 this implies that there exist reals
$\rho_{1},\dots,\rho_{s}$ such that $\mu_{j}=\exp(2\pi i\rho_{r})$ for all
$j\in\Pi_{r}$. Thus $\lVert t_{k}\theta_{j}-\rho_{r}\rVert\to 0$ as
$k\to\infty$ for all $j\in\Pi_{r}$. Finally, since $H$ has at least two
distinct eigenvalues, all the $\mu_{j}$s can’t be identical, which implies
that all the $\rho_{r}$ cannot be congruent mod 1. This proves the only if
part.

Conversely, if we are given $(\rho_{1},\dots,\rho_{s})$ that proves the non-
degenerateness of $\mathcal{P}_{K}$, let $t_{k}$ be the time for which (1)
holds for $\varepsilon=1/k$. Then, clearly, $\lim_{k\to\infty}\exp(2\pi
it_{k}\theta_{j})=\exp(2\pi i\rho_{r})$ for all $j\in\Pi_{r}$. Choose a
subsequence for which $\gamma_{j}=\lim_{k\to\infty}\exp(2\pi
it_{k}\theta_{j})$ also exists for all $j\in\Pi_{0}$. (This can be done simply
by compactness.) Then

$A=\lim_{k\to\infty}\exp(2\pi
it_{k}M)=\sum_{j\in\Pi_{0}}\gamma_{j}E_{j}+\sum_{r=1}^{s}\left(\exp(2\pi
i\rho_{r})\sum_{j\in\Pi_{r}}E_{j}\right)$ (2)

is block-diagonal by Lemma 2.5, and since not all the $\rho_{r}$s are
congruent mod 1, the restriction $H=A_{K\times K}$ is not a scalar multiple of
the identity.

To prove that the non-degeneracy of $\mathcal{P}_{K}$ implies the existence of
non-diagonal matrices in $\operatorname{cl}\\{\exp(itM)_{K\times K}:t\geq
0\\}\cap U(K)$, it suffices to show that $H=A_{K\times K}$ is non-diagonal.
The argument below is adapted from the proof of Lemma 2.9 in [1]. Let $x\in X$
and let $e_{x}$ denote the corresponding standard basis vector. Then, for any
$j\in\Pi_{r}$ we can write

$H(x,x)E_{j}e_{x}=E_{j}Ae_{v}=\exp(2\pi i\rho_{r})E_{j}e_{x},$

which implies that $E_{j}e_{x}=0$ unless $\exp(2\pi i\rho_{r})=H(x,x)$. By
symmetry the same holds for $e_{x}^{\mathrm{T}}E_{j}$.

Now suppose, for a contradiction, that $H$ is diagonal. Since it is not a
scalar multiple of the identity according to the first half of the theorem, we
can find two elements $x,y\in X$ such that $H(x,x)\neq H(y,y)$. Thus
$e_{y}^{\mathrm{T}}E_{j}e_{x}=0$ for all $j$, since $\exp(2\pi i\rho_{r})$
can’t equal to both $H(x,x)$ and $H(y,y)$. Then, however,
$e_{y}^{\mathrm{T}}M^{n}e_{x}=\sum_{j}\theta_{j}^{s}e_{y}^{\mathrm{T}}E_{j}e_{x}=0$
for any integer $n\geq 0$ which contradicts the primitivity of $M$. ∎

###  2.2 A number theoretic characterization

In this section we develop a method to verify that a partition is degenerate.
This can be considered as the generalization of previous results for the case
of pretty good state transfer. Its basis, as in earlier results, is the
following number theoretic lemma due to Kronecker.

######  Lemma 2.8 (Kronecker).

Let $\theta_{1},...,\theta_{k}$ and $\zeta_{1},...,\zeta_{k}$ be arbitrary
real numbers. For an arbitrarily small $\varepsilon>0$, the system of
inequalities

$\lVert\theta_{j}y-\zeta_{j}\rVert<\varepsilon\ (j=1,...,k),$

has a solution $y$ if and only if, for integers $\ell_{1},...,\ell_{k}$,

$\ell_{1}\theta_{1}+\cdots+\ell_{k}\theta_{k}=0,$

implies

$\lVert\ell_{1}\zeta_{1}+\cdots+\ell_{k}\zeta_{k}\rVert=0.$

######  Lemma 2.9.

Given a sequence of real numbers $\theta_{1},\dots,\theta_{k}$ and a partition
$\mathcal{P}=(\Pi_{1},\dots,\Pi_{s})$ of $\\{1,2,\dots,k\\}$, the following
are equivalent:

  1. i)

$\mathcal{P}$ is non-degenerate in the sense of Definition 2.6.

  2. ii)

There is a pair of indices $1\leq r_{1},r_{2}\leq s$ such that no sequence
integers $\ell_{1},\ell_{2},\dots,\ell_{k}$ satisfies all of the following:

    1. a)

$\sum_{1}^{k}\ell_{j}\theta_{j}=0$,

    2. b)

$\sum_{j\in\Pi_{r}}\ell_{j}=0$ for all $r\neq r_{1},r_{2}$

    3. c)

$\sum_{j\in\Pi_{r_{1}}}\ell_{j}=-1$ and $\sum_{j\in\Pi_{r_{2}}}\ell_{j}=1$.

######  Remark 2.10.

Note that this lemma is a direct generalization of [3][Theorem 2.4] which only
considers the $s=2$ case. The only essential new ingredient in the following
proof is the use of integer lattices in place of subgroups of $\mathbb{Z}$.

###### Proof.

First, suppose that $\mathcal{P}$ is non-degenerate as witnessed by the
sequence $(\rho_{1},\dots,\rho_{s})$ where
$\lVert\rho_{r_{1}}-\rho_{r_{2}}\rVert>0$ for some $r_{1},r_{2}$. Assume, for
the same $r_{1},r_{2}$, that there exists integers $\ell_{1},\dots,\ell_{k}$
satisfying the above criteria. Let
$\varepsilon=\varepsilon_{0}/(k\max\\{|\ell_{j}|\\})$, and choose $t$ such
that for any $1\leq r\leq s$

$\forall j\in\Pi_{r}:\lVert t\cdot\theta_{j}-\rho_{r}\rVert<\varepsilon.$

Then for all $j\in\Pi_{r}$

$\lVert t\ell_{j}\theta_{j}-\ell_{j}\rho_{r}\rVert<\varepsilon_{0}/k$

and thus

$\left\lVert\sum_{1}^{k}t\ell_{j}\theta_{j}-\sum_{r=1}^{s}\rho_{r}\sum_{j\in\Pi_{r}}\ell_{j}\right\rVert<\varepsilon_{0}.$

Here the left and the right sums are both $\rho_{r_{1}}-\rho_{r_{2}}$ and the
middle is 0. Thus $\lVert\rho_{r_{1}}-\rho_{r_{2}}\rVert<\varepsilon_{0}$.
Since this holds for any $\varepsilon_{0}$, we get that
$\lVert\rho_{r_{1}}-\rho_{r_{2}}\rVert=0$, contradicting the choice of
$r_{1},r_{2}$. This proves the $i)\Rightarrow ii)$ implication.

Next, we prove $ii)\Rightarrow i)$. Assume, without loss of generality, that
$r_{1}=1$ and $r_{2}=2$ is a pair of indices for which $ii)$ holds. Further
assume, again without loss of generality, that $\theta_{1}\in\Pi_{1}$. Let
$\tilde{\theta_{j}}=\theta_{j}-\theta_{1}:j=2,\dots,k$. Note that

$\sum_{j=2}^{k}\ell_{j}\tilde{\theta_{j}}=0\mbox{ implies
}\sum_{1}^{k}\ell_{j}\theta_{j}=0\mbox{ and }\sum_{1}^{k}\ell_{j}=0\mbox{
where $\ell_{1}=-\sum_{2}^{k}\ell_{j}$}.$ (3)

Consider the following integer lattice

$\mathcal{S}=\\{(a_{2},\dots,a_{s})\in\mathbb{Z}^{s-1}|\exists\mkern
2.0mu\ell_{2},\dots,\ell_{k}\in\mathbb{Z}:\sum_{2}^{k}\ell_{j}\tilde{\theta_{j}}=0\mbox{
and }a_{r}=\sum_{j\in\Pi_{r}}\ell_{j}\mbox{ for all $2\leq r\leq s$}\\}.$

By (3) we see that $(1,0,0,\dots,0)\not\in\mathcal{S}$ hence
$\mathcal{S}\subsetneq\mathbb{Z}^{s-1}$. This implies that the dual lattice
$\mathcal{S}^{*}\supsetneq\mathbb{Z}^{s-1}$. (For instance, because the
determinant of $\mathcal{S}$ has to be greater than 1, and thus the
determinant of the dual has to be less than 1 in absolute value so it can’t be
an integer lattice.) In other words, there exists vector
$(\tilde{\rho}_{2},\dots,\tilde{\rho}_{s})$ that is not congruent to
$(0,0,\dots,0)$ mod 1, such that
$\sum_{2}^{s}\tilde{\rho}_{r}a_{r}\in\mathbb{Z}$ for all
$(a_{2},\dots,a_{s})\in\mathcal{S}$.

Now let

$\zeta_{j}=\left\\{\begin{array}[]{lll}\tilde{\rho}_{r}&\mbox{ if
}&j\in\Pi_{r},r\geq 2\\\ 0&\mbox{ if }&j\in\Pi_{1}\end{array}\right.$

If $\ell_{2},\dots,\ell_{k}$ are integers such that
$\sum_{2}^{k}\ell_{j}\tilde{\theta_{j}}=0$, then
$(a_{2},\dots,a_{s})\in\mathcal{S}$ for
$a_{r}=\sum_{j\in\Pi_{r}}\ell_{r}:r=2,\dots,s$. Hence

$\sum_{2}^{k}\ell_{j}\zeta_{j}=\sum_{r=1}^{s}\left(\tilde{\rho}_{r}\sum_{j\in\Pi_{r}}\ell_{j}\right)=0+\sum_{r=2}^{s}\tilde{\rho}_{r}a_{r}\in\mathbb{Z}$

by the choice of $(\tilde{\rho}_{2},\dots,\tilde{\rho}_{s})\in\mathcal{S}*$.
So Lemma 2.8 implies that for any $\varepsilon>0$ there is a
$t=t(\varepsilon)$ such that we have $\lvert
t\tilde{\theta_{j}}-\tilde{\rho}_{r}\rvert<\varepsilon\pmod{1}$ for all
$j=2,\dots,k$ where $\Pi_{r}$ is the partition containing the index $j$. To
finish the proof, let $\rho_{1}$ be a mod 1 accumulation point of the sequence
$t(\varepsilon)\theta_{1}$ as $\varepsilon\to 0$. Then for any $\varepsilon>0$
there is a $t=t(\varepsilon)$ such that $\lVert
t\theta_{1}-\rho_{1}\rVert<\varepsilon$ and $\lvert
t\tilde{\theta_{j}}-\tilde{\rho}_{r}\rvert<\varepsilon\pmod{1}$ for all $j$,
simultaneously. Set $\rho_{r}=\tilde{\rho}_{r}+\rho_{1}$. Then
$(\rho_{1},\dots,\rho_{s})$ is not the constant vector mod 1. Since $\lVert
t(\varepsilon)\theta_{j}-\rho_{r}\rVert\leq\lVert
t(\varepsilon)\tilde{\theta_{j}}-\tilde{\rho}_{r}\rVert+\lVert
t(\varepsilon)\theta_{1}-\rho_{1}\rVert<2\varepsilon$ for $j=2,\dots,k$ where
$\Pi_{r}$ is the partition containing the index $j$. Thus the partition is
non-degenerate. ∎

In general, verifying ii) is difficult. Here we generalize a tool from [4]
that allows to show ii) holds under certain conditions that are easy to
verify.

######  Theorem 2.11.

Fix a field $\mathcal{F}$, a sequence of real numbers
$\theta_{1},\dots,\theta_{k}$ and a partition
$\mathcal{P}=(\Pi_{1},\dots,\Pi_{s})$ of $\\{1,2,\dots,k\\}$. Suppose there
are polynomials $P_{1},P_{2},\dots,P_{s}\in\mathcal{F}[x]$ that are
irreducible over $\mathcal{F}$ and such that $\Pi_{r}$ is exactly the set of
roots of $P_{r}$ for each $1\leq r\leq s$.

If for some pair of integers $1\leq r_{1},r_{2}\leq s$

$\frac{\operatorname{Tr}(P_{r_{1}})}{\deg(P_{r_{1}})}\neq\frac{\operatorname{Tr}(P_{r_{2}})}{\deg(P_{r_{2}})},$

where $\operatorname{Tr}$ denotes the trace (i.e. the sum of roots) of a
polynomial, then the partition $\mathcal{P}$ is non-degenerate.

###### Proof.

We verify ii) of Lemma 2.9 for $r_{1},r_{2}$ via the field trace a method
introduced in [8]. For a field extension $\mathcal{K}$ of $\mathcal{F}$, the
_field trace_ is a linear functional
$\operatorname{Tr}_{\mathcal{K}/\mathcal{F}}:\mathcal{K}\rightarrow\mathcal{F}$
defined for each element $\alpha\in\mathcal{K}$ as the trace of the
$\mathcal{F}$-linear map $x\mapsto\alpha x$. See [9] for details about the
field trace.

For each $1\leq r\leq s$ let $\mathcal{L}_{r}$ denote the splitting field of
$P_{r}$ over $\mathcal{F}$. Since $P_{r}$ is irreducible, for any
$j\in\Pi_{r}$ we have

$\operatorname{Tr}_{\mathcal{L}_{r}/\mathcal{F}}(\theta_{j})=\frac{[\mathcal{L}_{r}:\mathcal{F}]}{\deg
P_{j}}\sum_{j\in\Pi_{r}}\theta_{j}=\frac{[\mathcal{L}_{r}:\mathcal{F}]}{\deg
P_{j}}\operatorname{Tr}(P_{r})$

according to Lemma A.1 of [4]. Let us suppose for a contradiction that there
are integers $\ell_{j}\in\mathbb{Z}:j=1,\dots,k$ satisfying

$\displaystyle\sum_{j=1}^{k}\ell_{j}\theta_{j}$ $\displaystyle=0$ (4)
$\displaystyle\sum_{j\in\Pi_{r_{1}}}\ell_{j}$ $\displaystyle=1\text{ and
}\sum_{j\in\Pi_{r_{2}}}\ell_{j}=-1$ (5)
$\displaystyle\sum_{j\in\Pi_{r}}\ell_{j}$ $\displaystyle=0\text{ for all
}r\neq r_{1},r_{2}.$ (6)

Let $\mathcal{K}/\mathcal{F}$ the smallest field extension containing
$\mathcal{L}_{1},...,\mathcal{L}_{s}$. We apply
$\operatorname{Tr}_{\mathcal{K}/\mathcal{F}}$ to (4). Then, according to (5),
(6), and the basic properties of the field trace from Lemma A.1 in [4],

$\displaystyle 0$
$\displaystyle=\operatorname{Tr}_{\mathcal{K}/\mathcal{F}^{\prime}}\left(\sum_{j=1}^{k}\ell_{j}\theta_{j}\right)=\sum_{r=1}^{s}[\mathcal{K}:\mathcal{L}_{r}]\operatorname{Tr}_{\mathcal{L}_{r}/F}\left(\sum_{j\in\Pi_{r}}\ell_{j}\theta_{j}\right)$
$\displaystyle=\sum_{r=1}^{s}\left([\mathcal{K}:\mathcal{L}_{r}]\sum_{j\in\Pi_{r}}\ell_{j}\operatorname{Tr}_{\mathcal{L}_{r}/F}\left(\theta_{j}\right)\right)$
$\displaystyle=\sum_{r=1}^{s}\left(\frac{[\mathcal{K}:\mathcal{L}_{r}][\mathcal{L}_{r}:\mathcal{F}]}{\deg(P_{r})}\operatorname{Tr}(P_{r})\sum_{j\in\Pi_{r}}\ell_{j}\right)$
$\displaystyle=[\mathcal{K}:\mathcal{F}]\sum_{r=1}^{s}\frac{\operatorname{Tr}(P_{r})}{deg(P_{r})}\sum_{j\in\Pi_{r}}\ell_{j}=[\mathcal{K}:\mathcal{F}^{\prime}]\left(\frac{\operatorname{Tr}(P_{1})}{deg(P_{1})}-\frac{\operatorname{Tr}(P_{2})}{deg(P_{2})}\right).$

This contradicts
$\frac{\operatorname{Tr}(P_{1})}{deg(P_{1})}\neq\frac{\operatorname{Tr}(P_{2})}{deg(P_{2})}$
hence ii) of Lemma 2.9 holds for $r_{1},r_{2}$, and thus $\mathcal{P}$ is non-
degenerate.

∎

##  3 Generalized cospectrality

It is now a well-known fact that perfect state transfer can be characterized
by an eigenvector and an eigenvalue condition. The eigenvector condition is
called strong cospectrality (see [6]), a strengthening of the classical notion
of cospectrality of two nodes. In [8, 4] cospectrality was used as a starting
point to construct examples for pretty good state transfer.

In [1] the study of fractional revival between two nodes led to a
generalization of both cospectrality and strong cospectrality of two nodes to
the fractional setting. Further, decomposability was identified as the correct
generalization of strong (fractional) cospectrality to arbitrary subsets.
However, the analogous extension of the theory of cospectrality was not
discussed.

###  3.1 $H$-cospectrality

Since our goal is to construct examples that admit pretty good fractional
revival, in this section we complete the picture by developing the theory of
(fractional) cospectrality for arbitrary subsets. It turns out that most
features of the classical theory carry over to the general setting.

Let $K\subset X$ be a subset of indices. For the sake of the applications
later on, it turns out to be simpler to work in the more general setting of
complex vector spaces and normal matrices. We consider $\mathbb{C}^{X}$ and
$\mathbb{C}^{K}$ equipped the usual Hermitian scalar product $\langle
v,w\rangle:=v^{*}w\in\mathbb{C}$. For a vector $v\in\mathbb{C}^{X}$
(respectively a matrix $A\in\mathbb{C}^{X\times X}$) we denote
$\tilde{v}=v_{K}$ (respectively $\tilde{A}=A_{K\times K}$) its restriction to
the subset $K$. Conversely, given a vector $v\in\mathbb{C}^{K}$ we denote
$\hat{v}\in\mathbb{C}^{X}$ its extension by 0s to the other coordinates of
$X$. Note that

$\widetilde{A\hat{v}}=\tilde{A}v\mbox{ and }\langle
A\hat{v},\hat{w}\rangle=\langle\tilde{A}v,w\rangle$ (7)

for any $A\in\mathbb{C}^{X\times X}$ and $v,w\in\mathbb{C}^{K}$.

Let $M\in\mathbb{C}^{X\times X}$ and $H\in\mathbb{C}^{K\times K}$ be normal
matrices with spectral decompositions $M=\sum_{i=1}^{d}\theta_{i}E_{i}$ and
$H=\sum_{j=1}^{r}\rho_{j}F_{j}$. The $E_{i}$ and $F_{j}$ are self-adjoint
projections.

######  Definition 3.1.

We say that $K$ is $H$-cospectral in $M$ (or $H$-cospectral, for short) if
there is an orthonormal (with respect to the Hermitian scalar product)
eigenbasis $\psi_{1},\dots,\psi_{|X|}$ such that $\tilde{\psi_{j}}$ is either
0 or an eigenvector of $H$ for all $j=1,\dots,\lvert X\rvert$.

######  Remark 3.2.

  * •

Clearly, the dependence on $H$ is only via its spectral idempotents
$\\{F_{j}\\}$. However, it is often more convenient to refer to the matrix $H$
instead of a collection of projectors.

  * •

This generalizes fractional cospectrality between two nodes (the $\lvert
K\rvert=2$ case) introduced in [1] to subsets of arbitrary size. In
particular, if

$H=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ (8)

then we recover the classical notion of cospectrality.

######  Theorem 3.3.

Let $K\subset X,M,H$ as above. The following are equivalent:

  1. 1.

$K$ is fractionally cospectral with respect to $H$.

  2. 2.

$H\widetilde{M^{k}}=\widetilde{M^{k}}H$ for all $k$.

  3. 3.

$H\tilde{E_{i}}=\tilde{E_{i}}H$ for all $j$.

  4. 4.

$F_{j}\tilde{E_{i}}=\tilde{E_{i}}F_{j}$ for all $i,j$.

  5. 5.

For any $v,w$ eigenvectors of $H$ belonging to different eigenvalues,
$\langle\tilde{E_{i}}v,w\rangle=0$ for all $i$.

  6. 6.

For each $i$, there is an orthonormal basis of $\operatorname{Im}E_{i}$ that
contains exactly $\dim\\{E_{i}\hat{v}:v\in\operatorname{Im}F_{j}\\}$ vectors
that satisfy $0\neq\tilde{\psi}\in\operatorname{Im}F_{j}$ for each $j$, and
the rest of the basis elements satisfy $\tilde{\psi}=0$.

  7. 7.

For any $v,w$ eigenvectors of $H$ belonging to different eigenvalues, the
subspaces $\langle M^{k}\hat{v}:k=0,1,\dots\rangle$ and $\langle
M^{k}\hat{w}:k=0,1,\dots\rangle$ are orthogonal.

###### Proof.

We prove implications in a cyclic order.

#####  $\ref{it:frco}\implies\ref{it:HMk}$:

Let $\psi_{1},\dots,\psi_{\lvert X\rvert}$ as in Definition 3.1 with
corresponding eigenvalues $\lambda_{1},\dots,\lambda_{\lvert X\rvert}$. Then
$M^{k}=\sum\lambda_{i}^{k}\psi_{i}\psi_{i}^{*}$ and hence
$\widetilde{M^{k}}=\sum\lambda_{i}^{k}\tilde{\psi_{i}}\tilde{\psi_{i}}^{*}$.
The $\tilde{\psi_{i}}$s are all eigenvectors of $H$. If $Hv=\rho v$ then
$Hvv^{*}=\rho vv^{*}=v(\bar{\rho v})^{*}=v(H^{*}v)^{*}=vv^{*}H$ by the
normality of $H$. Thus $H$ commutes with all
$\tilde{\psi_{i}}\tilde{\psi_{i}}^{*}$ terms, and in turn also with
$\widetilde{M^{k}}$ for any $k$.

#####  $\ref{it:HMk}\implies\ref{it:HEi}$:

This follows since each $E_{i}$ is a polynomial of $M$, and thus
$\tilde{E_{i}}$ is a linear combination of the $\widetilde{M^{k}}$s.

#####  $\ref{it:HEi}\implies\ref{it:FjEi}$:

This follows since each $F_{j}$ is a polynomial of $H$.

#####  $\ref{it:FjEi}\implies\ref{it:Eivw}$:

Let $v$ be an eigenvector in the $F_{j}$ eigenspace of $H$. That is,
$F_{j}v=v$. Then $F_{j}w=0$ since $w$ is in a different eigenspace. Now, using
the self-adjointness of the $F_{j}$, we get
$\langle\tilde{E_{i}}v,w\rangle=\langle\tilde{E_{i}}F_{j}v,w\rangle=\langle
F_{j}\tilde{E_{i}}v,w\rangle=\langle\tilde{E_{i}}v,F_{j}w\rangle=0$ as
claimed.

#####  $\ref{it:Eivw}\implies\ref{it:basisrk}$:

Let $E=E_{i}$ for some fixed $i$. For any $j$ consider the subspace
$\mathcal{S}_{j}\subset\operatorname{Im}E$ defined as

$\mathcal{S}_{j}=\\{E\hat{v}:v\in\operatorname{Im}F_{j}\\}.$

If $v\in\operatorname{Im}F_{j_{1}}$ and $w\in\operatorname{Im}F_{j_{2}}$ for
some $j_{1}\neq j_{2}$, then $\langle E\hat{v},E\hat{w}\rangle=\langle
E\hat{v},\hat{w}\rangle=\langle\tilde{E}v,w\rangle=0$ by the condition. Hence
$\mathcal{S}_{j_{1}}\bot\mathcal{S}_{j_{2}}$. Thus we can pick an orthonormal
basis in each of the $\mathcal{S}_{j}$s as well as the orthogonal complement
of $\oplus\mathcal{S}_{j}$ in $\operatorname{Im}E$. We claim that the union of
these bases satisfies our requirements. To show this, consider
$u\in\operatorname{Im}E$ that is orthogonal to $\mathcal{S}_{j}$, and take any
$v\in\operatorname{Im}F_{j}$. Then

$0=\langle u,E\hat{v}\rangle=\langle Eu,\hat{v}\rangle=\langle
u,\hat{v}\rangle=\langle\tilde{u},v\rangle$

and thus $\tilde{u}$ is orthogonal to $\operatorname{Im}F_{j}$. This means
that if $u\in\mathcal{S}_{j}$ then it is orthogonal to $\mathcal{S}_{l}$ for
all $l\neq j$, hence $\tilde{u}$ is orthogonal to $\operatorname{Im}F_{l}$ for
all $l\neq j$. That is only possible if $\tilde{u}\in\operatorname{Im}F_{j}$.
If $u=E\hat{v}\neq 0$, then $0<\langle
E\hat{v},E\hat{v}\rangle=\langle\hat{v},E\hat{v}\rangle=\langle
v,\tilde{u}\rangle$, thus $\tilde{u}\neq 0$ as claimed. Lastly if
$u\in\operatorname{Im}E$ is orthogonal to all $\mathcal{S}_{j}$s, then by the
same argument $\tilde{u}$ is orthogonal to all $\operatorname{Im}F_{j}$s,
hence $\tilde{u}=0$ in this case.

#####  $\ref{it:basisrk}\implies\ref{it:frco}$:

The union for all $i$ of the bases defined in 6 obviously satisfies Definition
3.1.

#####  $\ref{it:Eivw}\Longleftrightarrow\ref{it:Mkw}$:

This follows since $\tilde{E_{i}}$ is a linear combination of the
$\widetilde{M^{k}}$s as we have seen before, and vice versa:
$\widetilde{M^{k}}$ is obviously a linear combination of the $\tilde{E_{i}}$s.
Finally $\langle M^{a}\hat{v},M^{b}\hat{w}\rangle=\langle
M^{a+b}\hat{v},\hat{w}\rangle=\langle\widetilde{M^{a+b}}v,w\rangle$, so the
latter is 0 for all $a,b$ if and only if the two subspaces are orthogonal. ∎

The following observation will be useful later, but we state it here since the
computation is essentially the same as in the proof of
$\ref{it:Eivw}\implies\ref{it:basisrk}$ above.

######  Claim 3.4.

Let $v\in\operatorname{Im}F_{j}$. Then

$\langle
E_{i}\hat{v},E_{i}\hat{v}\rangle=\langle\hat{v},E_{i}\hat{v}\rangle=\langle
v,\widetilde{E_{i}\hat{v}}\rangle=\langle v,\tilde{E_{i}}v\rangle,$

hence $E_{i}\hat{v}\neq 0$ if and only if its restriction to $K$ is non-zero.
And in this case the restriction, $\tilde{E_{i}}v$, contains a component
parallel to $v$.

######  Remark 3.5.

In the case when $|K|=2$ and $H$ is a 2-by-2 matrix with eigenvectors $(p,q)$
and $(-q,p)$, our definition of $K$-cospectral coincides with what is called a
fractionally cospectral pair of nodes in [1, Theorem 3.3, condition (ii)],
hence this is indeed a direct generalization of that notion.

###  3.2 A factorization of $\phi(M,t)$

######  Definition 3.6.

Let $K$ be $H$-cospectral in $M$. Define
$b_{i,j}=\dim\\{E_{i}\hat{v}:v\in\operatorname{Im}F_{j}\\}$ for any $1\leq
j\leq r$, and $b_{i,0}=\dim\\{u\in\operatorname{Im}E_{i}:\tilde{u}=0\\}$. Let
$P_{j}(t)=\prod_{i=1}^{d}(t-\theta_{i})^{b_{i,j}}:j=0,\dots,r$.

The following is immediate from 6 of Theorem 3.3:

######  Corollary 3.7.

Let $K$ be $H$-cospectral in $M$. Then the characteristic polynomial
$\phi(M)=\phi(M,t)$ can be decomposed as

$\phi(M,t)=\prod_{j=0}^{r}P_{j}(t).$

$H$-cospectrality becomes the most useful when $H$ has $|K|$ distinct
eigenvalues. That is, when all $F_{j}$s have rank 1. Let us fix an eigenvector
$v_{j}\in\operatorname{Im}F_{j}$ of $H$ for each $j$.

######  Lemma 3.8.

Assume $H$ has $K$ distinct eigenvalues. Then

  1. 1.

$P_{j}(t)=p_{v_{j}}(t)$ for all $1\leq j\leq r=|K|$, where $p_{v_{j}}$ is the
minimal polynomial of $M$ relative to $v_{j}$, that is, the smallest degree
monic polynomial $p$ such that $p(M)\hat{v_{j}}=0$, or, equivalently, the
characteristic polynomial of the action of $M$ restricted to the space
$\langle v_{j},Mv_{j},M^{2}v_{j},\dots,\rangle$.

  2. 2.

$(\theta_{a},\theta_{b})\in\Phi_{K}$, if and only if there is a $j$ such that
$\theta_{a},\theta_{b}$ are both roots of $P_{j}(t)$.

###### Proof.

It is well-known that the relative minimal polynomial has only simple roots,
and $p_{v_{j}}(\theta_{i})=0$ if and only if $E_{i}\hat{v_{j}}\neq 0$. Since
$\dim\operatorname{Im}F_{j}=1$, the integers $b_{i,j}$ can only be 0 or 1, so
$P_{j}$ has only simple roots. And $\theta_{i}$ is a root of $P_{j}(t)$ if and
only if $E_{i}\hat{v_{j}}\neq 0$. Thus $p_{v_{j}}$ and $P_{j}$ have exactly
the same roots, and they are both monic, so they are equal.

To prove the 2nd part, note that $(\theta_{a},\theta_{b})\in\Phi_{K}$ if and
only if $E_{a}D_{K}E_{b}\neq 0$, according to Definition 2.4. Here

$E_{a}D_{K}E_{b}w=E_{a}\widehat{\widetilde{D_{K}}\widetilde{E_{b}w}}=\sum_{j}E_{a}\widehat{F_{j}\widetilde{E_{b}w}}.$

Thus if $E_{a}D_{K}E_{b}w\neq 0$ then
$E_{a}\widehat{F_{j}\widetilde{E_{b}w}}\neq 0$ for some $j$. Since
$\operatorname{Im}F_{j}$ is 1-dimensional, this implies both that
$E_{a}\hat{v_{j}}\neq 0$ and that $F_{j}\widetilde{E_{b}w}\neq 0$. By Lemma
5.1 we see that there exists a vector $z\in\mathbb{R}^{K}$ such that
$\widetilde{E_{b}w}=\widetilde{E_{b}\hat{z}}=\tilde{E_{b}}z$, and so $0\neq
F_{j}\tilde{E_{b}}z=\tilde{E_{b}}F_{j}z$, which implies
$\tilde{E_{b}}v_{j}\neq 0$ and hence $E_{b}\hat{v_{j}}\neq 0$. We have already
established $E_{a}\hat{v_{j}}\neq 0$, and so we find that both $\theta_{a}$
and $\theta_{b}$ are roots of $P_{j}(t)$.

To see the converse direction, first note that Claim 3.4 implies
$E_{i}\hat{v_{j}}\neq 0$ if and only if
$\tilde{E_{i}}v_{j}=\widetilde{E_{i}\hat{v_{j}}}$ contains a component
parallel to $v_{j}$. But since
$\tilde{E_{i}}v_{j}=\tilde{E_{i}}F_{j}v_{j}=F_{j}\tilde{E_{i}}v_{j}\in\operatorname{Im}F_{j}$
the latter of which is 1-dimensional, we find that $E_{i}\hat{v_{j}}\neq 0$ if
and only if $v_{j}$ is an eigenvector of $\widetilde{E_{i}}$. If $\theta_{a}$
and $\theta_{b}$ are both roots of $P_{j}(t)$ then $E_{a}\hat{v_{j}}$ and
$E_{b}\hat{v_{j}}$ are both non-zero. Hence
$D_{K}E_{b}\hat{v_{j}}=\widehat{\tilde{E_{b}}v_{j}}$ is a non-zero constant
multiple of $\hat{v_{j}}$, and then $E_{a}D_{K}E_{b}\hat{v_{j}}$ is also non-
zero, implying that $(\theta_{a},\theta_{b})\in\Phi_{K}$ ∎

######  Corollary 3.9.

If $K$ is $H$-cospectral in $M$ for some $H$ that has no multiple eigenvalues,
then $\mathcal{P}_{K}$ is the most refined partition where for each $j$ the
roots of $P_{j}(t)$ fall in the same part.

###  3.3 Gluing and Diagonal perturbation

######  Theorem 3.10.

Let $X=X_{1}\cup X_{2}$ a splitting of the index set $X$ such that
$K=X_{1}\cap X_{2}$. Let $M_{1}$ and $M_{2}$ be two matrices supported on
$X_{1}$ and $X_{2}$ respectively. Suppose that $K$ is $H$-cospectral in both
$M_{1}$ and $M_{2}$ for some $H\in\mathbb{C}^{K\times K}$. Then $K$ is also
$H$-cospectral in $M=M_{1}+M_{2}$.

###### Proof.

By Theorem 3.3 it suffices to show that $H$ commutes with $\widetilde{M^{k}}$
for all $k$. Note that $M^{k}$ can be expressed as

$M^{k}=\sum_{\sum(a_{i}+b_{i})=k}M_{1}^{a_{1}}M_{2}^{b_{2}}M_{1}^{a_{2}}M_{2}^{b_{2}}\cdots$

Let $\Pi\in\mathbb{R}^{K\times X}$ denote the projection onto $K$, that is
$\Pi$ has 1 on its diagonal in $S$. Then $\tilde{N}=\Pi N\Pi^{\mathrm{T}}$ for
any $X\times X$ matrix $N$. Further, examining the matrix multiplication one
can see that for any matrices $N_{i}$ supported on $X_{i}$ for $i=1,2$, on has

$N_{1}N_{2}\Pi^{\mathrm{T}}=N_{1}\Pi^{\mathrm{T}}\Pi
N_{2}\Pi^{\mathrm{T}}\mbox{ and
}N_{2}N_{1}\Pi^{\mathrm{T}}=N_{2}\Pi^{\mathrm{T}}\Pi N_{1}\Pi^{\mathrm{T}}$

Thus

$\widetilde{M^{k}}=\Pi M^{k}\Pi^{\mathrm{T}}=\sum_{\sum(a_{i}+b_{i})=k}\Pi
M_{1}^{a_{1}}\Pi^{\mathrm{T}}\Pi M_{2}^{b_{2}}\Pi^{\mathrm{T}}\Pi
M_{1}^{a_{2}}\Pi^{\mathrm{T}}\Pi
M_{2}^{b_{2}}\Pi^{\mathrm{T}}\cdots=\sum_{\sum(a_{i}+b_{i})=k}\widetilde{M_{1}^{a_{1}}}\widetilde{M_{2}^{b_{2}}}\widetilde{M_{1}^{a_{2}}}\widetilde{M_{2}^{b_{2}}}\cdots.$

By assumption, $H$ commutes with both $\widetilde{M_{1}}$ and
$\widetilde{M_{2}}$, so it also commutes with $\widetilde{M^{k}}$. ∎

In the next section we will study a diagonal perturbation of $M$ in the form
of $M+QD_{K}$. Using the previous theorem with
$X_{1}=X,X_{2}=K,M_{1}=M,M_{2}=D_{K}$ and noting that $K$ is $H$-cospectral in
$D_{K}$ for any $H$, we get the following.

######  Corollary 3.11.

If $K$ is $H$-cospectral in $M$ then it is also $H$-cospectral in $M+QD_{K}$
for any $Q\in\mathbb{R}$.

###  3.4 Diagonal perturbation

Let $H$ be a normal matrix in $\mathbb{C}^{K\times K}$ that has $|K|$ distinct
eigenvalues and an orthonormal eigenbasis $v_{1},v_{2},\dots,v_{|K|}$. In this
section we assume that $K$ is $H$-cospectral in $M$. Let
$\mathcal{F}\geq\mathbb{Q}$ denote the smallest number field containing all
entries of $M$ and all roots of $\phi(H)$. Then
$v_{1},\dots,v_{|K|}\in\mathcal{F}^{|K|}$.

Let $Q\in\mathbb{R}$ denote a transcendental number that is algebraically
independent of $\mathcal{F}$, and consider $M^{K}=M+Q\cdot
D_{K}\in\mathbb{R}[Q]^{X\times X}$. Then, according to Corollary 3.11, $K$ is
also $H$-cospectral in $M^{K}$. In particular, according to Corollary 3.7 and
Lemma 3.8, we have the factorization

$\phi(M^{K})=P_{0}(t)\cdot\prod_{j=1}^{|K|}P_{j}(t)$ (9)

where $P_{j}$ is the minimal polynomial of $M^{K}$ relative to $\hat{v}_{j}$
for $j=1,\dots,|K|$.

######  Claim 3.12.

$P_{0}\in\mathcal{F}[t]$ and $P_{j}\in\mathcal{F}[Q,t]$ for $j=1,\dots,|K|$.
Furthermore, the $Q$-degree of $P_{j}$ is 1 and
$\operatorname{Tr}P_{j}-Q\in\mathcal{F}$ for each $j=1,\dots,|K|$

###### Proof.

Since $P_{j}$, for $j\geq 1$, is a relative minimal polynomial of a matrix in
$\mathcal{F}(Q)^{X\times X}$ relative to a vector with entries in
$\mathcal{F}$, it is automatic that $P_{j}\in\mathcal{F}(Q)[t]$. Then (9)
implies the same for $P_{0}$. However, $\mathcal{F}(Q)$ is the quotient field
of the ring $\mathcal{F}[Q]$ which is a UFD. Hence by Gauss’s Lemma the
factorization (9) is valid in $\mathcal{F}[Q,t]$.

The $Q$-degree of $\phi(M^{K})$ is obviously $|K|$. It is easy to see that
each $P_{j}:j\geq 1$ is at least linear in $Q$ and, in fact, satisfies
$\operatorname{Tr}P_{j}-Q\in\mathcal{F}$. (This follows from a short analysis
of the $Q^{s}$-terms in the linear combination of $(M^{K})^{s}\hat{v}_{j}$
vectors that define the coefficients of $P_{j}$. See [4, Lemma 3.3] for a
detailed proof.) This is only possible if the $Q$-degree of $P_{0}$ is 0, and
the $Q$-degree of the other $P_{j}$s are exactly 1. ∎

######  Theorem 3.13.

$P_{j}$ is irreducible in $\mathcal{F}[Q,t]$ (and hence also in
$\mathcal{F}(Q)[t]$) for all $j\geq 1$.

###### Proof.

Now suppose for a contradiction that, say, $P_{1}$ is reducible in
$\mathcal{F}[Q,t]$. Since its $Q$-degree is 1, this means
$P_{1}(t)=R(t)\cdot\tilde{P}_{1}(t)$ for some non-constant
$R\in\mathcal{F}[t]$ and $\tilde{P}_{1}\in\mathcal{F}[t,Q]$. Let $\theta$ be a
root of $R(t)$, and let $m$ denote the multiplicity of $\theta$ as a root of
$P_{0}(t)$. Then $\theta$ is a root of $\phi(M^{K})$ with multiplicity at
least $m+1$ for all values of $Q$. Since $D_{K}$ is a projection matrix, Lemma
5.2 implies that there are $m+1$ orthogonal eigenvectors of $M$ (corresponding
to the $\theta$ eigenvalue) that vanish on $K$. But then the multiplicity of
$\theta$ in $P_{0}$ should have been at least $m+1$ according to its
definition. This is a contradiction, hence each $P_{j}:j=1,2,\dots,|K|$ is
indeed irreducible. ∎

##  4 Examples

In this section we apply the above techniques to provide explicit families of
graphs where diagonal perturbation can be shown to induce pretty good
fractional revival. We use the graph and its adjacency matrix interchangeably.
The blueprint is the following:

  * •

Find a family of graphs and a subset $K$ of the nodes which are $H$-cospectral
for some normal matrix $H$ with only simple eigenvalues.

  * •

Add a transcendental diagonal perturbation to the nodes in $K$.

  * •

Identify the factorization $\Phi(M^{K},t)=\prod P_{j}(t)$ of the
characteristic polynomial as the relative minimal polynomials of the
eigenvectors of $H$.

  * •

Using this, compute the degrees and traces of the $P_{j}$ polynomials.

  * •

The transcendentality of the diagonal perturbation guarantees that the $P_{j}$
are irreducible over a certain field, hence each part in the partition
$\mathcal{P}_{K}$ coincides with the roots of one of the $P_{j}$s.

  * •

Show that for some choice of $i,j$ we have $\operatorname{Tr}P_{i}/\deg
P_{i}\neq\operatorname{Tr}P_{j}/\deg P_{j}$. This implies that
$\mathcal{P}_{K}$ is non-degenerate, and hence there is pretty good state
transfer relative to $K$.

###  4.1 PGFR on 2 nodes

Here we present two infinite families of graphs, both of which are built on a
path of length $2k+1$. In $S_{k,m}$ we take a path with $2k+1$ edges, and add
a loop edge of weight $m$ at the $k+1$st node, that is, one of the nodes
forming the middle edge. In $T_{k}$ we take a path with $2k+1$ edges and add
two new nodes. One is going to be adjacent to nodes $k$ and $k+1$ on the path,
the other is going to be adjacent to node $k+3$ on the path.

In both families, the endpoints of the path exhibit pretty good fractional
revival under magnetic fields applied to them.

$m$uv$k$$k$

uzxyabv$k$$k$

Figure 1: Two families of graphs with cospectral pairs: $S_{k,m}$ (left) and
$T_{k}$ (right).

######  Theorem 4.1.

Let $M$ denote the adjacency matrix of either $S_{k,m}$ or $T_{k}$, and
$K=\\{u,v\\}$ be the endpoints of the path, as in the figure. Let $Q$ be a
transcendental real number. Then $M^{K}=M+Q\cdot D_{K}$ exhibits PGFR with
respect to $K$.

The proof of this result requires some preparation, as outlined in the
blueprint above. The first step is to verify fractional cospectrality of $u$
and $v$.

######  Lemma 4.2.

Let $M$ denote the adjacency matrix of either $S_{k,m}$ or $T_{k}$. Then for
all $j\geq 0$

$(M^{j})_{u,u}-(M^{j})_{v,v}=c\cdot(M^{j})_{u,v}$

where $c=m$ in the case of $S_{k,m}$ and $c=2$ in the case of $T_{k}$.

###### Proof.

Let $n$ be the size of the matrix $M$. Since $M^{j}$, for any $j\geq n$ is a
linear combination of $M^{0},M^{1},\dots,M^{n-1}$, it suffices to check the
equation for $j=0,\dots,n-1$.

First we consider the $S_{k,m}$ case. Here $n=2k+2$. For $j=0,1,\dots,2k$ it
is clear that $(M^{j})_{u,u}=(M^{j})_{v,v}$ and $(M^{j})_{u,v}=0$. For
$j=2k+1$ we have $(M^{2k+1})_{u,u}-(M^{2k+1})_{v,v}=m$ and
$(M^{2k+1})_{u,v}=1$. Hence $c=m$ satisfies the equation for all
$j=0,\dots,n-1$.

Next we consider $T_{k}$. Here $n=2k+4$. Clearly, $(M^{j})$ counts walks of
length $j$ in the graph. This graph is symmetric, except for the $xy$ edge.
For $j<2k+4$ no $v\to v$ walk of length $j$ can use the $xy$ edge, and for
$j<2k+1$ no $u\to u$ walk of length $j$ can use the $xy$ edge. Hence for
$j<2k+1$ we have $(M^{j})_{u,u}=(M^{j})_{v,v}$, and for $2k+1\leq j\leq 2k+3$
we have $(M^{j})_{u,u}-(M^{j})_{v,v}=$ the number of such $u\to u$ walks of
length $j$ that do use the $xy$ edge. For $j=2k+1$ this number is 2 since you
can only use this edge once and thus the rest of the walk has to be straight
between $u$ and $x$ (respectively between $u$ and $y$). For $j=2k+2$ such a
walk will have to use the $xy$ edge twice due to parity constraints. But then
it has to use it twice back-to-back due to distance constraints. Hence there
are only two such walks: $u\to x\to y\to x\to u$ and $u\to y\to x\to y\to u$.
For both $j=2k+1$ and $j=2k+2$ there is clearly only a single $u\to v$ walk of
this length. Hence $(M^{j})_{u,u}-(M^{j})_{v,v}=2\cdot(M^{j})_{u,v}$ for
$j=0,1,\dots,2k+2$.

It remains to check the same expression holds for $j=2k+3$. Due to parity and
length constraints, each $u\to u$ walk of length $2k+3$ must “go around” the
$xyz$ triangle exactly once. Reversal of the walk yields a bijection between
those walks that go around clockwise and those that do so counterclockwise. By
similar considerations, each $u\to v$ walk of length $2k+3$ must traverse the
$zy$ edge in this direction. **** ∎

######  Corollary 4.3.

  1. 1.

According to Remark 3.5, in both families $K=\\{u,v\\}$ is $H$-cospectral for
a 2-by-2 matrix $H$ whose eigenvectors are $(p,q)$ and $(-q,p)$ where
$p/q-q/p=c$.

  2. 2.

Furthermore, according to Corollary 3.7 and Lemma 3.8, as long as
$c\neq\pm\infty$, the characteristic polynomial factors as
$\Phi(M,t)=P_{0}(t)P_{1}(t)P_{2}(t)$ where $P_{1}(t)$ (respectively
$P_{2}(t)$) is the minimal polynomial of $M$ relative to $v_{1}$ (respectively
$v_{2})$ where these vectors are defined as

$v_{1}(x)=\left\\{\begin{array}[]{cc}p/q:&x=u\\\ 1:&x=v\\\ 0:&x\neq
u,v\end{array}\right.\hskip
28.45274ptv_{2}(x)=\left\\{\begin{array}[]{cc}-q/p:&x=u\\\ 1:&x=v\\\ 0:&x\neq
u,v\end{array}\right.$

  3. 3.

Then, according to Corollary 3.11, $K$ is also $H$-cospectral in
$M^{K}=M+QD_{K}$, and $\Phi(M^{K},t)=P^{K}_{0}(t)P^{K}_{1}(t,Q)P^{K}_{2}(t,Q)$
where $P^{K}_{1}$ and $P^{K}_{2}$ are degree 1 in $Q$, and for any choice of a
transcendental $Q_{0}$, the one-variable polynomials
$P^{K}_{i}(t,Q_{0}):i=1,2$ are irreducible over $\mathbb{Q}(p/q,Q_{0})$.

Next, we show that the traces of $P^{K}_{1}$ and $P^{K}_{2}$ are not equal to
each other.

######  Lemma 4.4.

With the above notation, $\deg P^{K}_{1}=\deg P^{K}_{2}$ but
$\operatorname{Tr}P^{K}_{1}\neq\operatorname{Tr}P^{K}_{2}$. In particular they
are not the same polynomial.

###### Proof.

Since $c$ in Lemma 4.2 is always an integer and $p/q,-q/p$ are the roots of
$x-1/x=c$, it follows that $p/q$ and $-q/p$ are always quadratic integers that
are each others’ conjugates. This conjugation maps $v_{1}$ to $v_{2}$ in
Corollary 4.3 and thus also $P^{K}_{1}$ to $P^{K}_{2}$. So these polynomials
always have the same degree. They are both linear in $Q$, and, by definition,
substituting $Q=0$ into them we get $P_{1}$ and $P_{2}$ respectively. So it is
sufficient to show that $\operatorname{Tr}P_{1}\neq\operatorname{Tr}P_{2}$. We
will do so by examining specific coordinates of $M^{j}v_{1}$ and $M^{j}v_{2}$
for $j\leq k+2=n/2$.

We outline the computation for the $T_{k}$ case only, as the $S_{k,m}$ case
follows in a similar but simpler way. For $T_{k}$ we saw that $c=2$ and thus
$p/q=1+\sqrt{2}$ and $-q/p=1-\sqrt{2}$. The following are easily checked by
direct calculation of walk counts in $T_{k}$:

$\displaystyle(M^{j})_{u,y}=$
$\displaystyle\left\\{\begin{array}[]{cl}0:&j<k\\\ 1:&j=k\\\ 1:&j=k+1\\\
k+3:^{*}&j=k+2\end{array}\right.$ $\displaystyle(M^{j})_{v,y}=$
$\displaystyle\left\\{\begin{array}[]{cl}0:&j\leq k\\\ 1:&j=k+1\\\
0:&j=k+2\end{array}\right.$ $\displaystyle(M^{j})_{u,b}=$ $\displaystyle
0:j\leq k+2$ $\displaystyle(M^{j})_{v,b}=$
$\displaystyle\left\\{\begin{array}[]{cl}0:&j<k\\\ 1:&j=k\\\ 0:&j=k+1\\\
k+1:^{*}&j=k+2\end{array}\right.$

The entries marked by $(*)$ can be seen by noting that a walk of the
appropriate length will use exactly one edge more than once. Specifying which
edge this is determines the walk. In the $u\to y$ case this can be any of the
edges on the $uy$ path, the $xz$ edge, or any of the other two edges incident
to $y$. In the $v\to b$ case this can be any of the edges on the $va$ path, or
one of the other two edges incident to $a$.

Hence we get that

$\displaystyle M^{j}v_{1}|_{y,b}=$
$\displaystyle\left\\{\begin{array}[]{cl}(0,0):&j<k\\\ (1+\sqrt{2},1):&j=k\\\
(2+\sqrt{2},0):&j=k+1\\\ (k+3+(k+3)\sqrt{2},k+1):&j=k+2\end{array}\right.$
$\displaystyle M^{j}v_{2}|_{y,b}=$
$\displaystyle\left\\{\begin{array}[]{cl}(0,0):&j<k\\\ (1-\sqrt{2},1):&j=k\\\
(2-\sqrt{2},0):&j=k+1\\\ (k+3-(k+3)\sqrt{2},k+1):&j=k+2\end{array}\right.$

On one hand, these calculations imply that the degree of $P_{1}$ and $P_{2}$
are both at least $k+2$, but since their product has degree $2k+4$, their
degrees must, in fact, be equal to $k+2$. If
$P_{1}(t)=x^{k+2}+c_{1}x^{k+1}+c_{2}x^{k}+\dots$ then $k+1+c_{2}=0$ from the
$b$ coordinate of $0=P_{1}(M)v_{1}$, and
$(k+3)(1+\sqrt{2})+c_{1}(2+\sqrt{2})+c_{2}(1+\sqrt{2})=0$ from the $y$
coordinate of $0=P_{1}(M)v_{1}$. Hence $c_{2}=-k-1$ and
$c_{1}=(2+2\sqrt{2})/(2+\sqrt{2})=\sqrt{2}$. So
$\operatorname{Tr}P_{1}=-\sqrt{2}$. Similarly
$\operatorname{Tr}P_{2}=\sqrt{2}$. ∎

We are finally ready to prove the validity of our examples.

###### Proof of Theorem 4.1.

According to Corollary 4.3 the polynomials $P^{K}_{1}$ and $P^{K}_{2}$ are
irreducible and distinct from $P^{K}_{0}$. By Lemma 4.4 they are distinct from
each other, thus it follows that none of these polynomials share roots. Then
by part 2) of Lemma 3.8, the eigenvalue support consists entirely of pairs of
roots of $P^{K}_{1}$ and pairs of roots of $P^{K}_{2}$. Thus, the partition
$\mathcal{P}_{K}=(\Pi_{0},\Pi_{1},\Pi_{2})$ is such that $\Pi_{1}$ consists
exactly of all roots of $P^{K}_{1}$ and $\Pi_{2}$ consists exactly of all
roots of $P^{K}_{2}$. Now, Lemma 4.4 combined with Theorem 2.11 implies that
the partition $\mathcal{P}_{K}$ is non-degenerate. Hence, our result follows
by Theorem 2.7. ∎

###  4.2 Cyclic Symmetry

Consider a graph $G$ with an automorphism $T:V(G)\to V(G)$ of order $r$. We
denote its adjacency matrix by $M$. Let $K\subset V(G)$ be an orbit of
cardinality $r$. In this section we establish certain conditions under which
$M^{K}=M+Q\cdot D_{K}$ exhibits PGFR relative to $K$. This turns out to be
easier than for the previous examples.

Since $T$ has order $r$, it must act as a cyclic permutation on any orbit of
cardinality $r$, and on $K$ in particular. Let $H$ denote the $K\times K$
permutation matrix describing the (cyclic) action of $T$ on $K$. It is obvious
then that $H$ commutes with $\widetilde{M^{k}}$ for any $k\geq 0$, and thus
$K$ is $H$-cospectral in $M$, and also in $M^{K}$ by Corollary 3.11.

The eigenvalues of $H$ are simple, with eigenvectors

$v_{k}=(1,\rho^{k},\rho^{2k},\dots,\rho^{(r-1)k}):k=1,\dots,r,$ (10)

where $\rho=e^{\frac{2\pi i}{r}}$. Thus
$\Phi(M^{K},t)=P_{0}(t)\prod_{k=1}^{r}P_{k}(t,Q)$ and according to Lemma 3.8,
for each $k=1,\dots,p$ the polynomial $P_{k}(t)$ is the relative minimal
polynomial of $v_{k}$. According to Theorem 3.13 the polynomials
$P_{1},\dots,P_{r}$ are irreducible, so they have disjoint sets of roots
unless they are the same polynomial. Hence in the partition $\mathcal{P}_{K}$
each part is the set of roots of one of the $P_{k}$ polynomials.

######  Lemma 4.5.

Let $G,T,K$ be as above. If either $\deg P_{i}\neq\deg P_{j}$ or
$\operatorname{Tr}P_{i}\neq\operatorname{Tr}P_{j}$ for some $i,j\geq 1$ then
$M^{K}$ exhibits PGFR relative to $K$.

###### Proof.

By Claim 3.12 each of the $P_{k}:k=1,\dots,r$ polynomials have
$\operatorname{Tr}P_{k}-Q\in\mathbb{Q}(\rho)$. Since $Q$ is transcendental, if
$\deg P_{i}\neq\deg P_{j}$ then $\operatorname{Tr}P_{i}/\deg
P_{i}\neq\operatorname{Tr}P_{j}/\deg P_{j}$. On the other hand, if $\deg
P_{i}=\deg P_{j}$ but $\operatorname{Tr}P_{i}\neq\operatorname{Tr}P_{j}$ then
again clearly $\operatorname{Tr}P_{i}/\deg
P_{i}\neq\operatorname{Tr}P_{j}/\deg P_{j}$. Thus, we are done by Theorem
2.11. ∎

####  4.2.1 Orbits of Unequal Size

######  Theorem 4.6.

Let $G$ and $T$ as above. Let $d$ denote the largest distance between $K$ and
any other node of $G$. If $r(d+1)>|V(G)|$ then $M^{K}=M+Q\cdot D_{K}$ exhibits
PGFR with respect to $K$ for any transcendental $Q\in\mathbb{R}$.

######  Remark 4.7.

For this condition to hold, it is necessary that not all orbits of $T$ have
size $r$.

###### Proof.

Since there is a node of $G$ that is $d$ distance away from $K$, and since
$v_{r}=(1,1,\dots,1)$, we see that for each $j=0,1,\dots,d$ the vectors
$(M^{K})^{j}v_{p}$ have strictly growing support, hence they can’t be linearly
dependent. Thus $\deg P_{p}\geq d+1$. Let $j_{0}$ be such that $P_{j}$ has the
smallest degree among $P_{1},\dots,P_{r-1}$. We have $|V(G)|=\deg P_{0}+\deg
P_{p}+\sum_{j=1}^{r-1}\deg P_{j}\geq d+1+(p-1)\deg P_{j_{0}}$, hence $\deg
P_{j_{0}}\leq(|V(G)|-d-1)/(p-1)<d+1$ according to the conditions on $d$ and
$|V(G)|$. Thus $\deg P_{j_{0}}<d+1\leq\deg P_{p}$ and thus the statement
follows from Lemma 4.5. ∎

####  4.2.2 Cycles with Added Diamond Graphs

In this section, we consider another family with general cyclic symmetry. We
define $G_{r}$ to be the graph obtain by starting with a cycle $C_{r}$ of
order $r$, and attaching along each edge a diamond graph (two triangles
sharing an edge). The graph $G_{5}$ is pictured in Figure 2.

Figure 2: The graph $G_{5}$.

Note that $G_{r}$ has the cyclic group of order $r$ as its automorphism group.
There are three orbits of the automorphism group, consisting of the vertices
of degree 5, the vertices of degree 3, and the vertices of degree 2
respectively.

######  Theorem 4.8.

Let $K$ be the vertices of any single orbit, for instance the vertices of the
central cycle $C_{r}$, and as above $M^{K}=M+Q\cdot D_{K}$ for transcendental
$Q$. Then $M^{K}$ exhibits PGFR with respect to $K$.

###### Proof.

By Lemma 4.5 it suffices to show that
$\operatorname{Tr}P_{r}\neq\operatorname{Tr}P_{k}$ for some other $1\leq k\leq
r-1$, where $P_{k}$ is the minimal polynomial of $M^{K}$ relative to the
$v_{k}$ vector defined in (10).

Let us define $A_{C_{r}}$ to be the adjacency matrix for $C_{r}$ and

$R=\begin{bmatrix}1&1&0&0&\cdots&0\\\ 0&1&1&0&\cdots&0\\\ 0&0&1&1&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\ 0&0&0&\cdots&1&1\\\
1&0&0&\cdots&0&1\end{bmatrix}$

and note that

$M^{K}=\begin{bmatrix}A_{C_{r}}+Q\cdot I&R&I\\\ R^{T}&0&I\\\
I&I&0\end{bmatrix}.$

Let $\lambda_{k}=\rho^{-k}+\rho^{k}$ and observe that simple calculation
yields

$\displaystyle A_{C_{n}}v_{k}$ $\displaystyle=\lambda_{k}v_{k}$ $\displaystyle
Rv_{k}$ $\displaystyle=(1+\rho^{k})v_{k}$ $\displaystyle R^{T}v_{k}$
$\displaystyle=(1+\rho^{-k})v_{k}.$

Thus we can verify that

$\begin{bmatrix}A_{C_{r}}+Q\cdot I&R&I\\\ R^{T}&0&I\\\
I&I&0\end{bmatrix}\begin{bmatrix}av_{k}\\\ bv_{k}\\\
cv_{k}\end{bmatrix}=\begin{bmatrix}a^{\prime}v_{k}\\\ b^{\prime}v_{k}\\\
c^{\prime}v_{k}\end{bmatrix}$

as an equation of $3r\times 3$ matrices, where

$\begin{bmatrix}\lambda_{k}+Q&1+\rho^{k}&1\\\ 1+\rho^{-k}&0&1\\\
1&1&0\end{bmatrix}\begin{bmatrix}a\\\ b\\\
c\end{bmatrix}=\begin{bmatrix}a^{\prime}\\\ b^{\prime}\\\
c^{\prime}\end{bmatrix}.$

It is easy to see that $\\{(av_{k},bv_{k},cv_{k}):a,b,c\in\mathbb{R}\\}$ is
the subspace generated by $(M^{K})^{j}v_{k}:j=0,1,\dots$, hence the roots of
the relative minimal polynomial $P_{k}$ are exactly the eigenvalues of

$N_{k}:=\begin{bmatrix}\lambda_{k}+Q&1+\rho^{k}&1\\\ 1+\rho^{-k}&0&1\\\
1&1&0\end{bmatrix}.$

Thus $P_{k}$ has degree 3, and we can see that
$\operatorname{Tr}P_{k}=\operatorname{Tr}N_{k}=\lambda_{k}+Q$ for each $k$.
Note in particular that $\operatorname{Tr}P_{r}\neq\operatorname{Tr}P_{k}$ for
$k\neq r$, and hence Lemma 4.5 finishes the proof. ∎

##  5 Appendix

######  Lemma 5.1.

Let $E\in\mathbb{R}^{X\times X}$ be a projection, and $K\subset X$. If $v=Ew$
then there is a $w^{\prime}\in\mathbb{R}^{K}$ such that
$\tilde{v}=\tilde{E}w^{\prime}$. Here we used the notation from Section 3.1.

###### Proof.

Let $F=E_{K\times X}$. Then
$\\{\tilde{v}:v\in\operatorname{Im}E\\}=\operatorname{Im}F$. So it is
sufficient to show that $\operatorname{Im}\tilde{E}=\operatorname{Im}F$. In
fact, $\operatorname{Im}\tilde{E}\leq\operatorname{Im}F$ by definition, so
showing $\operatorname{Im}F\leq\operatorname{Im}\tilde{E}$ suffices.

Since $E$ is a projection, we can write $E=\sum_{j}v_{j}v_{j}^{\mathrm{T}}$,
and $\tilde{E}=\sum_{j}\tilde{v_{j}}\tilde{v_{j}}^{\mathrm{T}}$. Thus

$\ker\tilde{E}=\\{w\in\mathbb{R}^{K}|\forall
j:\tilde{v_{j}}^{\mathrm{T}}w=0\\}.$ (11)

Since $\tilde{E}$ is self-adjoint, $\operatorname{Im}\tilde{E}$ is the
orthogonal complement of $\ker\tilde{E}$. From (11) it is clear that
$\operatorname{Im}\tilde{E}=\langle\tilde{v_{1}},\tilde{v_{2}},\dots\rangle$.
On the other hand, $F=\sum\tilde{v_{j}}v_{j}^{\mathrm{T}}$, so clearly
$\operatorname{Im}F\leq\langle\tilde{v_{1}},\tilde{v_{2}},\dots\rangle=\operatorname{Im}\tilde{E}$.
∎

######  Lemma 5.2.

Let $M$ be a symmetric matrix and $N$ be a projection matrix. Suppose $\theta$
has multiplicity $k$ as an eigenvalue of $M+Q\cdot N$ for all
$Q\in\mathbb{R}$. Then $\ker N$ contains $k$ orthonormal eigenvectors of $M$,
each with eigenvalue $\theta$.

###### Proof.

Suppose we already exhibited $0\leq j\leq k-1$ such orthonormal vectors,
$w_{1},\dots,w_{j}\in\ker N$. We exhibit one more as follows. Let $v_{q}$ be a
unit length eigenvector of $M+Q\cdot N$ with eigenvalue $\theta$ that is
orthogonal to $w_{1},\dots,w_{j}$. Such vectors exist because the multiplicity
of $\theta$ is more than $j$, and $w_{1},\dots,w_{j}$ are all eigenvectors of
$M+Q\cdot N$ as well. Let $w$ be a subsequential limit of the $v_{q}$s as
$Q\to 0$. Such a limit exists because of compactness. From now on we restrict
$Q$ to such a subsequence. Clearly $w$ is unit length and orthogonal to
$w_{1},\dots,w_{j}$. Taking limit as $Q\to 0$ in $(M+Q\cdot N)v_{q}=\theta
v_{q}$ gives that $Mw=\theta w$. It remains to show that $w\in\ker N$.

Taking the scalar product of both sides with $w$, and using the symmetry of
$M$ as well as $Mw=\theta w$, we obtain

$\theta\langle v_{q},w\rangle=\langle Mv_{q},w\rangle+Q\langle
Nv_{q},w\rangle=\langle v_{q},Mw\rangle+Q\langle Nv_{q},w\rangle=\langle
v_{q},\theta w\rangle+Q\langle Nv_{q},w\rangle.$

Hence $\langle Nv_{q},w\rangle=0$, from which we get $\langle
Nw,w\rangle=\langle Nw,Nw\rangle=0$ after passing to the limit and using that
$N$ is idempotent. Thus $Nw=0$ as claimed. ∎

## References

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#  Generating 4-dimensional Wormholes with Yang-Mills Casimir Sources

A. C. L. Santos<EMAIL_ADDRESS>Universidade Federal do Ceará
(UFC), Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  R. V. Maluf
<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC), Departamento de
Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  C. R. Muniz
<EMAIL_ADDRESS>Universidade Estadual do Ceará (UECE), Faculdade de
Educação, Ciências e Letras de Iguatu, Av. Dário Rabelo s/n, Iguatu-CE,
63.500-00 - Brazil.

(August 28, 2024)

###### Abstract

This work presents a new wormhole solution in General Relativity supported by
the quantum vacuum fluctuations of the Casimir effect between perfect
chromometallic mirrors in $(3+1)$ dimensions, which was recently fitted using
first-principle numerical simulations. Initially, we employ a perturbative
approach for $x=mr\ll 1$, where $m$ represents the Casimir mass. This approach
has proven to be a reasonable approximation when compared with the exact case
in this regime. To find well-behaved redshift functions, we impose constraints
on the free parameters. As expected, this solution recovers the
electromagnetic-like Casimir solution for $m=0$. Analyzing the traversability
conditions, we graphically find that all will be satisfied for $0\leq m\leq
0.20$. On the other hand, all the energy conditions are violated, as usual in
this context. Stability from Tolman-Oppenheimer-Volkov (TOV) equation is
guaranteed for all $r$ and from the speed of sound for $0.16\leq m\leq 0.18$.
Therefore, for $0.16\leq m\leq 0.18$, we will have a stable solution that
satisfies all traversability conditions.

General Relativity. Yang-Mills Field. Casimir Effect. Wormholes.

##  I Introduction

Some inconsistencies within the framework of General Relativity have demanded
severe efforts from physicists. Singularities and the incompatibility with
observations in the cosmological scenario, among others, have guided different
proposals: adding new invariants or fields, exploring alternative geometries
and quantization (a great compilation can be found in CANTATA:2021ktz ).
However, faced with these problems, we inquire: Are we adequately
comprehending and describing the vacuum structure? What effects could arise
when we include experimentally confirmed quantum vacuum effects?

One of the most established effects is called the Casimir effect and is
associated with quantum vacuum fluctuations when we impose boundary conditions
Casimir:1948dh . Over the years, studies have emerged to investigate the
effect of curved spaces on Casimir energy density Sorge:2019kuh ;
Santos:2020taq ; Santos:2021jjs . On the other hand, following Garattini’s
work, this energy density and the associated pressure was incorporated as
source in Einstein’s equations, resulting in the formation of wormholes for
the Casimir effect of the electromagnetic field in $(3+1)$ Garattini:2019ivd ,
$(2+1)$ Alencar:2021ejd  and $D$ dimensions Oliveira:2021ypz . Also with the
Casimir effect of the Yang-Mills field in $(2+1)$ dimensions Santos:2023zrj
and extensions of General Relativity Cruz:2024ihb ; Zubair:2023abs ;
Hassan:2022hcb ; Tripathy:2020ehi ; Azmat:2023ygn ; Mishra:2023bfe .

Wormholes are hypothetical structures that would connect two distinct regions
of spacetime. However, unlike Black Holes, which already have observational
confirmation and well-established sources provided by stellar evolution
EventHorizonTelescope:2019ths ; EventHorizonTelescope:2022wkp , wormholes
suffer from a series of questions, including a natural formation process. In
1988, Morris and Thorne conducted a detailed study of the characteristics this
hypothetical structure would need to possess in order to be traversable
Morris:1988cz . They concluded that it would require negative energy density -
exotic matter - which at the time increased the skepticism related to this
solution. Interestingly, one of the characteristics of the Casimir energy
density is that for certain configurations, it can assume negative values.
This is why it has been proposed as a potential source for this solution.

In this sense, based on the previously mentioned results, this work aims to
reaffirm the possibility of a wormhole formation with a new source given by
the Casimir energy density of the Yang-Mills field in $(3+1)$ dimensions, that
was recently fitted through first-principles numerical simulations, where was
identified that the Casimir interaction between perfect chromometallic mirrors
reveals the presence of a new gluonic state with a mass of $m=0.49(5)$ GeV
Chernodub:2023dok . On the other hand, due to the numerical difficulties in
treating the solution exactly and taking into account that this effect occurs
at small distances for small masses in a gravitational context, we will
consider a perturbative approach for $x=mr\ll 1$ and in order to eliminate the
singularities that arise at the throat - common in this context - we will
impose a constraint on the free parameters.

Our paper is organized as follows: In section II, considering a perturbative
approach, we find the shape and redshift functions, analyzing how the Casimir
mass influences the wormhole characteristics. In section III, to identify the
physical consistency of this solution, we investigated the traversability
conditions, energy conditions, stability from the speed of sound, and TOV
equation. Finally, in section IV we outline our conclusions.

##  II Wormhole Solution

Since we are going to consider General Relativity, we begin by defining the
well-known Einstein-Hilbert action in $(3+1)$ dimensions

$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}(R+\mathcal{L}_{m}),$ (1)

where $g$ stands for the determinant of the metric $g_{\mu\nu}$, $R$ is the
Ricci scalar and $\mathcal{L}_{m}$ is the Lagrangian density of matter.
Varying (1) concerning the metric, we obtain the equations of motion

$R^{\mu}_{\nu}-\frac{1}{2}g^{\mu}_{\nu}R=\kappa T^{\mu}_{\nu},$ (2)

with $\displaystyle\kappa=8\pi$, where $G=\hbar=c=1$. Assuming that the
Casimir energy density and pressures effectively act as a fluid with density
$\rho(r)$, radial pressure $p_{r}(r)$, and tangential pressure $p_{t}(r)$, we
will consider:

$T_{\nu}^{\mu}=\mbox{diag (-$\rho(r),p_{r}(r),p_{t}(r),p_{t}(r)$)}.$ (3)

Let us assume a spherically symmetric and static ansatz for the spacetime
metric with the line element given by

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{1}{1-\frac{b(r)}{r}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (4)

which represents a $(3+1)$-dimensional Morris-Thorne wormhole Morris:1988cz ,
where the redshift function $\Phi(r)$ and the shape function $b(r)$ are
arbitrary functions of the polar coordinate $r\in\left[r_{0},+\infty\right)$.
Thus, the coordinate $r$ must be decreased from infinity to a minimum value
$r_{0}$, the radius of the throat. Replacing the ansatz to the metric (4) and
the energy-momentum tensor (3) in the Einstein field equations (2), we find

$\frac{b^{\prime}(r)}{r^{2}}=\kappa\rho(r),$ (5) $\kappa
p_{r}(r)=-\frac{b(r)}{r^{3}}+\frac{2\Phi^{\prime}(r)}{r}-\frac{2b(r)\Phi^{\prime}(r)}{r^{2}},$
(6)
$\displaystyle\left(1-\frac{b(r)}{r}\right)\left(\Phi^{\prime\prime}(r)+\Phi^{\prime
2}(r)+\frac{\Phi^{\prime}(r)}{r}\right)-\frac{b^{\prime}(r)r-b(r)}{2r^{2}}\left(\Phi^{\prime}(r)+\frac{1}{r}\right)=\kappa
p_{t}(r).$ (7)

The prime $(^{\prime})$ stands for the total derivative concerning the radial
coordinate $r$. On the other hand, the covariant energy-momentum conservation
law leads to

$p_{r}^{\prime}(r)=\frac{-2p_{r}(r)+2p_{t}(r)-r(p_{r}(r)+\rho(r))\Phi^{\prime}(r)}{r}.$
(8)

We will consider as a source the Casimir energy in non-Abelian gauge theory
described as the Casimir energy of a massive scalar particle with certain mass
$m$ which was recently fitted using first-principles numerical simulations
Chernodub:2023dok

$\frac{\mathcal{E}_{Cas}}{S}=\sum_{n=1}^{\infty}-\frac{2C_{0}m^{2}}{\pi^{2}r}\frac{K_{2}(2nmr)}{n^{2}},$
(9)

where $S$ is the surface of the plates, $C_{0}$ is a phenomenological
parameter, $r$ is the distance between the plates and $\displaystyle K_{2}(z)$
gives the modified Bessel function of the second kind. The analysis provides
the following best-fit parameters: $C_{0}=5.60(7)$ and $m=0.49(5)$ GeV. Due to
the numerical difficulties encountered when dealing with this expression
exactly, we will use the perturbative approach considering $\displaystyle
x=mr$, for $x\ll 1$. Let us consider the following expansion Maluf:2019ujv :

$\frac{\mathcal{E}_{Cas}}{S}=\sum_{n=1}^{\infty}\left(-\frac{\lambda}{2n^{4}r^{3}}+\frac{\lambda
x^{2}}{2n^{2}r^{3}}+O(x^{4})\right)=-\frac{\pi^{2}\lambda\left(\pi^{2}-15x^{2}\right)}{180r^{3}}+O(x^{4}),$
(10)

where,

$\lambda=\frac{2C_{0}}{\pi^{2}}.$ (11)

Which leads to the Casimir energy density

$\rho(m\ll r^{-1})=\frac{\mathcal{E}_{Cas}(m\ll
r^{-1})}{Sr}=-\frac{\pi^{2}\lambda\left(\pi^{2}-15m^{2}r^{2}\right)}{180r^{4}}.$
(12)

To confirm the accuracy of this density as an approximation of the exact
function within this regime, we generated a comparative graph depicted in
Figure 1. This graph juxtaposes the exact densities with those expanded using
the best fit parameters $C_{0}$ and $m=5\times 10^{-2}$ Chernodub:2023dok .
Through this comparison, we can confidently assert its validity as a reliable
approximation.

Figure 1: The graphical representation of the radial dependence for $\rho(r)$
exact and expanded, with $n=1,...,100,C_{0}=5.6$ and $m=5$x$10^{-2}$.  Figure
2: Representation of the embedding diagram with different values of $m$, with
$C_{0}=5.6$ and $r_{0}$ given by (18), in natural units where $G=\hbar=c=1$.
---
Figure 3: Embedded shape of the Casimir-Yang-Mills wormhole with
$m=5$x$10^{-2},C_{0}=5.6$ and $r_{0}$ given by (18), in natural units where
$G=\hbar=c=1$.

From (5) and (12) we obtain the shape function

$\displaystyle b(m\ll
r^{-1})=\frac{\pi^{2}\kappa\lambda(r-r_{0})\left(15r_{0}m^{2}r-\pi^{2}\right)}{180r_{0}r}+r_{0}.$
(13)

The constant of integration was fixed such that the throat condition
$b(r_{0})=r_{0}$ is satisfied. When $m\to 0$, we recover the profile of the
wormhole solution sustained by the Casimir effect of the electromagnetic
field, as expected Garattini:2019ivd . Analyzing Figure 2, which depicts
embedding diagrams for different values of $m$, we identified that the larger
the Casimir mass, the smaller the throat radius, which aligns with the
constraint given by the equation (18) and can be interpreted as an increase in
$m$ leading to more intense quantum effects, causing pronounced distortions in
the geometry of the wormhole. However, very small masses have a throat radius
that is very close due to the dependence of $r_{0}$ with the inverse of
$m^{2}$. Figure 3 represents this three-dimensional embedding, highlighting
its asymptotic flatness.

Let us turn our attention to the redshift function associated with the
Casimir-Yang Mills wormhole. The radial pressure is given by

$\displaystyle p_{r}(m\ll
r^{-1})=-\frac{1}{S}\frac{d\mathcal{E}_{Cas}}{dr}=-\frac{\pi^{2}\lambda\left(\pi^{2}-5m^{2}r^{2}\right)}{60r^{4}}.$
(14)

This allows us to conclude the following equation of state:

$p_{r}(m\ll r^{-1})=\omega(m\ll r^{-1})\rho(m\ll r^{-1});\;\;\omega(m\ll
r^{-1})=\frac{2\pi^{2}}{\pi^{2}-15m^{2}r^{2}}+1,$ (15)

which recovers the $\omega$ of the electromagnetic case for $m=0$, as
expected. From (6), (13) and (14) we obtain

$\displaystyle\Phi(m\ll r^{-1})=\frac{1}{2\left(\pi^{2}\kappa\lambda
m^{2}-12\right)\left(\pi^{4}\kappa\lambda-15{r_{0}}^{2}\left(\pi^{2}\kappa\lambda
m^{2}-12\right)\right)}$
$\displaystyle\times\left(3\ln(r-r_{0})\left(\pi^{2}\kappa\lambda
m^{2}-12\right)\left(5{r_{0}}^{2}\left(\pi^{2}\kappa\lambda
m^{2}+12\right)-\pi^{4}\kappa\lambda\right)\right.$
$\displaystyle+\left.\left(45{r_{0}}^{2}\left(\pi^{2}\kappa\lambda
m^{2}-12\right)^{2}-\pi^{4}\kappa\lambda\left(\pi^{2}\kappa\lambda
m^{2}+12\right)\right)\ln\left(15r_{0}r\left(\pi^{2}\kappa\lambda
m^{2}-12\right)-\pi^{4}\kappa\lambda\right)\right)$
$\displaystyle+\ln(r)+c_{1},$ (16)

where $c_{1}$ is a constant. In order to have a traversable wormhole we need
to eliminate the divergence in the logarithmic term at $r\to r_{0}$, which
implies the constraint

$5{r_{0}}^{2}\left(\pi^{2}\kappa\lambda
m^{2}+12\right)-\pi^{4}\kappa\lambda=0,$ (17)

which leads

$\displaystyle r_{0}=\pi^{2}\sqrt{\frac{\kappa\lambda}{5\pi^{2}\kappa\lambda
m^{2}+60}}.$ (18)

Considering (17), the redshift function becomes:

$\displaystyle\Phi(m\ll r^{-1})$ $\displaystyle=$
$\displaystyle\frac{2\left(\pi^{2}\kappa\lambda
m^{2}-6\right)\ln\left(10{r_{0}}^{2}\left(\pi^{2}\kappa\lambda
m^{2}-24\right)\right)}{\pi^{2}\kappa\lambda
m^{2}-12}+\ln\left(\frac{r}{r_{0}}\right)+1$ (19) $\displaystyle-$
$\displaystyle\frac{2\left(\pi^{2}\kappa\lambda
m^{2}-6\right)\ln\left(-5\pi^{2}r_{0}\kappa\lambda
m^{2}(r_{0}-3r)-60r_{0}(r_{0}+3r)\right)}{\pi^{2}\kappa\lambda m^{2}-12},$

which is a finite quantity when $r\to r_{0}$. The constant of integration
$c_{1}$ was fixed such that $\Phi(r_{0})=1$ is satisfied. This behavior is
illustrated in Figure 4. Note that for the argument of the first logarithm we
have the condition $\pi^{2}\kappa\lambda m^{2}\neq 24$ and for the second
logarithm to be null, we need

$5\pi^{2}r_{0}\kappa\lambda m^{2}(3r-r_{0})-60r_{0}(r_{0}+3r)=0,$ (20)

which implies,

$r=r_{0}\left(\frac{8}{\pi^{2}\kappa\lambda
m^{2}-12}+\frac{1}{3}\right)<r_{0}.$ (21)

It is valid to mention that for $m\to 0$, we recover the profile of the
solution sustained by the electromagnetic Casimir effect Garattini:2019ivd .
Certainly, the redshift function will diverge as $r\to\infty$ due to the
presence of the second logarithmic term. However, in this limit, the
approximation $x=mr\ll 1$ is no longer valid. Therefore, we cannot use (12) to
describe the Casimir energy density and, consequently, (19) as a solution to
the redshift function. Since the Kretschmann scalar is an involved expression,
to ensure the absence of singularities in this spacetime for this regime we
plotted Figure 5.

Figure 4:  Redshift function from equation (19) with $m=5$ x
$10^{-4},C_{0}=5.6$ and $r_{0}$ given by (18), in natural units where
$G=\hbar=c=1$. Figure 5: Kretschmann scalar with $m=5$x$10^{-4},C_{0}=5.6$ and
$r_{0}$ given by (18), in natural units where $G=\hbar=c=1$.

##  III Source Properties

###  III.1 Traversability Conditions

Let’s investigate whether the conditions established for traversability
Morris:1988cz  are satisfied by the redshift and shape functions found.
$(i)$ A flaring-out condition, associated with the minimality of the wormhole
throat, implies that

$\frac{b(r)-rb^{\prime}(r)}{b(r)^{2}}=-\frac{180r_{0}r\left(-15{r_{0}}^{2}r\left(12-\pi^{2}\kappa\lambda
m^{2}\right)-2\pi^{4}r_{0}\kappa\lambda+\pi^{4}\kappa\lambda
r\right)}{\left(180{r_{0}}^{2}r+\pi^{2}\kappa\lambda(r_{0}-r)\left(\pi^{2}-15r_{0}m^{2}r\right)\right)^{2}}>0.$
(22)

And on the throat

$b^{\prime}(r_{0})=\frac{1}{180}\pi^{2}\kappa\lambda\left(15m^{2}-\frac{\pi^{2}}{{r_{0}}^{2}}\right)<1.$
(23)

$(ii)$ Another condition is given by

$\displaystyle
1-\frac{b(r)}{r}=\frac{(r-r_{0})\left(15r_{0}r\left(12-\pi^{2}\kappa\lambda
m^{2}\right)+\pi^{4}\kappa\lambda\right)}{180r_{0}r^{2}}\geq 0.$ (24)

We plotted a graph in Figure 6 to analyze which values of $m$ allow the
conditions (22), (23), and (24) to be satisfied for all $r$ (within the
approach), since $r_{0}$ must be given by (18). Thus, we can conclude that for
$0\leq m\leq 0.20$, all traversability conditions will be satisfied, which is
consistent with the approximation we are employing.

Figure 6: Region where the traversability conditions given by (22), (23) and
(24) are satisfied with $C_{0}=5.6$ and $r_{0}$ given by (18), in natural
units where $G=\hbar=c=1$.

On the other hand, the shape function is asymptotically flat-like, since

$\lim_{r\rightarrow\infty}\left(1-\frac{b(r)}{r}\right)=1-\frac{\pi^{2}\kappa\lambda
m^{2}}{12}.$ (25)

Despite exhibiting the desired behavior, as mentioned earlier, the asymptotic
limit $r\to\infty$ does not fit within the approximation we are using.
Finally, from (7), (13) and (II) and imposing the constraint (18), we can
obtain the tangential pressure:

$\displaystyle p_{t}(m\ll r^{-1})$ $\displaystyle=$
$\displaystyle-\frac{\pi^{2}\lambda\left(3\pi^{2}r\left(5\kappa\lambda
m^{2}\left(\pi^{2}-3m^{2}r^{2}\right)-36\right)-\alpha\left(15m^{2}r^{2}+\pi^{2}\right)\right)}{180r^{4}\left(\alpha+r\left(36-3\pi^{2}\kappa\lambda
m^{2}\right)\right)},$ (26)

where,

$\alpha=\pi^{2}\sqrt{\frac{\kappa\lambda(\pi^{2}\kappa\lambda m^{2}+12)}{5}}.$
(27)

It is worth mentioning that the same result is found using the equation (8),
as expected. Therefore, the Casimir source that generates a wormhole
effectively acts as an anisotropic fluid, since the radial and tangential
pressures are distinct as evidenced by the the equations (14) and (26). This
is in agreement with analysis in more general contexts Kim:2019ojs .

###  III.2 Energy Conditions

The energy conditions state that for a given fluid with density $\rho(r)$,
radial pressure $p_{r}(r)$, and lateral pressures $p_{t}(r)$, the following
relationships must be satisfied: $\rho(r)\geq 0$ and $\rho(r)+p_{i}(r)\geq 0$
(Weak Energy Condition), $\rho(r)+p_{i}(r)\geq 0$ (Null Energy Condition),
$\rho(r)+\sum_{i}p_{i}(r)\geq 0$ and $\rho(r)+p_{i}(r)\geq 0$ (Strong Energy
Condition) and $\rho(r)-|p_{i}(r)|\geq 0$ (Dominant Energy Condition).

As we can see in Figure 7, all the energy conditions will be violated. This
fact is common in other contexts of wormholes sustained by the Casimir source
and is reasonable since these conditions have a classical nature, whereas the
analyzed source has a quantum origin.

---
(a) (b)
Figure 7: The graphical representation of the radial dependence for a
combination of density and pressures with $r_{0}$ given by (18), $m=5$ x
$10^{-4}$ and $C_{0}=5.6$, in natural units where $G=\hbar=c=1$.

###  III.3 Stability from sound velocity

In order to evaluate the stability of the Casimir wormhole, we must first
analyze the condition described by the expression Capozziello:2020zbx ;
Capozziello:2022zoz :

$v_{s}^{2}(r)=\frac{1}{3}\left[\frac{d(p_{r}+2p_{t})}{d\rho}\right]=\frac{1}{3}\left[\frac{p_{r}^{\prime}(r)+2p_{t}^{\prime}(r)}{\rho^{\prime}(r)}\right]\geq
0,$ (28)

with $v_{s}$ representing the sound velocity in the medium. Considering (12),
(14) and (26) we obtain

$\displaystyle
v_{s}^{2}(r)=\frac{-15m^{2}r^{2}+\gamma(r)}{3\left(2\pi^{2}-15m^{2}r^{2}\right)}\geq
0,$ (29)

where,

$\displaystyle\gamma(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\pi^{2}\sqrt{5(\kappa\lambda\pi^{2}\kappa\lambda
m^{2}+12)}-15r\left(\pi^{2}\kappa\lambda m^{2}-12\right)\right)^{2}}$ (30)
$\displaystyle\times$
$\displaystyle\left(10\pi^{2}\left(45\sqrt{5\kappa\lambda}m^{2}r^{3}\left(\pi^{2}\kappa\lambda
m^{2}-18\right)\sqrt{\pi^{2}\kappa\lambda m^{2}+12}+675\kappa\lambda
m^{4}r^{4}\left(\pi^{2}\kappa\lambda m^{2}-12\right)\right.\right.$
$\displaystyle+$ $\displaystyle\pi^{4}\kappa\lambda\left(\pi^{2}\kappa\lambda
m^{2}+12\right)+12\pi^{2}\sqrt{5\kappa\lambda}r\left(\pi^{2}\kappa\lambda
m^{2}-3\right)\sqrt{\pi^{2}\kappa\lambda m^{2}+12}$ $\displaystyle-$
$\displaystyle\left.\left.30r^{2}\left(\pi^{2}\kappa\lambda
m^{2}-6\right)\left(11\pi^{2}\kappa\lambda m^{2}-108\right)\right)\right).$

Since stability must be analyzed in the throat, let’s consider $r\approx
r_{0}$, where $r_{0}$ is given by (18), and examine in the Figure 8 which
values of $m$ allow for a stable solution. As we can identify, to satisfy the
condition given by (28) and avoid superluminal velocities, we need $0.16\leq
m\leq 0.18$. Therefore, for $0.16\leq m\leq 0.18$ we have a stable solution
with the traversability conditions satisfied.

Figure 8: Graphical representation of $v^{2}_{s}$ with $r\approx r_{0}$ given
by (18) and $C_{0}=5.6$, in natural units where $G=\hbar=c=1$.

###  III.4 Stability from TOV equation

Figure 9: The graphical representation of $\mathcal{F}_{g},\mathcal{F}_{h}$
and $\mathcal{F}_{a}$ as functions of the radial coordinate $r$, with $r_{0}$
given by (18), $m=5$x$10^{-4}$ and $C_{0}=5.6$, in natural units where
$G=\hbar=c=1$.

From (8), we can identify:

$\mathcal{F}_{g}+\mathcal{F}_{h}+\mathcal{F}_{a}=0,$ (31)

where,

$\mathcal{F}_{g}=-\Phi^{\prime}(r)(\rho(r)+p_{r}(r));\;\mathcal{F}_{h}=-\frac{dp_{r}(r)}{dr};\;\mathcal{F}_{a}=\frac{-2(p_{r}(r)-p_{t}(r))}{r},$
(32)

with $\mathcal{F}_{h}$ the hydrostatic force, $\mathcal{F}_{g}$ the
gravitational force and $\mathcal{F}_{a}$ the anisotropic force. It is
straightforward to verify that our traversable wormhole solution satisfies the
TOV equation with $r_{0}$ given by (18). The profiles of $\mathcal{F}_{g}$,
$\mathcal{F}_{h}$ and $\mathcal{F}_{a}$ are depicted in Figure 9.
$\mathcal{F}_{a}$ and $\mathcal{F}_{g}$ take positive values near the throat,
while $\mathcal{F}_{h}$ is negative, clearly indicating that to maintain the
system in an equilibrium state, the hydrostatic force is balanced by the
combined effect of gravitational and the anisotropic forces, in a similar
behavior of $(2+1)$ dimensions Santos:2023zrj .

##  IV Conclusion

In summary, we have found a wormhole solution supported by the density and
Casimir pressures of the Yang-Mills field, which has recently been simulated
using first-principles numerical simulations in $(3+1)$ dimensions. We
considered a perturbative approach for $x=mr\ll 1$, where $m$ is the Casimir
mass. To eliminate the divergence in the throat, we imposed a constraint on
the free parameter $r_{0}$. We identified in the shape function that a larger
Casimir mass implies a smaller wormhole throat. In order to the traversability
conditions to be satisfied, it is necessary $0\leq m\leq 0.20$, which is
consistent with the approximation considered. In turn, the solution is stable
for all $r$ from the point of view of TOV equation, as well as of the speed
sound for $0.16\leq m\leq 0.18$. Thus, $0.16\leq m\leq 0.18$ implies that we
will have a stable solution that satisfies all traversability conditions. All
the energy conditions are violated, this is a common fact in this context
given by a source of quantum origin. Finally, for $m=0$ we recover the Casimir
electromagnetic solution, as expected.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grants no. 88887.822058/2023-00 (ACLS), no. 308268/2021-6
(CRM), and no. 200879/2022-7 (RVM) for financial support.

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