# On Noisy Duplication Channels
with Markov Sources

Brendon McBain, James Saunderson, and Emanuele Viterbo  Department of
Electrical & Computer Systems Engineering, Monash University, Clayton,
Australia

###### Abstract

Channels with noisy duplications have recently been used to model the nanopore
sequencer. This paper extends some foundational information-theoretic results
to this new scenario. We prove the asymptotic equipartition property (AEP) for
noisy duplication processes based on ergodic Markov processes. A consequence
is that the noisy duplication channel is information stable for ergodic Markov
sources, and therefore the channel capacity constrained to Markov sources is
the Markov-constrained Shannon capacity. We use the AEP to estimate lower
bounds on the capacity of the binary symmetric channel with Bernoulli and
geometric duplications using Monte Carlo simulations. In addition, we relate
the AEP for noisy duplication processes to the AEP for hidden semi-Markov
processes.

##  I Introduction

Recently, the nanopore sequencer in DNA storage systems has motivated the
study of channels with noisy duplications [10]. In this application, a
sequence of nucleotide molecules
$\\{\mathsf{A},\mathsf{T},\mathsf{C},\mathsf{G}\\}$ pass through a microscopic
pore that outputs a current response dependent on the nucleotides inside the
pore. The mechanism that feeds the nucleotides through the pore varies in
speed such that the sampled current signal contains a random number of samples
per nucleotide. In the presence of measurement noise, this imperfect mechanism
results in a channel with noisy duplications. This paper is an information-
theoretic analysis of generic noisy duplication channels, not specifically
related to the model of the nanopore sequencer.

Numerical capacity bounds of noisy duplication channels were computed in [11,
13] using Monte Carlo simulations to characterize the theoretical performance
of the nanopore sequencer. However, the results were based on the assumption
that the asymptotic equipartition property (AEP) holds for the channel
processes of noisy duplication channels. The key contribution of this paper is
proving that the AEP holds for discrete-time ergodic Markov sources on a
finite state space. A consequence of our results is that the Markov-
constrained channel capacity is the Markov-constrained Shannon capacity
$C_{\mathsf{Markov}}$, which is a lower bound on the channel capacity $C$ for
arbitrary sources. We compute numerical lower bounds on $C_{\mathsf{Markov}}$
for the binary symmetric channel (BSC) with Bernoulli and geometric
duplications by choosing a $\mathsf{Ber}(1/2)$ source. These extend the
numerical capacity bounds for the sticky channels in [14] that correspond to
the noiseless cases. In addition, we relate the AEP for noisy duplication
processes with the AEP for hidden semi-Markov processes (HSMPs), which has
largely only been studied in the special case of semi-Markov processes (SMPs)
[5, 6].

###  I-A Noisy duplication channel

The noisy duplication channel for the special case of Markov sources is
conveniently described using SMPs. For a sequence of channel inputs
$\\{X_{i}\\}$, the sequence of channel states is the discrete-time homogeneous
Markov process (MP) $\\{S_{\ell}\\}$ on a finite state space $\Omega$. Since
there is a one-to-one correspondence between the channel inputs and the
channel states, we will conveniently consider the latter as being the input to
the channel. This Markov source is ergodic if its Markov transition
probability matrix $P$ is irreducible. For an arbitrary initial state $S_{0}$,
their relationship is visualised as

$\displaystyle
S_{1}\overset{X_{1}}{\longrightarrow}S_{2}\overset{X_{2}}{\longrightarrow}\ldots\overset{X_{m}}{\longrightarrow}S_{m}$

for $m$ channel-uses, where each arrow corresponds to an input that triggers a
change in channel state. Next, the channel states are duplicated according to
the i.i.d. state duration process $\\{K_{\ell}\\}$ on a discrete support
$\Lambda$, and then sequentially concatenated together to form the SMP
$\\{Z_{t}\\}$ on $\Omega$. For $m$ channel-uses, we have the duplicated states

$\displaystyle\underbrace{Z_{1},Z_{2},\ldots,Z_{T_{1}}}_{K_{1}},\underbrace{Z_{T_{1}+1},\ldots,Z_{T_{2}}}_{K_{2}},\ldots,\underbrace{Z_{T_{m-1}+1},\ldots,Z_{T_{m}}}_{K_{m}}$

where for convenience we additionally define the jump time
$T_{\ell}=\sum_{i\leq\ell}K_{i}$ for the index of the last sample in the
$\ell$-th segment, forming the jump time process $\\{T_{\ell}\\}$. This
process is semi-Markov since the Markov property only holds at the jump time
between each segment of duplications. Each duplicated state in $\\{Z_{t}\\}$
undergoes the channel mapping $f:\Omega\rightarrow\mathbb{R}$, which is
identical for each duplicated state. Then it is corrupted by memoryless noise
to give the HSMP $\\{Y_{t}\\}$. For a channel input $S_{1}^{m}$ of $m$
channel-uses, the channel output is $Y_{1}^{T_{m}}$ and the channel joint
input-output is $(S_{1}^{m},Y_{1}^{T_{m}})$.

###  I-B Entropy rates

For the MP $\mathbb{S}=\\{S_{\ell}\\}$, the entropy rate is
$H(\mathbb{S})=\lim_{m\rightarrow\infty}\frac{1}{m}H(S_{1}^{m})$. The HSMP
$\mathbb{Y}=\\{Y_{t}\\}$ has entropy rate
$H(\mathbb{Y})=\lim_{t\rightarrow\infty}\frac{1}{t}H(Y_{1}^{t})$. When this
HSMP is randomly indexed as $Y_{1}^{T_{m}}$ using i.i.d. indexing process
$\mathbb{T}=\\{T_{\ell}\\}$, the entropy rate is
$H(\mathbb{Y}^{\mathbb{T}})=\lim_{m\rightarrow\infty}\frac{1}{m}H(Y_{1}^{T_{m}})$.
Similarly, we have the entropy rate of the joint process
$(S_{1}^{m},Y_{1}^{T_{m}})$ as
$H(\mathbb{S},\mathbb{Y}^{\mathbb{T}})=\lim_{m\rightarrow\infty}\frac{1}{m}H(S_{1}^{m},Y_{1}^{T_{m}})$.
The existence of these two randomly-indexed entropy rates is proven in Lemma
1. Finally, let the (mutual) information rate of the noisy duplication channel
with a Markov source be
$I(\mathbb{S};\mathbb{Y}^{\mathbb{T}})=H(\mathbb{S})+H(\mathbb{Y}^{\mathbb{T}})-H(\mathbb{S},\mathbb{Y}^{\mathbb{T}})$.

######  Lemma 1.

The entropy rates $H(\mathbb{Y}^{\mathbb{T}})$ and
$H(\mathbb{S},\mathbb{Y}^{\mathbb{T}})$ exist.

###### Proof.

Let $H_{m}=\frac{1}{m}H(Y_{1}^{T_{m}})$. Observe that

$\displaystyle mH_{m}$ $\displaystyle=H(Y_{1}^{T_{m}})$ (1) $\displaystyle\leq
H(Y_{1}^{T_{\ell}},Y_{T_{\ell+1}}^{T_{m}})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for any}\leavevmode\nobreak\ \ell<m$ (2)
$\displaystyle=H(Y_{1}^{T_{\ell}})+H(Y_{T_{\ell}+1}^{T_{m}}|Y_{1}^{T_{\ell}})$
(3) $\displaystyle\leq H(Y_{1}^{T_{\ell}})+H(Y_{T_{\ell}+1}^{T_{m}})$ (4)
$\displaystyle=\ell H_{\ell}+(m-\ell)H_{m-\ell}$ (5)

then $\\{mH_{m}\\}$ is sub-additive. By Fekete’s lemma [4, Lemma 4A.2], we
have that $\lim_{m\rightarrow\infty}H_{m}=\liminf_{m\rightarrow\infty}H_{m}$
exists. Each of the above steps similarly applies when setting
$H_{m}=\frac{1}{m}H(S_{1}^{m},Y_{1}^{T_{m}})$. ∎

##  II Asymptotic equipartition property

###  II-A Noisy duplication processes

In this section, we consider the AEP for the output entropy rate
$H(\mathbb{Y}^{\mathbb{T}})$ and the joint input-output entropy rate
$H(\mathbb{S},\mathbb{Y}^{\mathbb{T}})$. These results will be proven using
almost identical techniques based on genie-aided markers. Since we cannot
directly use ergodicity to prove the AEP through the Shannon-McMillan-Breiman
theorem (SMB) [1], we use the side-information of genie-aided markers to force
ergodicity in the noisy duplication processes with respect to the sub-
sequences between the markers in order to apply SMB and indirectly prove the
AEP.

######  Theorem 1 (Output AEP).

If $\\{S_{\ell}\\}$ is ergodic, then

$\displaystyle-\frac{1}{m}\log\mathbb{P}(Y_{1}^{T_{m}})\xrightarrow[]{\mathbb{P}}H(\mathbb{Y}^{\mathbb{T}}).$
(6)

The full proof of Theorem 1 is given in Appendix B, which we now summarise as
a proof sketch:

###### Proof sketch.

Let $\mathcal{M}_{m,d}=\\{T_{nd}:nd\leq m\\}$ be the markers up to the $m$-th
symbol separated by $d$ symbols. Let $n=|\mathcal{M}_{m,d}|$ be the number of
markers up to $m$, then we have $m=nd+i$ for $0\leq i<d$. Let the sample
entropy rate of $Y_{1}^{T_{m}}$ be
$g_{m}=-\frac{1}{m}\log\mathbb{P}(Y_{1}^{T_{m}})$, and let the entropy rate be
$H_{m}=\mathbb{E}[g_{m}]=\frac{1}{m}H(Y_{1}^{T_{m}})$ with limit
$H=\lim_{m\rightarrow\infty}H_{m}$ (which exists from Lemma 1). For a fixed
$d$, let the sample entropy rate of $(Y_{1}^{T_{m}},\mathcal{M}_{m,d})$ be
$g_{m,d}=-\frac{1}{m}\log\mathbb{P}(Y_{1}^{T_{m}},\mathcal{M}_{m,d})$, and let
the entropy rate be
$H_{m,d}=\mathbb{E}[g_{m,d}]=\frac{1}{m}H(Y_{1}^{T_{m}},\mathcal{M}_{m,d})$
with limit $H_{\infty,d}=\lim_{m\rightarrow\infty}H_{m,d}$.

  * •

Split problem into three parts: We need to show that $g_{m}\rightarrow H$ as
$m\rightarrow\infty$. By the triangle inequality, we need only show the chain
of convergences $g_{m}\rightarrow g_{m,d}\rightarrow H_{\infty,d}\rightarrow
H$ for some $d=d(m)$ as $m\rightarrow\infty$.

  * •

Convergence of entropy rates with markers: Adding markers can only increase
the entropy rate such that $H_{m,d}-H_{m}\geq 0$. This increase in entropy
rate is bounded as $H_{m,d}-H_{m}\leq\frac{2}{d}H(T_{d})$ by Lemma A1 and
converges to zero as the marker distance $d$ increases by Lemma A2.

  * •

Convergence of sample entropy rates with markers: Observe that $g_{m,d}$ is
decreasing towards $g_{m}$ for increasing $d$ since the side-information of
markers reduces the number of terms in the marginalisation of $T_{1}^{m}$.
Then $g_{m,d}-g_{m}\geq 0$ and Markov’s inequality says they converge since
$\mathbb{E}[g_{m,d}-g_{m}]=H_{m,d}-H_{m}$.

  * •

Shannon-McMillan-Breiman theorem with markers: Form a hidden MP (HMP)
$\\{W_{n}\\}$ with $W_{n}=Y_{T_{(n-1)d}+1}^{T_{nd}}$ based on the MP
$\\{V_{n}\\}$ with $V_{n}=S_{(n-1)d+1}^{nd}$ in between the markers. The HMP
is ergodic since the MP is ergodic [9]. Then $g_{m,d}\rightarrow H_{\infty,d}$
as $m\rightarrow\infty$ by SMB.

∎

######  Theorem 2 (Joint AEP).

If $\\{S_{\ell}\\}$ is ergodic, then

$\displaystyle-\frac{1}{m}\log\mathbb{P}(S_{1}^{m},Y_{1}^{T_{m}})\xrightarrow[]{\mathbb{P}}H(\mathbb{S},\mathbb{Y}^{\mathbb{T}}).$
(7)

###### Proof.

This proof only differs from Theorem 1 in that instead of showing ergodicity
of the HMP $\\{W_{n}\\}$ we need only show ergodicity of the process
$\\{U_{n}\\}$ for $U_{n}=(V_{n},W_{n})$. This is also an ergodic HMP with
respect to ergodic MP $\\{V_{n}\\}$. It is actually a MP, however it is more
convenient to invoke the ergodicity of HMPs [9]. ∎

There are many consequences of these AEP results. In Section III, we will use
them to derive achievable lower bounds of channel capacity and demonstrate how
they can be estimated in practice. Before that, let us explore another AEP
with an interesting connection to the output AEP.

###  II-B Hidden semi-Markov processes

For noisy duplication channels, we were interested in the case where there are
$m$ inputs and $T_{m}$ duplicated samples at the output. Alternatively, we can
choose the number of outputs to be $t$ samples, and therefore the number of
inputs is a random variable $M(t)$ much smaller than $t$. We consider the AEP
for discrete-time HSMPs over a finite state-space, extending the AEP for
discrete-time SMPs over a finite state-space [5] (and was later extended to
the Borel state-space [6], which is not considered in this paper). The central
idea behind proving our AEP will be based on embedding the SMP in a MP, for
which we can leverage existing results.

######  Definition 1 (Embedded SMP [7]).

The embedded SMP is a MP with transition probabilities

$\displaystyle\begin{split}P(s,k|s^{\prime},k^{\prime})&=\begin{cases}P(s|s^{\prime})\mathbb{P}(K=k)&k=1\\\
\mathbb{P}(K>k|K>k-1)&k=k^{\prime}+1,s=s^{\prime}\\\
0&\text{otherwise}\end{cases}\end{split}$ (8)

for all $k^{\prime},k\in\Lambda$ and for all $s^{\prime},s\in\Omega$.

An important property of the embedded duplication process is that it preserves
ergodicity. Observe that the embedded duration for each channel input form a
sequence of states with one possible transition, the “extending” transition,
until it reaches the “terminating” transition. Therefore, the embedding MP is
irreducible and thus obeys the ergodic theorem. This embedding of the SMP will
be used to prove the AEP for HSMPs in Theorem 3. Further, this HSMP AEP will
be related to the output AEP in Theorem 1 through its entropy rate using Lemma
2.

######  Lemma 2 (Randomly indexed entropy rate).

Let
$H(\mathbb{Y}^{\mathbb{T}})=\lim_{m\rightarrow\infty}\frac{1}{m}H(Y_{1}^{T_{m}})$
be the entropy rate for fixed-length inputs and variable-length outputs, and
let $H(\mathbb{Y})=\lim_{t\rightarrow\infty}\frac{1}{t}H(Y_{1}^{t})$ be the
entropy rate for variable-length inputs and fixed-length outputs. Then
$H(\mathbb{Y}^{\mathbb{T}})=\mathbb{E}[K]H(\mathbb{Y})$.

###### Proof.

Let $\tilde{Z}_{1}^{t}$ be an embedding of the randomly indexed MP
$S_{1}^{M(t)}$. Observe that the number of random inputs for $t$ output
samples is $M(t)=\sum_{t}\mathbf{1}_{\mathcal{D}}[\tilde{Z}_{t}]$ where
$\mathcal{D}=\\{(s,k)\in\mathcal{S}\times\Lambda:k=1\\}$ is the set of
embedding states that begin a new segment. With this, we will show that the
entropy rate $H(M(t))/t$ goes to zero as $t\rightarrow\infty$.

By Hoeffding’s inequality for MPs [3], we have the concentration inequality
$\mathbb{P}(|M(t)-\mathbb{E}[M(t)]|>t)\leq 2\exp(-Ct)$ for some constant $C>0$
that depends on the convergence speed of the MP to its stationary
distribution. Let $\mathcal{A}_{t}=\\{|M(t)-\mathbb{E}[M(t)]|>t\\}$, then we
can partition the entropy of $M(t)$ and bound it as

$\displaystyle\begin{split}\frac{1}{t}H(M(t))&=\frac{1}{t}H(M(t)|\mathcal{A}_{t})\mathbb{P}(\mathcal{A}_{t})\\\
&\quad\quad+\frac{1}{t}H(M(t)|\mathcal{A}^{c}_{t})(1-\mathbb{P}(\mathcal{A}_{t}))\end{split}$
(9)
$\displaystyle\leq\frac{1}{t}H(M(t))2e^{-Ct}+\frac{1}{t}\log(Ct)(1-2e^{-Ct})$
(10)

which uses the conditioning property, Hoeffding’s inequality, and that
$H(M(t)|\mathcal{A}^{c}_{t})\leq\log(2Ct)$ since $M(t)$ conditioned on
$\mathcal{A}^{c}_{t}$ is supported on the interval
$[\mathbb{E}[M(t)]-t,\mathbb{E}[M(t)]+t]$.

Observe that $|t-T_{M(t)}|\leq a$ for constant $a=\max\Lambda$ and for all
$t$. Then $H(Y_{1}^{t})/t$ is arbitrarily close to $H(Y_{1}^{T_{M(t)}})/t$ for
sufficiently large $t$, and

$\displaystyle H(\mathbb{Y})$
$\displaystyle=\lim_{t\rightarrow\infty}\frac{1}{t}H(Y_{1}^{T_{M(t)}})$ (11)
$\displaystyle=\lim_{t\rightarrow\infty}\frac{1}{t}[H(Y_{1}^{T_{M(t)}}|M(t))+H(M(t))]$
(12)
$\displaystyle=\lim_{t\rightarrow\infty}\frac{1}{t}H(Y_{1}^{T_{M(t)}}|M(t))$
(13)
$\displaystyle=\lim_{t\rightarrow\infty}\sum_{m}\mathbb{P}(M(t)=m)\left[\frac{H(Y_{1}^{T_{m}})}{m\mathbb{E}[K]}\right]$
(14)

where (14) uses a sandwich bound on the conditional entropy
$mH_{m}=H(Y_{1}^{T_{m}})=H(Y_{1}^{T_{M(t)}}|M(t)=m)$, given by

$\displaystyle\left(\frac{m}{T_{M(t)}+a}\right)H_{m}$
$\displaystyle\leq\frac{1}{t}H(Y_{1}^{T_{m}})\leq\left(\frac{m}{T_{M(t)}-a}\right)H_{m}$
(15)

which is derived from $|t-T_{M(t)}|\leq a$. By the (randomly-indexed) law of
large numbers $T_{M(t)}/M(t)\rightarrow\mathbb{E}[K]$ as $t\rightarrow\infty$,
we can squeeze $H(Y_{1}^{T_{m}})/t$ and show it has the limit
$H(\mathbb{Y}^{\mathbb{T}})/\mathbb{E}[K]$. Since $\mathbb{E}[M(t)]$ is
increasing in $t$, the left-hand tail of $\mathbb{P}(M(t)=m)$ is arbitrarily
small up to any given $m$ for sufficiently large $t$, showing convergence of
(14).∎

######  Theorem 3 (HSMP AEP).

If $\\{Z_{t}\\}$ is ergodic, then

$\displaystyle-\frac{1}{t}\log\mathbb{P}(Y_{1}^{t})\xrightarrow[]{a.s.}H(\mathbb{Y})=\frac{H(\mathbb{Y}^{\mathbb{T}})}{\mathbb{E}[K]}.$
(16)

###### Proof.

Embed the SMP $Z_{1}^{t}$ on $\Omega$ into the MP $\tilde{Z}_{1}^{t}$ on
$\Omega\times\Lambda$, such that the corresponding embedding output process
$\tilde{Y}_{1}^{t}$ is a HMP. Since the embedding MP is ergodic, the AEP
follows from the ergodicity of HMPs [9] and SMB, combined with Lemma 2. ∎

This theorem combined with the AEP for the output of the noisy duplication
channel shows equivalence between the two AEPs. An interesting consequence of
this observation is that it implies the true block length $M(t)$, which is
unknown from the $t$ observed samples, dominates the marginalisation of
$T_{1}^{M(t)}$ in the entropy rate. It is not clear if there are any practical
implications from this, but it is nonetheless an interesting observation. It
does, however, allow us to use the classical forward algorithm to estimate
$H(\mathbb{Y}^{\mathbb{T}})$ with the embedding using techniques for finite-
state channels [15, 2].

##  III Markov-constrained channel capacity

An open problem of the noisy duplication channel is its channel capacity. If
the channel were ergodic, the channel capacity would be the Shannon capacity
formula. However, this must be proven directly since we do not have ergodicity
as a given. In the special case of ergodic Markov sources, this result turns
out to be a corollary of the AEPs in Theorem 1 and Theorem 2 since they imply
information stability.

######  Theorem 4.

If the Markov source $\\{S_{\ell}\\}$ with Markov transition matrix $P$ is
ergodic, then the Markov-constrained channel capacity is the Markov-
constrained Shannon capacity

$\displaystyle
C_{\text{Markov}}=\sup_{P\in\mathcal{P}}I(\mathbb{S};\mathbb{Y}^{\mathbb{T}})$
(17)

where $\mathcal{P}$ is the set of all ergodic Markov transition matrices.

###### Proof.

Consider the information density

$i_{m}=\frac{1}{m}\log\mathbb{P}(S_{1}^{m},Y_{1}^{T_{m}})-\frac{1}{m}\log\mathbb{P}(Y_{1}^{T_{m}})-\frac{1}{m}\log\mathbb{P}(S_{1}^{m}),$

whose expectation is the finite-letter mutual information rate
$\frac{1}{m}I(S_{1}^{m};Y_{1}^{T_{m}})$. Observe that the first and second
terms converge in probability to $H(\mathbb{S},\mathbb{Y}^{\mathbb{T}})$ and
$H(\mathbb{Y}^{\mathbb{T}})$, respectively, due to Theorem 1 and Theorem 2,
and that the third term converges to $H(\mathbb{S})$ due to SMB. Consequently,
the noisy duplication channel with an ergodic Markov source is information
stable. This is a special case of the general capacity formula in [16], and
implies that the Markov-constrained Shannon capacity is achievable using the
capacity-achieving Markov source. ∎

For channel capacity $C$ with an arbitrary source, we have
$C_{\text{Markov}}\leq C$ since the optimal source may not be Markov. Even in
the restricted case of Markov sources, finding the Markov source that achieves
$C_{\text{Markov}}$ is an open problem. However, there exist heuristic
algorithms [11, 13] based on the generalised Blahut-Arimoto algorithm (GBAA)
[8, 20] for finite-state channels that can significantly improve upon the
benchmark Markov source with independent and uniformly distributed inputs
(i.e., the maximum-entropy Markov source).

In the following examples, we consider some simple capacity bounds of a noisy
duplication channel that concatenates the BSC with the sticky channel in [14]
for Bernoulli and geometric duplications.

###  III-A BSC with Bernoulli duplications

The simplest non-trivial noisy duplication channel is a BSC with Bernoulli
duplications. In particular, we consider a BSC with crossover probability $p$
and durations $K_{\ell}\sim\mathsf{Ber}(p_{d})$ on $\Lambda=\\{1,2\\}$ with
duplication probability $p_{d}$. Each source bit is duplicated with
probability $p_{d}$, and then each bit goes through the BSC that corrupts it
with probability $p$.

In Fig. 1, we use the Monte Carlo technique from [13] with a block length of
$m=10^{6}$ to compute the achievable information rate
$I_{\text{BSCD,Ber}}(p,p_{d})$ of this channel in the case of a
$\mathsf{Ber}(1/2)$ source. When $p_{d}=0$, the rate is equal to the capacity
of the BSC, $C_{\text{BSC}}(p)$. When $p_{d}=1$, the rate is equal to the
capacity of the BSC with $2$ looks, $C_{\text{BSC}^{2}}(p)$. The red curve is
when $p=0.1$, which starts at $C_{\text{BSC}}(0.1)=0.5310$ and ends at
$C_{\text{BSC}^{2}}(0.1)=0.7421$. The blue curve is when $p=0.01$, which
starts at $C_{\text{BSC}}(0.01)=0.9192$ and ends at
$C_{\text{BSC}^{2}}(0.01)=0.9787$. When $p=0$ the BSC with Bernoulli
duplications simplifies into a sticky channel with Bernoulli duplications with
rate $I_{\text{SC,Ber}}(p_{d})$ for a $\mathsf{Ber}(1/2)$ source and channel
capacity $C_{\text{SC,Ber}}(p_{d})$ for the capacity-achieving source, which
can be computed numerically [14]. The capacity of the sticky channel with
Bernoulli duplications is an upper bound on the capacity of the BSC with
Bernoulli duplications.

###  III-B BSC with geometric duplications

We now consider a BSC with geometric duplications, which is a slightly more
complex channel compared to the BSC with Bernoulli duplications. In
particular, we consider a BSC with crossover probability $p$ and durations
$K_{\ell}\sim\mathsf{Geom}(p_{d})$ on $\Lambda=\\{1,2,\ldots\\}$ with
duplication probability $p_{d}$. Each source bit is repeatedly duplicated with
probability $p_{d}$ until the first non-duplication with probability
$1-p_{d}$, and then each bit goes through the BSC that corrupts it with
probability $p$.

In Fig. 2, we once again use the Monte Carlo technique from [13] with a block
length of $m=10^{6}$ to compute the achievable information rate
$I_{\text{BSCD,geom}}(p,p_{d})$ of this channel in the case of a
$\mathsf{Ber}(1/2)$ source. As noted in [14], accurately computing rates with
geometric duplications and a high $p_{d}$ is challenging, and therefore we
stop at $p_{d}=0.6$ (just over $2$ duplications on average) as they did. In
addition, the geometric distribution is truncated after $15$ samples.
Analogously to the previous example, when $p=0$ the BSC with geometric
duplications simplifies into a sticky channel with geometric duplications with
rate $I_{\text{SC,geom}}(p_{d})$ for a $\mathsf{Ber}(1/2)$ source and channel
capacity $C_{\text{SC,geom}}(p_{d})$ for the capacity-achieving source, which
can be computed numerically [14]. The capacity of the sticky channel with
geometric duplications is an upper bound on the capacity of the BSC with
geometric duplications.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p_{d}$Information
rate [bits/symbol]BSC with Bernoulli
duplications$C_{\text{SC,Ber}}(p_{d})$$I_{\text{SC,Ber}}(p_{d})=I_{\text{BSCD,Ber}}(0,p_{d})$$I_{\text{BSCD,Ber}}(0.01,p_{d})$$I_{\text{BSCD,Ber}}(0.1,p_{d})$
Figure 1: Monte Carlo estimates of the information rate
$I_{\text{BSCD,Ber}}(p,p_{d})$ of the BSC with error probability $p$,
Bernoulli duplications with probability $p_{d}$, and a $\mathsf{Ber}(1/2)$
source. The information rate $I_{\text{SC,Ber}}(p_{d})$ is the case when
$p=0$, which corresponds to a sticky channel with capacity
$C_{\text{SC,Ber}}(p_{d})$ and is computed numerically [14].
$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$p_{d}$Information
rate [bits/symbol]BSC with geometric
duplications$C_{\text{SC,geom}}(p_{d})$$I_{\text{SC,geom}}(p_{d})=I_{\text{BSCD,geom}}(0,p_{d})$$I_{\text{BSCD,geom}}(0.01,p_{d})$$I_{\text{BSCD,geom}}(0.1,p_{d})$
Figure 2: Monte Carlo estimates of the information rate
$I_{\text{BSCD,geom}}(p,p_{d})$ of the BSC with error probability $p$,
geometric duplications with probability $p_{d}$, and a $\mathsf{Ber}(1/2)$
source. The information rate $I_{\text{SC,geom}}(p_{d})$ is the case when
$p=0$, which corresponds to a sticky channel with capacity
$C_{\text{SC,geom}}(p_{d})$ and is computed numerically [14].

##  IV Conclusion

This paper studied the noisy duplication channel and established its channel
capacity as the Markov-constrained Shannon capacity in the special case of
ergodic Markov sources. It was proven through the AEP for noisy duplication
processes, which was related to the AEP for HSMPs. The motivation for these
results was to provide a theoretical underpinning to the numerical results in
[11, 13]. A significant open problem that remains is the construction of codes
that can achieve the Markov-constrained capacity in practice, which is still
in its early stages [17, 18, 12, 19]. Further progress in this area could
greatly advance DNA storage systems based on nanopore sequencing.

##  Appendix A Preliminary lemmas

######  Lemma A1.

$H(Y,A)-H(Y)\leq 2H(A)$.

###### Proof.

We start from the inequality $|H(Y)-H(Y|A)|\leq H(A)$ [4, Lemma 4A.1]. Then
$H(Y,A)-H(Y)=H(Y|A)+H(A)-H(Y)=|H(Y|A)+H(A)-H(Y)|\leq|H(Y|A)-H(Y)|+H(A)\leq
2H(A)$. ∎

######  Lemma A2.

$\lim_{d\rightarrow\infty}\frac{1}{d}H(T_{d})=0$.

###### Proof.

By Hoeffding’s inequality, we have the concentration inequality
$\mathbb{P}(|T_{d}-\mathbb{E}[T_{d}]|>d)\leq 2\exp(-Cd)$ for some constant
$C>0$. Then the proof follows using the same argument as in Lemma 2.
Alternatively, we could use the Gaussian approximation.∎

##  Appendix B Proof of Theorem 1

Split problem into three parts. We want to show that for any $\epsilon>0$ and
$\delta>0$ we have $\mathbb{P}(|g_{m}-H|>\epsilon)<\delta$ for sufficiently
large $m$. Let $\Delta_{1}=\\{|g_{m,d}-g_{m}|>\epsilon/3\\}$,
$\Delta_{2}=\\{|g_{m,d}-H_{\infty,d}|>\epsilon/3\\}$, and
$\Delta_{3}=\\{|H_{\infty,d}-H|>\epsilon/3\\}$. Then we have the bound

$\displaystyle\mathbb{P}(|g_{m}-H|>\epsilon)$ (18)
$\displaystyle\leq\mathbb{P}(|g_{m,d}-g_{m}|+|g_{m,d}-H_{\infty,d}|+|H_{\infty,d}-H|>\epsilon)$
(19) $\displaystyle\leq\underbrace{\mathbb{P}(\Delta_{1})}_{\text{Term
1}}+\underbrace{\mathbb{P}(\Delta_{2})}_{\text{Term
2}}+\underbrace{\mathbb{P}(\Delta_{3})}_{\text{Term 3}}$ (20)

and we want to show that each term is bounded by $\delta/3$, where $d$ is
fixed and does not affect our choice of $m$. We will prove this in three parts
and then combine the results.

Convergence of entropy rates with markers. Observe that $g_{m,d}$ is
decreasing towards $g_{m}$ for increasing $d$ (with equality at $d=m$), then
$g_{m,d}-g_{m}\geq 0$ and $\mathbb{E}[g_{m,d}-g_{m}]\geq 0$. From Lemma A1,
for a fixed $d$ we have the bound
$\mathbb{E}[g_{m,d}-g_{m}]=H_{m,d}-H_{m}\leq\frac{2}{m}H(\mathcal{M}_{m,d})=\frac{2}{d}H(T_{d})$
uniformly in $m$. Then take the limit in $m$ to get
$H_{\infty,d}-H\leq\frac{2}{d}H(T_{d})$. By Lemma A2, there exists a $D$ such
that $H_{m,d}-H\leq\epsilon/3$ and then $\Delta_{3}$ occurs with probability
$0$ for all $d\geq D$. This proves almost sure convergence for Term 3.

Convergence of sample entropies with markers. For any $\delta^{\prime}>0$
there exists a $D^{\prime}$ such that such that
$H_{m,d}-H_{m}\leq\delta^{\prime}$ and $\delta^{\prime}/\epsilon\leq\delta/3$
for all $d\geq D^{\prime}$. Recalling that $g_{m,d}-g_{m}\geq 0$, we can apply
Markov’s inequality to get

$\displaystyle\mathbb{P}(\Delta_{1})$ $\displaystyle\leq
3\mathbb{E}[g_{m,d}-g_{m}]/\epsilon$ (21) $\displaystyle\leq
3(H_{m,d}-H_{m})/\epsilon$ (22)
$\displaystyle\leq\delta^{\prime}/\epsilon\leq\delta/3$ (23)

for all $d\geq D^{\prime}$. This proves convergence in probability for Term 1,
which will now be linked with the convergence result of Term 3 by showing the
convergence of Term 2.

Shannon-McMillan-Breiman theorem with markers. Consider the $d$-step MP
$\\{V_{n}\\}$ where $V_{n}=S_{(n-1)d+1}^{nd}$. Since it embeds the ergodic MP
$\\{S_{\ell}\\}$, which is a one-to-one mapping between probability spaces
with identical probability measure, it is also ergodic. Then we have the HMP
$\\{W_{n}\\}$ where $W_{n}=Y_{T_{(n-1)d}+1}^{T_{nd}}$. Since the $d$-step MP
is ergodic, then the resulting HMP is ergodic [9]. Apply SMB to get
$g_{nd,d}\rightarrow H_{\infty,d}$ almost surely as $n\rightarrow\infty$ for a
fixed $d$ (hence $m\rightarrow\infty$). Then there exists a sequence
$\\{N(d)\\}$ such that $|g_{nd,d}-H_{\infty,d}|\leq\epsilon/6$ for all
$n\geq\overline{N}(d)=\max\\{N(n_{1}):n_{1}\leq d\\}$ with probability $1$. We
force monotonicity in $\\{\overline{N}(d)\\}$ in order to later specify a
lower bound on $m$ in terms of a lower bound on $d$.

By applying the chain rule, observe that $g_{m,d}=g_{nd,d}+g^{\prime}_{i}$ for
a sample entropy $g^{\prime}_{i}\geq 0$ with mean bounded as
$\mathbb{E}[g^{\prime}_{i}]\leq\frac{1}{m}H(Y_{1}^{T_{i}})\leq C/m$ for some
constant $C>0$. By Markov’s inequality, we have
$\mathbb{P}(g^{\prime}_{i}\geq\epsilon/6)\leq 6C/m\epsilon$ and hence there
exists an $M$ such that $6C/m\epsilon\leq\delta/3$ for all $m\geq M$. Then

$\displaystyle\mathbb{P}(\Delta_{2})$
$\displaystyle\leq\mathbb{P}(|g_{nd,d}-H_{\infty,d}|+g^{\prime}_{i}>\epsilon/3)$
(24)
$\displaystyle\leq\mathbb{P}(|g_{nd,d}-H_{\infty,d}|>\epsilon/6)+\mathbb{P}(g^{\prime}_{i}>\epsilon/6)$
(25) $\displaystyle\leq 6C/m\epsilon\leq\delta/3$ (26)

for all $m\geq\max\\{\overline{N}(d)d,M\\}$ and for any fixed $d$, where we
use $m\geq nd$ to specify a lower bound on $m$ in terms of $\overline{N}(d)$.
This proves convergence in probability for Term 2.

Combining all three terms, we have $\mathbb{P}(|g_{m}-H|>\epsilon)\leq
2\delta/3$ for all $m\geq\max\\{\overline{N}(d)d,M\\}$ and for all
$d\geq\overline{D}=\max\\{D,D^{\prime}\\}$. Hence, for any $\epsilon>0$ and
$\delta>0$ we choose $d=\overline{D}$ so that
$\mathbb{P}(|g_{m}-H|>\epsilon)<\delta$ for all
$m\geq\max\\{\overline{N}(\overline{D})\overline{D},M\\}$.

## References

  * [1] Paul H. Algoet and Thomas M. Cover.  A sandwich proof of the Shannon-McMillan-Breiman theorem.  Annals of Probability, 16:899–909, 1988.
  * [2] Dieter-Michael Arnold and Hans-Andrea Loeliger.  On the information rate of binary-input channels with memory.  ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240), 9:2692–2695 vol.9, 2001.
  * [3] Jianqing Fan, Bai Jiang, and Qiang Sun.  Hoeffding’s inequality for general Markov chains and its applications to statistical learning.  Journal of machine learning research : JMLR, 22, 2021.
  * [4] Robert G. Gallager.  Information theory and reliable communication.  1968\.
  * [5] Valerie Girardin and Nikolaos Limnios.  Entropy rate and maximum entropy methods for countable semi-Markov chains.  Communications in Statistics - Theory and Methods, 33(3):609–622, 2004.
  * [6] Valérie Girardin and Nikolaos Limnios.  Entropy for semi-Markov processes with Borel state spaces: asymptotic equirepartition properties and invariance principles.  Bernoulli, 12:515–533, 2006.
  * [7] Matthew J. Johnson.  Bayesian time series models and scalable inference.  PhD thesis, Massachusetts Institute of Technology, 2014.
  * [8] Aleksandar Kavcic.  On the capacity of Markov sources over noisy channels.  GLOBECOM’01. IEEE Global Telecommunications Conference (Cat. No.01CH37270), 5:2997–3001 vol.5, 2001.
  * [9] Brian G. Leroux.  Maximum-likelihood estimation for hidden Markov models.  Stochastic Processes and their Applications, 40(1):127–143, 1992.
  * [10] Brendon McBain and Emanuele Viterbo.  An information-theoretic approach to nanopore sequencing for DNA storage.  IEEE BITS the Information Theory Magazine, pages 1–13, 2024.
  * [11] Brendon McBain, Emanuele Viterbo, and James Saunderson.  Finite-state semi-Markov channels for nanopore sequencing.  2022 IEEE International Symposium on Information Theory (ISIT), pages 216–221, 2022.
  * [12] Brendon McBain, Emanuele Viterbo, and James Saunderson.  Homophonic coding for the noisy nanopore channel with constrained Markov sources.  2023 IEEE International Symposium on Information Theory (ISIT), 2023.
  * [13] Brendon McBain, Emanuele Viterbo, and James Saunderson.  Information rates of the noisy nanopore channel.  IEEE Transactions on Information Theory, Accepted: May 2024.
  * [14] Michael Mitzenmacher.  Capacity bounds for sticky channels.  IEEE Transactions on Information Theory, 54(1):72–77, 2008.
  * [15] Henry D. Pfister, Joseph B. Soriaga, and Paul H. Siegel.  On the achievable information rates of finite state ISI channels.  GLOBECOM’01. IEEE Global Telecommunications Conference (Cat. No.01CH37270), 5:2992–2996 vol.5, 2001.
  * [16] Sergio Verdú and Te Sun Han.  A general formula for channel capacity.  IEEE Trans. Inf. Theory, 40:1147–1157, 1994.
  * [17] A. Vidal, V. B. Wijekoon, and E. Viterbo.  Error bounds for decoding piecewise constant nanopore signals in DNA storage.  Int. Conference on Communications (ICC), May 2023.
  * [18] Adrian Vidal, V. B. Wijekoon, and Emanuele Viterbo.  Union bound for generalized duplication channels with DTW decoding.  2023 IEEE International Symposium on Information Theory (ISIT), pages 358–363, 2023.
  * [19] Adrian Vidal, V. B. Wijekoon, and Emanuele Viterbo.  Concatenated nanopore DNA codes.  IEEE Transactions on NanoBioscience, 23(2):310–318, 2024.
  * [20] Pascal O. Vontobel, Aleksandar Kavcic, Dieter-Michael Arnold, and Hans-Andrea Loeliger.  A generalization of the Blahut–Arimoto algorithm to finite-state channels.  IEEE Transactions on Information Theory, 54:1887–1918, 2008.


11institutetext:  Department of Mathematics, Birla Institute of Technology and
Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.
Department of Physics, Zhejiang Normal University, Jinhua, 321004, People’s
Republic of China.
UFCG - Universidade Federal de Campina Grande - Unidade Acadêmica de Física,
58429-900 Campina Grande, PB, Brazil.

# Embedding procedure and wormhole solutions in $f(Q)$ gravity

Zinnat Hassan111E-mail<EMAIL_ADDRESS>11 G. Mustafa222E-mail:
<EMAIL_ADDRESS>22 Joao R.L. Santos333E-Mail<EMAIL_ADDRESS>33 P. K. Sahoo444E-mail<EMAIL_ADDRESS>11112233

###### Abstract

An intriguing solution that appears in General Relativity (GR) but has not
been observed so far is the wormhole. This exotic solution describes a
topological bridge connecting two distinct universes or two different points
in the same universe. It is known that the traversable wormhole solutions
violate all the energy conditions in GR, resulting in their instability. In
this work, we are going to unveil new wormhole solutions for $f(Q)$ gravity
where $Q$ is the non-metricity scalar, which is responsible for the
gravitational interaction. The energy conditions to constraint these wormhole
solutions were derived using the embedding procedure. This procedure consists
of rewriting the density and the pressures of the solutions as those presented
by General Relativity. Then, the nontrivial contributions coming from new
theories of gravity are embedded into the effective equations for density and
pressures. Along with our approach, we carefully analyze two families of
$f(Q)$ models and we used two different shape functions to build the wormholes
solutions for each of these $f(Q)$ models. We are going to present new
scenarios with the possibility of traversable wormholes satisfying SEC or DEC
energy conditions in the presence of exotic matter.

###### pacs:

04.50.Kd

##  1 Introduction

Today we are facing one of the most exciting times in astrophysics and
cosmology. Experiments such as, Laser Interferometer Gravitational-Wave
Observatory (LIGO) [1], Event Horizon Telescope (EHT) [2, 3], Virgo [4],
Advanced Telescope for High-Energy Astrophysics (ATHENA) [5], and Swift [6],
have been responsible for changing the bars of our knowledge about the nature
of gravity and astrophysical objects. They have also been used to set new
constraints over different theories of gravity, selecting viable models among
the enormous amount of proposals presented in the literature.
An exotic type of solution that appears in General Relativity but has not been
observed so far is the wormhole (WH) [7]. The wormhole is a topological bridge
connecting points in two distinct universes or two different points in the
same universe. Although they have never been observed, several investigations
in the literature propose the viability of finding wormholes. One possible
strategy of observation was recently proposed by Bueno et al. [8], using
gravitational waves measurements to study echoes of the gravitational wave
signal at the horizon scale of black holes. These gravitational waves would be
related to the postmerger ringdown phase in binary coalescences. Another
interesting idea for observation was introduced by Paul et al. [9], using
distinctive features of the accretion disk images to distinguish between a
wormhole geometry and a black hole.
Beyond the observations, it is also relevant to a search for traversable
wormholes, which means wormholes that are big enough that a person could
traverse them and survive the tidal forces. This class of wormholes was
proposed in the seminal paper of Morris and Thorne [10] and is still under
investigation, as we can see in the recent manuscript of Maldacena and
Milekhin [11].
The main issue related to traversable wormholes is that these solutions
violate all energy conditions in General Relativity. Therefore we are impelled
to search for traversable wormhole solutions in different theories of gravity
that could obey energy conditions, allowing their existence.
As it is known, wormholes can also be used as prototypes to test several
theories of gravity [12, 13]. A new and promising theory of gravity is the so-
called $f(Q)$ gravity, introduced by Jimenez et al. [14], where the
nonmetricity $Q$ is responsible for the gravitational interaction. In the last
few years, several observational data have been tested in $f(Q)$ gravity. Some
interesting constraints over $f(Q)$ gravity were proposed by Lazkoz et al.
[15]. There the authors tested the viability of $f(Q)$ gravity using data from
the expansion rate, Type Ia Supernovae, Quasars, Gamma-Ray Bursts, Baryon
Acoustic Oscillations data, and Cosmic Microwave Background distance.
Moreover, the viability of $f(Q)$ gravity models in respect to energy
conditions was successfully proven by Mandal et al. [18], where the authors
introduced an embedding procedure to include non-trivial contributions of the
nonmetricity function into the energy constraints.
In this work, we apply the embedding procedure to derive traversable wormhole
solutions in $f(Q)$ gravity. This procedure consists of rewriting the density
and the pressures of the solutions as those presented by General Relativity.
Then, the nontrivial contributions coming from new theories of gravity are
embedded into the effective equations for density and pressures. The advantage
of such a methodology is that we can find the energy conditions using the
standard constraints from the Raychaudhuri equation. The embedding procedure
allowed us to study the null, dominant, and strong energy conditions for
wormholes in the presence of an anisotropic fluid. We carefully analyze two
families of $f(Q)$ models, and we also used two different shape functions, a
$r^{\,n}$ and $r\,e^{1-\frac{r}{r_{0}}}$ ones, to build the wormholes
solutions for each of these $f(Q)$ models. It is relevant to point that some
of us had recently reported new traversable wormhole families of solutions in
$f(Q)$ [19]. However, the energy constraints previously determined in [19] did
not consider the embedding procedure, so the current work presents an updated
approach for finding traversable wormholes compatible with investigations
recently introduced on cosmological constraints for $f(Q)$, and $f(Q,T)$
gravity [18, 20]. Along with our discussions, we are going to present
scenarios with the possibility of traversable wormholes, satisfying at least
one set of energy conditions in the presence of exotic matter.

##  2 The $f(Q)$ gravity

In this section, we are going to briefly present some generalities about the
$f(Q)$ gravity. The so-called symmetric teleparallel gravity or $f(Q)$ gravity
was introduced by Jimenez et al. [14]. The action for this modified gravity is
given by

$\mathcal{S}=\int\frac{1}{2}\,f(Q)\sqrt{-g}\,d^{4}x+\int\mathcal{L}_{m}\,\sqrt{-g}\,d^{4}x\,,$
(1)

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$, and
$\mathcal{L}_{m}$ is Lagrangian density of matter.
The nonmetricity tensor is given by

$Q_{\lambda\mu\nu}=\bigtriangledown_{\lambda}g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}-\Gamma^{\beta}_{\,\,\,\lambda\mu}g_{\beta\nu}-\Gamma^{\beta}_{\,\,\,\lambda\nu}g_{\mu\beta},$
(2)

where $\Gamma^{\beta}_{\,\,\,\mu\nu}$ is the affine connection and it can be
decomposed into the following components,

$\Gamma^{\beta}_{\,\,\,\mu\nu}=\\{^{\beta}_{\,\,\,\mu\nu}\\}+K^{\beta}_{\,\,\,\mu\nu}+L^{\beta}_{\,\,\,\mu\nu},$
(3)

where $\\{^{\beta}_{\,\,\,\mu\nu}\\}$ is the Levi-Civita connection defined by

$\\{^{\beta}_{\,\,\,\mu\nu}\\}=\frac{1}{2}g^{\beta\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right),$
(4)

the disformation $L^{\beta}_{\,\,\,\mu\nu}$ is defined by

$L^{\beta}_{\,\,\,\mu\nu}=\frac{1}{2}Q^{\beta}_{\,\,\,\mu\nu}-Q_{(\mu\,\,\,\,\,\,\nu)}^{\,\,\,\,\,\,\beta},$
(5)

and $K^{\beta}_{\,\,\,\mu\nu}$ is the contortion written as

$K^{\beta}_{\,\,\,\mu\nu}=\frac{1}{2}T^{\beta}_{\,\,\,\mu\nu}+T_{(\mu\,\,\,\,\,\,\nu)}^{\,\,\,\,\,\,\beta},$
(6)

with the torsion tensor $T^{\beta}_{\,\,\,\mu\nu}$ defined as the anti-
symmetric part of the affine connection,
$T^{\beta}_{\,\,\,\mu\nu}=2\Gamma^{\lambda}_{\,\,\,[\mu\nu]}$.
One can also establish a superpotential related to the nonmetricity tensor as

$\displaystyle P^{\alpha}\;_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[-Q^{\alpha}\;_{\mu\nu}+2Q_{(\mu}\;^{\alpha}\;{}_{\nu)}+Q^{\alpha}g_{\mu\nu}\right.$
(7)
$\displaystyle\left.-\tilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu)}\right],$

where

$Q_{\alpha}=Q_{\alpha}\;^{\mu}\;{}_{\mu},\;\tilde{Q}_{\alpha}=Q^{\mu}\;_{\alpha\mu}.$
(8)

are two independence traces.
which enable us to define the nonmetricity scalar as

$Q=-Q_{\alpha\mu\nu}\,P^{\alpha\mu\nu}.$ (9)

In order to find the field equations for this theory of gravity, we can set
that the action (1) is constant in respect to variations over the metric
tensor $g_{\mu\nu}$, resulting in

$\frac{2}{\sqrt{-g}}\bigtriangledown_{\gamma}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f\\\
+f_{Q}\left(P_{\mu\gamma i}\,Q_{\nu}\;^{\gamma i}-2\,Q_{\gamma
i\mu}\,P^{\gamma i}\;_{\nu}\right)=-T_{\mu\nu},$ (10)

where $f_{Q}=\frac{df}{dQ}$, and $T_{\mu\,\nu}$ is the standard energy-
momentum tensor, whose form is

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\,\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}}.$ (11)

Moreover, by taking action (1) constant in respect to variations over the
connection, we are able to derive the extra constraint

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)=0.$
(12)

The torsionless and curvatureless make the affine connection $\Gamma$ have the
following form [14]

$\Gamma^{\lambda}_{\,\,\,\mu\nu}=\left(\frac{\partial
x^{\lambda}}{\partial\xi^{\beta}}\right)\partial_{\mu}\partial_{\nu}\xi^{\beta}.$
(13)

where $\xi$ is the frame of transformation. This transformation of
$\Gamma^{\lambda}_{\,\,\,\mu\nu}$ is the core of the covariant formulation of
$f(Q)$ gravity, which preserves the Lorentz covariance under the nonmetricity
framework. Two excellent reviews about this calculation were nicely presented
by Xu et al. in [21], and by Jimenez et al. [22]
In this study, we have considered a special coordinate choice, the so-called
coincident gauge, so that $\Gamma^{\lambda}_{\,\,\,\mu\nu}=0$.
Then, the nonmetricity (2) reduces to

$Q_{\lambda\mu\nu}=\partial_{\lambda}g_{\mu\nu},$ (14)

and thereby vastly simplifies the calculation since only the metric is the
fundamental variable. However, in this case, the action no longer remains
diffeomorphism invariant, except for standard General Relativity [22]. One can
use the covariant formulation of $f(Q)$ gravity to avoid such an issue. Since
the affine connection in Eq. (13) is purely inertial, one could use the
covariant formulation by first determining affine connection in the absence of
gravity [23].

##  3 Wormholes in $f(Q)$ gravity

Let us consider the standard static spherically symmetric Morris and Thorne
metric in Schwarzschild coordinates [10], given by

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

where $b(r)$, and $\Phi(r)$ denote the shape function and redshift function,
respectively. Moreover, the radial coordinate $r$ varies between $r_{0}\leq
r<\infty$, with $r_{0}$ as the radius of the throat. To avoid the presence of
event horizons, the redshift function $\Phi(r)$ is must be finite everywhere.
Besides, the shape function $b(r)$ must obey the following conditions to
ensure a traversable wormhole:

  * •

Throat condition: at the throat i.e. at $r=r_{0}$, the shape function must
satisfy the constraint $b(r_{0})=r_{0}$, and for $r>r_{0}$ we have
$1-\frac{b(r)}{r}>0$.

  * •

Flaring out condition: $b^{\,\prime}(r_{0})<1$, at wormhole’s throat.

  * •

Asymptotically flatness condition: $\frac{b(r)}{r}\rightarrow o$ as
$r\rightarrow\infty$.

In this study, we will work with the energy-momentum tensor for an anisotropic
fluid which is define by

$T_{\mu}^{\nu}=\left(\rho+P_{t}\right)u_{\mu}\,u^{\nu}+P_{t}\,\delta_{\mu}^{\nu}+\left(P_{r}-P_{t}\right)v_{\mu}\,v^{\nu},$
(16)

where $\rho$ and $u_{\mu}$ are the density and the four-velocity vector,
respectively. This energy-momentum tensor was introduced by Letelier [24] to
describe a two fluid model in plasma physics, and has been applied in several
distinct scenarios so far, as for instance, in models for magnetized neutron
stars [25]. Besides $v_{\mu}$ is the unitary space-like vector in the radial
direction, $P_{r}$ is the radial pressure in the direction of $u_{\mu}$, and
$P_{t}$ is the tangential pressure orthogonal to $v_{\mu}$.
Using Eq. (9), we can determine the nonmetricity scalar for the metric (15) is
given by

$Q=-\frac{2}{r}\left(1-\frac{b(r)}{r}\right)\left(2\phi{{}^{\prime}}(r)+\frac{1}{r}\right).$
(17)

So, by taking the previous energy-momentum tensor together with the metric
(15), into the field equations (10), we obtain

$\frac{b^{{}^{\prime}}}{r^{2}}=-\frac{1}{f_{Q}}\left(\rho-\frac{f}{2}\right)+\frac{r-b}{r^{2}}\left[\frac{f_{QQ}}{f_{Q}}\,Q^{{}^{\prime}}+2\phi^{{}^{\prime}}+\frac{1}{r}\right],$
(18)
$-\frac{b}{r^{3}}=-\frac{1}{2\,f_{Q}}\left(p_{r}+\frac{f}{2}\right)-\frac{1}{2\,r^{2}}\left[1+4(r-b)\phi^{{}^{\prime}}\right],$
(19)
$-\frac{rb^{{}^{\prime}}-b}{2r^{3}}=-\frac{1}{f_{Q}}\left(P_{t}+\frac{f}{2}\right)-\\\
\frac{r-b}{r^{2}}\left[\frac{f_{QQ}}{f_{Q}}\,Q^{{}^{\prime}}+\frac{1}{1+r\,\phi{{}^{\prime}}}\left(\frac{1}{r}+\phi{{}^{\prime}}(3+r\,\phi{{}^{\prime}})+r\,\phi{{}^{\prime\prime}}\right)\right],$
(20)

where ′ represents $\frac{d}{dr}$.
Since the redshift function $\phi(r)$ should be finite and non-vanishing at
the WH throat. So we have considered $\phi(r)=constant$ in this current study
and hence $\phi{{}^{\prime}}(r)=0$.

###  3.1 The Embedding Procedure

In order to successfully embed the non-trivial contributions from $f(Q)$
gravity into the energy density and pressures related to the traversable
wormhole solution, let us rewrite Eqs. (18) - (20) in the following forms

$\rho=\frac{1}{r}\left(1-\frac{b}{r}\right)\left[2f_{QQ}Q^{{}^{\prime}}+f_{Q}\left(\frac{1}{r}-\frac{b^{{}^{\prime}}}{r-b}\right)+\right.\\\
\left.f\,\frac{r^{2}}{2(r-b)}\right],$ (21)
$p_{r}=\frac{1}{r}\left(1-\frac{b}{r}\right)\left[f_{Q}\left(\frac{2\,b-r}{r(r-b)}\right)-f\,\frac{r^{2}}{2(r-b)}\right],$
(22)
$p_{t}=\frac{1}{2\,r}\left(1-\frac{b}{r}\right)\left[-2\,f_{QQ}Q^{{}^{\prime}}+f_{Q}\,\left(\frac{r\,b^{{}^{\prime}}+b-2\,r}{r(r-b)}\right)-\right.\\\
\left.f\,\frac{r^{2}}{2(r-b)}\right].$ (23)

Comparing Eqs. (18) - (20) with the Morris-Throne traversable wormhole
equations for General Relativity [26], we are able to find the following
effective density and pressures

$\tilde{\rho}=\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{2\,f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)-\frac{1}{f_{Q}}\left(\rho-\frac{f}{2}\right)\,,$
(24)
$\tilde{P_{r}}=-\frac{1}{2}\left(\frac{1}{r^{2}}+\frac{1}{f_{Q}}\left(P_{r}+\frac{f}{2}\right)\right)\,,$
(25)
$\tilde{P_{t}}=-\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)-\frac{1}{f_{Q}}\left(P_{t}+\frac{f}{2}\right)\,.$
(26)

These previous relations unveil that the nonmetricity contributions are
embedded into $\tilde{\rho}$, $\tilde{P_{r}}$, and $\tilde{P_{t}}$, and are
crucial to properly determine the different set of energy conditions for the
wormholes.

###  3.2 Energy Conditions

The energy conditions are fundamental tools for modified theories of gravity,
since they enable us to carefully analyze the casual and geodesic structure of
space-time. One path to derive such conditions is through the Raychaudhuri
equations, which describe the action of congruence and attractiveness of the
gravity for timelike, spacelike, or lightlike curves. Since we are working
with an anisotropic fluid, the energy conditions for standard GR are such that

$\bullet$ Strong energy conditions (SEC) if $\tilde{\rho}+\tilde{P_{j}}\geq
0$, $\tilde{\rho}+\sum_{j}\tilde{P_{j}}\geq 0$, $\forall j$.
$\bullet$ Dominant energy conditions (DEC) if $\tilde{\rho}\geq 0$,
$\tilde{\rho}\pm\tilde{P_{j}}\geq 0$, $\forall j$.
$\bullet$ Weak energy conditions (WEC) if $\tilde{\rho}\geq 0$,
$\tilde{\rho}+\tilde{P_{j}}\geq 0$, $\forall j$.
$\bullet$ Null energy condition (NEC) if $\tilde{\rho}+\tilde{P_{j}}\geq 0$,
$\forall j$.
where $j=r,t$ and $P_{j}$ and $\rho$ denotes the pressures and energy density,
respectively.

So, taking the effective density and pressures from Eqs. (24)-(26), we yield
to

$\tilde{\rho}+\tilde{P_{r}}=-\frac{1}{2\,f_{Q}}\left(\rho+P_{r}\right)+\frac{1}{2r}\left(-\frac{2}{r}+\frac{r\,b^{{}^{\prime}}+b}{r^{2}}\right)\\\
+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right);$
(27)
$\tilde{\rho}+\tilde{P_{t}}=-\frac{1}{f_{Q}}\left(\rho+P_{t}\right)+\frac{1}{r}\left(1-\frac{b}{r}\right)\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}};$
(28) $\tilde{\rho}-\tilde{P_{r}}=-\frac{1}{2\,f_{Q}}\left(\rho-
P_{r}\right)+\frac{f}{2\,f_{Q}}+\frac{r\,b^{{}^{\prime}}+b}{2\,r^{3}}\\\
+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right);$
(29) $\displaystyle\tilde{\rho}-\tilde{P_{t}}=-\frac{1}{2\,f_{Q}}$ (30)
$\displaystyle\times\left(\rho-
P_{t}\right)+\frac{f}{f_{Q}}+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{2}{r}+\frac{3\,f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right);$
$\displaystyle\tilde{\rho}+\tilde{P_{r}}+2\tilde{P_{t}}=-\frac{1}{2\,f_{Q}}$
$\displaystyle\times\,\left(\rho+P_{r}+2\,P_{t}\right)+\frac{b}{r^{3}}-\frac{f}{2\,f_{Q}}.$
(31)

Therefore, the previous equations enable us to compute the following set of
energy conditions for wormholes at $f(Q)$ gravity:

WEC: $\,\,\,\,\,\tilde{\rho}\geq 0\,\,\Rightarrow\,\,\,\,\,\,\rho\geq 0\,\,\,$
with $\,\,f_{Q}\leq 0$,
$\,\,\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{2\,f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)+\frac{f}{2\,f_{Q}}\geq
0.$
NEC: $\,\,\,\,\,\tilde{\rho}+\tilde{P_{r}}\geq
0\,\,\Rightarrow\,\,\,\,\,\,\rho+P_{r}\geq 0\,\,\,$ with $f_{Q}\leq
0$,$\,\,\,\,\,\frac{1}{2r}\left(-\frac{2}{r}+\frac{r\,b^{{}^{\prime}}+b}{r^{2}}\right)+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)\geq
0\,\,\,\,$ and $\,\,\,\tilde{\rho}+\tilde{P_{t}}\geq 0\,\,\Rightarrow\,\,\,\,\
\rho+P_{t}\geq 0\,\,\,$ with $\,\,\,f_{Q}\leq
0$,$\,\,\,\frac{1}{r}\left(1-\frac{b}{r}\right)\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\geq
0$.
DEC: $\,\,\,\,\,\tilde{\rho}-\tilde{P_{r}}\geq
0\,\,\Rightarrow\,\,\,\,\,\,\rho-P_{r}\geq 0\,\,\,$ with $\,\,\,f_{Q}\leq
0$,$\,\,\,\,\frac{r\,b^{{}^{\prime}}+b}{2\,r^{3}}+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{1}{r}+\frac{f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)\geq
0\,\,\,\,$ and $\,\,\,\,\,\tilde{\rho}-\tilde{P_{t}}\geq
0\,\,\Rightarrow\,\,\,\,\,\,\rho-P_{t}\geq 0\,\,\,$ with $\,\,\,f_{Q}\leq
0$,$\,\,\,\,\frac{f}{f_{Q}}+\frac{1}{r}\left(1-\frac{b}{r}\right)\left(\frac{2}{r}+\frac{3\,f_{QQ}Q^{{}^{\prime}}}{f_{Q}}\right)\geq
0$.
SEC: $\,\,\,\,\,\tilde{\rho}+\tilde{P_{r}}+2\tilde{P_{t}}\geq
0\,\,\Rightarrow\,\,\,\,\,\,\rho+P_{r}+2P_{t}\geq 0\,\,\,$ with
$\,\,\,f_{Q}\leq 0$,$\,\,\,\,\frac{b}{r^{3}}-\frac{f}{2\,f_{Q}}\geq 0$.

##  4 Wormholes in $f(Q)$ Gravity

After presenting the generalities and the constraints over the energy
conditions, we can test the viability of different families of $f(Q)$ gravity
to properly describe stable wormhole solutions. In this section, we present
our analyzes considering two specific forms of $f(Q)$ models, (i) the non-
linear model, and (ii) the logarithmic model.

###  4.1 Model-I: $f(Q)=\alpha Q+\beta Q^{n}$

Let us firstly consider a non-linear form of $f(Q)$, i.e., $f(Q)=\alpha
Q+\beta Q^{n}$ [27], with $\alpha$, $\beta$, and $n$ are free parameters. In
this article, we study with the particular form of this model by considering
$n=4$. In order to build wormhole solutions, we are going to work with the
shape functions $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{m}$, and
$b(r)=r\,e^{1-\frac{r}{r_{0}}}$. The first shape function was presented in the
seminal work of Lobo et al. [28], where it was verified the viability of
wormhole solutions for different $f(R)$ gravity theories. The second shape
function was introduced by Moraes et al. [29], in order to fulfill the energy
conditions constraints for wormholes in $f(R,T)$ gravity theories. Moreover,
these shape functions satisfies all the basic criteria of traversable
wormhole.

###  4.2 Shape function (SF-1): $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{m}$

By taking this $f(Q)$ gravity family, as well as the shape function
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$ into Eqs. (21)-(23), we get the
following density and pressures,

$\displaystyle\rho=\frac{\beta
2^{n-1}\left(\frac{r_{0}\left(\frac{r}{r_{0}}\right)^{m}-r}{r^{3}}\right)^{n}}{r-r_{0}\left(\frac{r}{r_{0}}\right)^{m}}\left(r_{0}(n(2n(m-3)-m+7)-1)\right.$
$\displaystyle\left.\left(\frac{r}{r_{0}}\right)^{m}+(n-1)(4n-1)r\right)-\frac{\alpha
mr_{0}\left(\frac{r}{r_{0}}\right)^{m}}{r^{3}},$ (32) $\displaystyle
P_{r}=\frac{\beta
2^{n-1}\left(\frac{r_{0}\left(\frac{r}{r_{0}}\right)^{m}-r}{r^{3}}\right)^{n}}{r-r_{0}\left(\frac{r}{r_{0}}\right)^{m}}\left((n-1)r-(2n-1)r_{0}\left(\frac{r}{r_{0}}\right)^{m}\right)$
$\displaystyle+\frac{\alpha r_{0}\left(\frac{r}{r_{0}}\right)^{m}}{r^{3}},$
(33)
$P_{t}=\frac{\alpha(m-1)r_{0}\left(\frac{r}{r_{0}}\right)^{m}}{2\,r^{3}}-\\\
\frac{\beta\,2^{n-2}(2n-1)\left(\frac{r_{0}\left(\frac{r}{r_{0}}\right)^{m}-r}{r^{3}}\right)^{n}}{r-r_{0}\left(\frac{r}{r_{0}}\right)^{m}}\left(r_{0}(n(m-3)+2)\left(\frac{r}{r_{0}}\right)^{m}\right.\\\
\left.+2(n-1)r\right).$ (34)

Now, combining Eqs. (4.2)-(34) with the constraints found at the end of
section 3, we have shown the behavior of energy conditions which is depicted
in Fig. 1. In the last section, we will discuss the behavior of our obtained
graphs.

Figure 1: Evolution of energy conditions with $n=4,\,\,\beta=0.8\,\,m=0.7$ and
$r_{0}=1$ for the shape function $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$.

###  4.3 Shape function (SF-2): $b(r)=r\,e^{1-\frac{r}{r_{0}}}$

As a next example, let us work with the quadratic form of $f(Q)$ together with
the exponential shape function $b(r)=r\,e^{1-\frac{r}{r_{0}}}$. So, by taking
these ingredients into Eqs. (21)-(23) we derive our density and pressures for
these wormhole solutions as follows

$\rho=\frac{\beta
2^{n-1}\left(\frac{e^{1-\frac{r}{r_{0}}}-1}{r^{2}}\right)^{n}}{r_{0}(e^{r/r_{0}}-e)}\left((n-1)(4n-1)\,r_{0}e^{r/r_{0}}-\right.\\\
\left.e\left(2n^{2}(r+2\,r_{0})-n(r+6r_{0})+r_{0}\right)\right)+\frac{\alpha
e^{1-\frac{r}{r_{0}}}(r-r_{0})}{r_{0}\,r^{2}},$ (35) $\displaystyle
P_{r}=\frac{\beta
2^{n-1}\left((n-1)e^{r/r_{0}}-2en+e\right)\left(\frac{e^{1-\frac{r}{r_{0}}}-1}{r^{2}}\right)^{n}}{e^{r/r_{0}}-e}$
$\displaystyle+\frac{\alpha e^{1-\frac{r}{r_{0}}}}{r^{2}},$ (36)
$\displaystyle P_{t}=\frac{\beta
2^{n-1}\left((n-1)e^{r/r_{0}}-2en+e\right)\left(\frac{e^{1-\frac{r}{r_{0}}}-1}{r^{2}}\right)^{n}}{e^{r/r_{0}}-e}$
$\displaystyle+\frac{\alpha e^{1-\frac{r}{r_{0}}}}{r^{2}}.$ (37)

The behavior of energy conditions have been shown in Fig. 2 by using the above
equations along with the constraints found in section-3.

Figure 2: Evolution of energy conditions with $\beta=5,\,\,n=4$ and
$r_{0}=0.5$ for the shape function $b(r)=r\,e^{1-\frac{r}{r_{0}}}$.

###  4.4 Model-II: $f(Q)=Q+\alpha\,Q^{2}+\beta\,Q^{2}Log(\beta\,Q)$

As a second path to find wormhole solutions in $f(Q)$ gravity, we carefully
investigated a logarithmic form of $f(Q)$, i.e.,
$f(Q)=Q+\alpha\,Q^{2}+\beta\,Q^{2}Log(\beta\,Q)$. This model is analog to the
$R^{2}$ Starobinsky inflation model [16, 17], which may clarify both the
early-time and the late-time acceleration cycles. In $f(Q)$ gravity, this
model combines several nonlinear nonmetricity functions, generalizing the
$f(Q)=Q^{m}$ family introduced by Mandal et al. [18]. Here $\alpha$ and
$\beta$ are the free parameters which are going to be constrained through the
energy conditions.
Analogously to Model-I, the wormhole solutions are determined using the shape
functions $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$ and
$b(r)=r\,e^{1-\frac{r}{r_{0}}}$.

###  4.5 Shape function (SF-1): $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{m}$

By taking the present $f(Q)$ gravity family, together with the shape function
$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{n}$ into Eqs. (21)-(23), we were able
to derive analytic expressions for $\rho$, $P_{r}$, and $P_{t}$ which we
omitted for the sake of simplicity. The graphical behavior of the energy
conditions derived through these parameters are depicted in Fig. 3.

Figure 3: Behavior of energy conditions with $\beta=3,\,\,\,m=2$ and $r_{0}=1$
for the shape function $b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{m}$.

###  4.6 Shape function (SF-2): $b(r)=r\,e^{1-\frac{r}{r_{0}}}$

As a final example, let us work with the exponential form of $f(Q)$ together
with the exponential shape function $b(r)=r\,e^{1-\frac{r}{r_{0}}}$. Thus, by
taking them into Eqs. (21)-(23) we were able to derive analytic forms for
density and pressures $P_{r}$ and $P_{t}$ which were suppressed for
simplicity. The energy conditions obtained using these parameters are depicted
in Fig. 4.

Figure 4: Behavior of energy conditions with $\beta=-2$ and $r_{0}=1$ for the
shape function $b(r)=r\,e^{1-\frac{r}{r_{0}}}$.

##  5 Results and discussions

This letter carefully investigated the viability of wormholes solutions in
$f(Q)$ gravity, considering specific sets of shape functions, and an energy-
momentum tensor for an anisotropic fluid.
In order to find the energy conditions, we made use of the constraints imposed
by the Raychaudhuri Eq. using the embedding procedure introduced by Mandal et
al. [18]. The constraints derived together with the restrictions over the
shape functions allowed us to fix our free parameters, yielding to wormholes
solutions that satisfy at least one energy condition for positive energy
density. The present methodology using the embedding procedure also
complements the discussions recent carried by some of us in [19].
Firstly we derive the energy conditions for model I, considering both shape
functions (SF-1 and SF-2). For SF-1, we saw that NEC is violated for the
radial pressure for any value of $\beta$ with $m<1$. For $m>1$, NEC will be
satisfied. Also, DEC is satisfied in the entire spacetime and SEC is violated
for bigger values of $r$. For SF-2, we have found that all the energy
conditions are violated close to the throat for $r<1$. Such features are the
same presented by wormholes solutions in GR non-minimally coupled with scalar
fields. These behaviors indicate the possibility of stable traversable
wormhole solutions for $f(Q)$ gravity, with the exotic matter at their
throats.
For model-II, we consider logarithmic function of $Q$ and studied the energy
conditions for two different shape functions. For SF-1, we observed that NEC
is violated for both pressures close to the throat, for $\alpha\geq-3.5$ and
$m\geq 2$. Besides, we find that DEC is violated in the neighborhood of the
throat and SEC is violated for $\alpha\leq-3.5$. Again for SF-2, we found that
NEC is violated for the tangential pressure close to the throat if we work
with a small range of positive values of $\alpha$ and for negative values of
$\beta$. DEC is also violated in the vicinity of the WH throat whereas SEC is
satisfied for $\alpha\approx 1$ or for $r$ close to the throat. Wormholes that
do not obey DEC were found as solutions of the so-called $f(R,T)$ gravity
[30]. Although, the wormholes derived in [30] also satisfy NEC, indicating
that they do not have exotic matter at their throats. Consequently, our work
presents an interesting scenario with the possibility of stable exotic
traversable wormhole solutions which satisfy SEC in the presence of exotic
matter. The standard General Relativity approach can be recovered if we take
$\alpha=\beta=0$ in both $f(Q)$ families [14]. In such cases, all energy
conditions for traversable wormholes would be violated as pointed by Morris
and Thorne [10] .

The methodology here presented can be easily applied to new families of $f(Q)$
as well as for several other theories of gravity, opening a new path in
deriving wormhole solutions, and to constraint them through energy conditions.
Moreover, we could also work with a different equation of state using the
Chaplygin gas [30] or the Van der Waals fluid. It would also be interesting to
constraint our free parameters by combining the energy conditions together
with another noncommutative geometry, as recently approached by Hassan et al.
[31].

###### Acknowledgements.

Z.H. thanks the Department of Science and Technology (DST), Government of
India, New Delhi, for awarding a Senior Research Fellowship (File No.
DST/INSPIRE Fellowship/2019/IF190911). JRLS thanks CNPq (Grant nos.
420479/2018-0, and 309494/2021-4), and PRONEX/CNPq/FAPESQ-PB (Grant nos.
165/2018, and 0015/2019). PKS thanks National Board for Higher Mathematics
(NBHM) under Department of Atomic Energy (DAE), Govt. of India for financial
support to carry out the Research project No.: 02011/3/2022 NBHM(R.P.)/R&D
II/2152 Dt.14.02.2022. The authors also would like to thank the anonymous
referee for comments which improved the quality of this work.

## References

  * [1] E. Gourgoulhon et al., Astronomy and Astrophysics 627, A92 (2019).
  * [2] A. A. Chael et al., Astrophysical Journal 829, 11 (2016).
  * [3] K. Akiyama et al., Astro. J. Lett. 875, L1 (2019).
  * [4] R. Abuter et al., Astronomy and Astrophysics 636, L5 (2020).
  * [5] X. Barcons et al., Astronomische Nachrichten, 338, 153 (2017).
  * [6] D. N. Burrows et al., SSR 120, 165 (2005).
  * [7] M. Visser, Lorentzian wormholes: From Einstein to Hawking (1995).
  * [8] Pablo Bueno, Pablo A. Cano, Frederik Goelen, Thomas Hertog, and Bert Vercnocke, Phys. Rev. D, 97, 024040 (2018).
  * [9] Suvankar Paul, Rajibul Shaikh, Pritam Banerjee, and Tapobrata Sarkar, JCAP, 03, 055 (2020).
  * [10] M.S. Morris, and K.S. Thorne, Am. J. of Phys., 56, 395 (1988).
  * [11] Juan Maldacena and Alexey Milekhin, Phys. Rev. D, 103, 066007 (2021).
  * [12] S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167 (2011).
  * [13] S. Capozziello, F. S. N. Lobo and J. P. Mimoso, Phys. Rev. D 91, 124019 (2015).
  * [14] J. B. Jimenez, L. Heisenberg, and T. Koivisto, Phys. Rev. D 98, 044048 (2018).
  * [15] R. Lazkoz et al., Phys. Rev. D, 100, 104027 (2019).
  * [16] I. Fayyaz and M. Farasat Shamir, Eur. Phys. J. C, 80, 430 (2020).
  * [17] E. Elizalde et al. J. Cosmo. Astro. Phys. 2019, 017–017 (2019).
  * [18] Sanjay Mandal, P.K. Sahoo, and J.R.L. Santos, Phys. Rev. D, 102, 024057 (2020).
  * [19] G. Mustafa, et al., Phys. Lett. B, 821, 136612 (2021).
  * [20] Simran Arora, J.R.L. Santos, P.K. Sahoo, Phys. Dark Univ., 31, 100790 (2021).
  * [21] Yixin Xu et al., Eur. Phys. J. C, 79, 8, 708 (2019).
  * [22] J. B. Jimenez, L. Heisenberg, T. Koivisto, and S. Pekar, Phys. Rev. D 101, 103507 (2020).
  * [23] D. Zhao, Eur. Phys. J. C 82, 303 (2022).
  * [24] Patricio S. Letelier, Phys. Rev. D 22, 807 (1980).
  * [25] Debabrata Deb, Banibrata Mukhopadhyay, Fridolin Weber, ApJ 922 149 (2021).
  * [26] Francisco S. N. Lobo,  Phys. Rev. D, 71, 124022 (2005).
  * [27] Raja Solanki, Avik De, P. K. Sahoo, Phys. Dark Univ., 36, 100996 (2022).
  * [28] F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D  80, 104012 (2009)
  * [29] P. H. R. S. Moraes et al., Chin. Phys. Lett., 36, No. 12 120401 (2019).
  * [30] Emilio Elizalde, and Martiros Khurshudyan, Phys. Rev. D, 98, 123525 (2018).
  * [31] Zinnat Hassan, Ghulam Mustafa, and Pradyumn Kumar Sahoo, Symmetry, 13, 1260 (2021).