# Secure Semantic Communication over Wiretap Channel ††thanks: This research
was supported by the 5/6GIC, Institute for Communication Systems (ICS),
University of Surrey.

Denis Kozlov, Mahtab Mirmohseni and Rahim Tafazolli  5/6GIC, Institute for
Communication Systems (ICS), University of Surrey, Guildford, United Kingdom

###### Abstract

Semantic communication is an emerging feature for future networks like 6G,
which emphasizes the meaning of the message in contrast to the traditional
approach where the meaning of the message is irrelevant. Yet, the open nature
of a wireless channel poses security risks for semantic communications. In
this paper, we derive information-theoretic limits for the secure transmission
of semantic source over wiretap channel. Under separate equivocation and
distortion conditions for semantics and observed data, we present the general
outer and inner bounds on the rate-distortion-equivocation region. We also
reduce the general region to a case of Gaussian source and channel and provide
numerical evaluations.

###### Index Terms:

Semantic Communications, Wiretap Channel, Rate-Distortion-Equivocation Region.

##  I Introduction

Semantic communications represents a promising approach for the next
generation of wireless networks, particularly within the realm of 6G
technology. In this paradigm, the semantic content of messages is given
significant consideration, marking a departure from traditional communication
methods [1, 2, 3].

Despite its potential, wireless semantic communication still faces substantial
security challenges due to the inherent openness of communication channels,
leaving them susceptible to eavesdropping. In addition, semantics can carry
more sensitive information. Thus, the security requirement could be different
for the semantics compared with the observed source. This makes the problem
challenging and novel, requiring attention and exploration by researchers [3],
[4].

In this work, we derive the information-theoretic limits governing the secrecy
of semantic communication. To achieve this, we model a source as the intrinsic
(semantic) part and extrinsic (observed) part, building upon previous work
that introduced this source model [5, 6]. To illustrate this model with a
concrete example, a semantic part may be represented by a textual description
of an image, coupled with an observed part generated by a neural network in
response to the text prompt.

Our research centers on a wiretap channel scenario, wherein we consider a
passive eavesdropper as the adversary. Within the model of wiretap channel and
general semantic source (which is modeled as two correlated random variables
(r.v.s) with joint distribution), we analyze the trade-off between
equivocation and distortion for the semantic and observed components
separately, particularly in the context of joint source-channel coding (JSCC).
Specifically, we derive inner and outer bounds for rate, equivocations, and
distortions pairs.

Related works. Wyner’s work in 1975 laid the groundwork for secure
communication over wiretap channels [7], while subsequent advancements
generalized this model to broadcast channels with common and confidential
messages [8]. In [9], the wiretap model is extended to incorporate JSCC and
the one-time pad technique, bringing an important result that a separation
principle holds: first a rate-distortion achieving code can be applied, then
one-time pad can be applied for a given key-rate, and it is finished by using
a wiretap code. Further extensions to the JSCC model, including scenarios with
side information at the decoders, have been explored in [10, 11]. The lossy
compression aspect of semantic sources was covered in [5].

Challenges of secure semantic communications from a machine learning (ML)
perspective were covered in [12], [13]. Authors of [14] proposed a way for
encrypting semantic data in a deep learning JSCC scenario. Also, encryption
and obfuscation algorithm for semantic communication was presented in [15].

To the best of our knowledge, there is no work on secure JSCC of semantic
sources over wiretap channel with passive eavesdroppers under separate
equivocation and distortion conditions for semantics and observed source.

##  II Problem statement

Consider a model shown in Fig. 1, where a transmitter wishes to send the
semantic and observed data to a receiver (Bob), subject to some distortion
constraints while keeping them hidden from an eavesdropper (Eve) with some
equivocation constraints. Thus, the source consists of intrinsic (semantics)
state and extrinsic observation which are modeled as a sequence of i.i.d.
r.v.s $S^{k}$ and $U^{k}$, respectively. They are correlated through joint
distribution $p_{S,U}(s,u)$ defined on product alphabet
$\mathcal{S}\times\mathcal{U}$.

Figure 1: System Model

Main (Bob’s) and wiretap (Eve’s) channels are modeled as a discrete memoryless
channel (DMC) with input $X$ on $\mathcal{X}$ and outputs $Y$ on $\mathcal{Y}$
and $Z$ on $\mathcal{Z}$ given transition probability $p_{Y,Z|X}$. In this
work, we consider a degraded channel model: $p_{Y,Z|X}=p_{Y|X}p_{Z|X}$.

There are two cases for encoder input. In the first case, the encoder has
access to both semantics $s^{k}$ and observation $u^{k}$, while in the second
case, the encoder is only given $u^{k}$ and has no access to a semantic sample
$s^{k}$.

The case 1 encoder $f_{1}:S^{k}\times U^{k}\times K\rightarrow X^{n}$ maps the
semantic and observed sequence and a key to a channel input sequence $X^{n}$.
The case 2 encoder $f_{2}:U^{k}\times K\rightarrow X^{n}$, acts only on the
observed sequence and key.

The transmitter and receiver have access to a shared key which is modeled as
random variable $K$ with alphabet $\mathcal{K}$. This key is used to secure
part of the data with a one-time pad technique.

We define decoding function as $\hat{f}:Y^{n}\times
K\rightarrow(\hat{S}^{k},\hat{U}^{k})$ which maps Bob’s received signal
$Y^{n}$ and the shared key to estimated semantic and observed sources
$\hat{S}^{k}$ and $\hat{U}^{k}$ defined on alphabets $\hat{\mathcal{S}}$,
$\hat{\mathcal{U}}$.

From the decoder function definition we have:

$H(\hat{S}^{k},\hat{U}^{k}|Y^{n},K)=0.$ (1)

Additionally, we set distortion measure for semantics
$d_{S}:\mathcal{S}\times\mathcal{\hat{S}}\rightarrow\mathbb{R}^{+}$ and for
observation
$d_{U}:\mathcal{U}\times\mathcal{\hat{U}}\rightarrow\mathbb{R}^{+}$. We follow
the same naming for the block average distortion:
$d_{S}(s^{k},\hat{s}^{k})=\frac{1}{k}\sum^{k}_{i=1}d_{S}(s_{i},\hat{s}_{i})$
and
$d_{U}(u^{k},\hat{u}^{k})=\frac{1}{k}\sum^{k}_{i=1}d_{U}(u_{i},\hat{u}_{i})$.

The goal of this work is to characterize the following region:

$\mathcal{R}\doteq\\{(R,R_{k},D_{S},D_{U},\Delta_{S},\Delta_{U},\Delta_{SU})\text{
is achievable}\\},$

where a tuple $(R,R_{k},D_{S},D_{U},\Delta_{S},\Delta_{U},\Delta_{SU})$ is
achievable if there exist source-channel $(k,n)$-code $(f_{1},\hat{f})$ (or
$(f_{2},\hat{f})$ for the second encoder input type) s.t.:

$\displaystyle n/k$ $\displaystyle\leq R+\epsilon,$ (2)
$\displaystyle\frac{1}{k}\log|\mathcal{K}|$ $\displaystyle\leq
R_{k}+\epsilon,$ (3)
$\displaystyle\operatorname{\mathbb{E}}d_{S}(S^{k},\hat{S}^{k})$
$\displaystyle\leq D_{S}+\epsilon,$ (4)
$\displaystyle\operatorname{\mathbb{E}}d_{U}(U^{k},\hat{U}^{k})$
$\displaystyle\leq D_{U}+\epsilon,$ (5)
$\displaystyle\frac{1}{k}H(S^{k}|Z^{n})$
$\displaystyle\geq\Delta_{S}-\epsilon,$ (6)
$\displaystyle\frac{1}{k}H(U^{k}|Z^{n})$
$\displaystyle\geq\Delta_{U}-\epsilon,$ (7)
$\displaystyle\frac{1}{k}H(S^{k},U^{k}|Z^{n})$
$\displaystyle\geq\Delta_{SU}-\epsilon,$ (8)

are satisfied for any $\epsilon>0$. Condition (2) restricts channel expansion
ratio (inverse of rate), that is channel uses per source symbol. Equation (3)
restricts the rate of the key which is used to protect data with the one-time
pad technique. Average distortion for semantics and observation is restricted
by conditions (4) and (5). Equivocation for semantics and observation as well
as joint one, restricted by conditions (6), (7) and (8), respectively.

##  III Preliminaries

In this section, we provide some useful lemmas.

######  Lemma 1 ([16, Theorem 3.5])

The rate-distortion function for discrete memoryless source (DMS) $U$ has the
following form:

$R_{U}(D_{U})=\inf_{\begin{subarray}{c}p_{\hat{U}|U}\\\
\operatorname{\mathbb{E}}d_{U}(U,\hat{U})\leq
D_{U}\end{subarray}}I(U;\hat{U}).$

To meet distortion conditions optimally in terms of rate for both semantics
and observation, we can employ the multiple-description rate-distortion
function.

###### Definition 1

The multiple-description rate-distortion function:

$R(D_{S},D_{U})\doteq\inf\\{R:(R,D_{S},D_{U})\text{ is achievable}\\},$

where tuple $(R,D_{S},D_{U})$ is considered to be achievable if there exists a
code such that (2), (4) and (5) are satisfied for any $\epsilon>0$.

######  Lemma 2 ([17, Theorem 2])

$R(D_{S},D_{U})=\inf_{\begin{subarray}{c}p_{\hat{S},\hat{U}|S,U}\\\
\operatorname{\mathbb{E}}d_{U}(U,\hat{U})\leq D_{U}\\\
\operatorname{\mathbb{E}}d_{S}(S,\hat{S})\leq
D_{S}\end{subarray}}I(S,U;\hat{S},\hat{U}),$

######  Lemma 3 ([5, Theorem 1])

The case 2 of the encoder input rate-distortion function can be rewritten as
follows:

$R(D_{S},D_{U})=\inf_{\begin{subarray}{c}p_{\hat{S},\hat{U}|U}\\\
\operatorname{\mathbb{E}}d_{U}(U,\hat{U})\leq D_{U}\\\
\operatorname{\mathbb{E}}\hat{d}_{S}(U,\hat{S})\leq
D_{S}\end{subarray}}I(U;\hat{S},\hat{U}),$

where $\hat{d}_{S}(U,\hat{S})=\sum_{s\in S}p_{S|U}(s|U)d_{S}(s,\hat{S})$ is a
modified distortion metric.

##  IV Main Result

In this section, we present the general outer and inner bound for a system
model defined in Section II.

###### Theorem 1

(Converse). For both cases of encoder input, any achievable tuple
$(R,R_{k},D_{S},D_{U},\Delta_{S},\Delta_{U},\Delta_{SU})$ must satisfy:

$\displaystyle R_{U}(D_{U})\leq RI(X;Y),$ (9) $\displaystyle R_{S}(D_{S})\leq
RI(X;Y),$ (10) $\displaystyle R(D_{S},D_{U})\leq RI(X;Y),$ (11)
$\displaystyle\Delta_{U}\leq R_{k}+R[I(X;Y)-I(X;Z)]$ $\displaystyle\qquad-
R_{U}(D_{U})+H(U),$ (12) $\displaystyle\Delta_{S}\leq R_{k}+R[I(X;Y)-I(X;Z)]$
$\displaystyle\qquad-R_{S}(D_{S})+H(S),$ (13) $\displaystyle\Delta_{SU}\leq
R_{k}+R[I(X;Y)-I(X;Z)]$ $\displaystyle\qquad-R(D_{S},D_{U})+H(S,U).$ (14)

The proof of Theorem 1 is provided in the Appendix.

One can see that the equivocation in the converse region consists of three
basic terms. The first one, $R_{k}$, shows how much equivocation we achieve
using the one-time pad technique with a key rate $R_{k}$. The second one is
the secrecy capacity ($R[I(X;Y)-I(X;Z)]$). And the last one (i.e.
$H(U)-R_{U}(D_{U})$), describes the part of equivocation due to loss in source
encoding.

###### Theorem 2

(Inner bound). When the transmitter has access to both semantics and
observation (case 1 of encoder input), a tuple
$(R,D_{S},D_{U},\Delta_{S},\Delta_{U},\Delta_{SU})$ is achievable if there
exist auxiliary r.v.s $A_{c},A_{p},B_{c},B_{p},Q_{c},Q_{p},W_{c},X$ with joint
distribution $p(a_{c},a_{p},b_{c},b_{p},q_{c},q_{p},w_{c},x)$ and functions
$\tilde{S}:A^{k}_{c}\times A^{k}_{p}\rightarrow\hat{S}^{k}$ and
$\tilde{U}:A^{k}_{c}\times A^{k}_{p}\times B^{k}_{c}\times
B^{k}_{p}\rightarrow\hat{U}^{k}$ such that the following inequalities hold:

$\begin{cases}I(S;A_{c})<RI(Q_{c};Y),\\\
I(S;A_{c},A_{p})<RI(Y;Q_{c},Q_{p}),\\\
I(U;B_{c}|S,A_{c})<RI(W_{c};Y|Q_{c}),\\\
I(U;B_{c}|S,A_{c})+I(U;B_{p}|S,A_{c},A_{p},B_{c})<\\\
\qquad<R[I(W_{c};Y|Q_{c})+I(X;Y|Q_{c},Q_{p},W_{c})]\\\
D_{S}\geq\operatorname{\mathbb{E}}d_{S}(S,\tilde{S}(A_{c},A_{p})),\\\
D_{U}\geq\operatorname{\mathbb{E}}d_{U}(U,\tilde{U}(A_{c},A_{p},B_{c},B_{p})),\\\
\Delta_{S}\leq H(S|Ac,Ap)+R(H(Z|X)-H(Z|Qc))\\\ \qquad+RI(Qp;Y|Qc),\\\
\Delta_{U}\leq H(U)-I(A_{c},A_{p};S)-I(B_{c};U|A_{c})\\\
\qquad-I(B_{p};U|S,A_{c},A_{p},B_{c})+RH(Z|X)\\\ \qquad-
RH(Z|Q_{c},W_{c})+RI(Q_{p};Y|Q_{c})\\\ \qquad+RI(X;Y|Q_{c},Q_{p},W_{c}),\\\
\Delta_{SU}\leq H(S,U)-I(S;A_{c},A_{p})-I(U;B_{p}|S,A_{c},A_{p},B_{c})\\\
\qquad-I(U;B_{c}|S,A_{c})+R(H(Z|X)-H(Z|Qc,Wc))\\\
\qquad+RI(Q_{p};Y|Q_{c})+RI(X;Y|Q_{c},Q_{p},W_{c}).\end{cases}$

Achievability proof outline. Consider a codebook with source, channel, and
wiretap codes. Wiretap code is embedded in the channel encoder and introduces
additional random noise to cover private parts of the data from the
eavesdropper. We introduce source encoder auxiliary r.v.s $A_{c}$, $A_{p}$,
$B_{c}$, $B_{p}$, and channel encoder auxiliary r.v.s $Q_{c}$, $Q_{p}$,
$W_{c}$ for codebook generation. The r.v.s $A_{c}$ and $A_{p}$ reflect the
distribution of i.i.d. codewords (denoted by $a^{k}_{c}$ and $a^{k}_{p}$) for
the common and private parts of semantics, while $B_{c}$ and $B_{p}$ are used
for the common and private parts of the observation, for codewords denoted as
$b^{k}_{c}$ and $b^{k}_{p}$, correspondingly.

Then we use a technique similar to superposition coding: for each sequence
$a^{k}_{c}$ we generate $a^{k}_{p}$ and $b^{k}_{c}$, then for each
$a^{k}_{c}$, $a^{k}_{p}$, $b^{k}_{c}$ we generate $b^{k}_{p}$. All of these
sequences are selected from typical sets.

The channel encoder codebook has a layered structure related to the source
encoder. That is, we generate sequence $q^{n}_{c}$ from $Q_{c}$ distribution,
for each $q^{n}_{c}$ we generate $q^{n}_{p}$ and $w^{n}_{c}$ using $Q_{p}$ and
$W_{c}$, given $q^{n}_{c}$, $q^{n}_{p}$, $w^{n}_{c}$ we generate $x^{n}$
according to $X$. Sequences $q^{n}_{p}$ and $x^{n}$ additionally covered with
noise.

To encode data, we choose source encoder sequences that are jointly typical
with encoder input. Then using the mapping, we obtain the channel code
sequences and transmit $x^{n}$. The decoding is based on the joint typically
with channel output $y^{n}$.

For this codebook and encoding/decoding procedure, we show that the
probability of encoding/decoding errors goes to zero under some rate
conditions with $k\rightarrow\infty$.

Finally, we bound the average distortion and equivocations.

For the complete proof of achievability see Appendix.

##  V Gaussian Case

In this section, we consider the Gaussian source and channel with quadratic
distortion measure.

###  V-A System model.

Let source be distributed according to normal distribution
$(S,U)\sim\mathcal{N}(0,K)$ with covariance matrix,

$K=\begin{pmatrix}P_{S}&\sigma_{SU}\\\ \sigma_{SU}&P_{U}\end{pmatrix}.$

We set distortion measure $d_{s}(x,y)=d_{u}(x,y)=(x-y)^{2}$, and we model the
channel as,

$\displaystyle\operatorname{\mathbb{E}}(X^{2})\leq P,$ $\displaystyle
Y=X+N_{1},\quad N_{1},\sim\operatorname{\mathcal{N}}(0,P_{N_{1}}),$
$\displaystyle Z=Y+N_{2},\quad
N_{2},\sim\operatorname{\mathcal{N}}(0,P_{N_{2}}),$

where $P$ is the power constraint for channel input, $P_{N_{1}}$ is the noise
power for the main channel, and $P_{N_{2}}$ is the noise power for the
eavesdropper channel, we define $P_{N}=P_{N_{1}}+P_{N_{2}}$.

###  V-B Rate-equivocation-distortion region

For the Gaussian system model, we derive the following outer bound from
Theorem 1.

###### Proposition 1

(Converse). In the case when the encoder has access only to observation
$u^{k}$ (case 2 of encoder input), for the Gaussian source and channel, to
fulfill conditions (2)-(8) any code must satisfy:

$\begin{cases}D_{S}\geq\eta,\\\ D_{U}\geq
P_{U}\left(\frac{P_{N_{1}}}{P+P_{N_{1}}}\right)^{R},\\\ D_{S}\geq
P_{S}\left(\frac{P_{N_{1}}}{P+P_{N_{1}}}\right)^{R},\\\
\max\left[\frac{P_{U}}{D_{U}},\frac{\sigma_{SU}^{2}}{P_{U}(D_{S}-\eta)}\right]\leq\left(1+\frac{P}{P_{N_{1}}}\right)^{R},\\\
\Delta_{U}\leq R_{k}+RC_{s}+\frac{1}{2}\log(2\pi eD_{U}),\\\ \Delta_{S}\leq
R_{k}+RC_{s}+\frac{1}{2}\log(2\pi eD_{S}),\\\ \Delta_{SU}\leq
R_{k}+RC_{s}+\frac{1}{2}\log\left[(2\pi e)^{2}|K|\right]-\\\
\qquad-\frac{1}{2}\max\left[\log\frac{P_{U}}{D_{U}},\log\frac{\sigma_{SU}^{2}}{P_{U}(D_{S}-\eta)}\right],\end{cases}$

where $\eta=P_{S}-\frac{\sigma_{SU}^{2}}{P_{U}}$, and
$C_{s}=\frac{1}{2}\log\frac{P_{N}(P+P_{N_{1}})}{P_{N_{1}}(P+P_{N})}$ is the
secrecy channel capacity.

###### Proposition 2

(Inner bound). When the encoder has access to both $u^{k}$ and $s^{k}$ (case 1
of encoder input), for the Gaussian source and channel, the tuple
$(R,D_{S},D_{U},\Delta_{S},\Delta_{U},\Delta_{SU})$ is achievable if:

$\begin{cases}\left(1+\frac{\alpha_{1}^{2}P_{s}}{\beta_{1}^{2}P_{\tilde{A}_{p}}}\right)\leq\left(1+\frac{P_{Q_{c}}+P_{\tilde{Q}_{p}}}{P_{\tilde{W}_{c}}+P_{\tilde{X}+P_{N_{1}}}}\right)^{R},\\\
\left(1+\frac{\alpha_{2}^{2}P_{U}}{\beta_{2}^{2}P_{\tilde{B}_{p}}}\right)\leq\left(1+\frac{P_{\tilde{W}_{c}}}{P_{\tilde{X}}+P_{\tilde{Q}_{p}}+P_{N_{1}}}\right)^{R}\left(1+\frac{P_{\tilde{X}}}{P_{N_{1}}}\right)^{R},\\\
D_{S}\geq
P_{S}-\frac{\alpha_{1}^{2}P_{S}}{(\alpha_{1}^{2}P_{S}+\beta_{1}^{2}P_{\tilde{A}_{p}})^{2}},\\\
D_{U}\geq
P_{U}-\frac{(\alpha_{2}^{2}P_{U}+\gamma\sigma_{SU})^{2}}{P_{U}(\alpha_{2}^{2}P_{U}+\gamma^{2}P_{S}+2\alpha_{2}\gamma\sigma_{SU})^{2}},\\\
\Delta_{S}\leq\frac{R}{2}\log\left(\frac{P+P_{N_{1}}}{P_{\tilde{W}_{c}}+P_{\tilde{X}}+P_{N_{1}}}\frac{P_{N}}{P-P_{Q_{c}}+P_{N}}\right)\\\
\qquad+\frac{1}{2}\log\left(2\pi
e\frac{\beta_{1}^{2}P_{S}P_{\tilde{A}_{p}}}{\alpha_{1}^{2}P_{S}+\beta_{1}^{2}P_{\tilde{A}_{p}}}\right),\\\
\Delta_{U}\leq\frac{1}{2}\log\left(\frac{\beta_{2}^{2}P_{U}}{\beta_{1}^{2}P_{S}}\frac{P_{\tilde{B}_{p}}}{P_{\tilde{A}_{p}}}\frac{\alpha_{1}^{2}P_{S}+\beta_{1}^{2}P_{\tilde{A}_{p}}}{\alpha_{2}^{2}P_{U}+\beta_{2}^{2}P_{\tilde{B}_{p}}}\right)\\\
\qquad+\frac{R}{2}\log\left(\frac{P_{N_{1}}+P_{N_{2}}}{P_{\tilde{Q}_{p}+P_{\tilde{X}}+P_{N_{1}}+P_{N2}}}\frac{P+P_{N_{1}}}{P_{\tilde{W}_{c}}+P_{\tilde{X}}+P_{N_{1}}}\frac{P_{\tilde{X}}+P_{N_{1}}}{P_{N_{1}}}\right),\\\
\Delta_{SU}\leq\frac{1}{2}\log\left((2\pi
e)^{2}|K|\frac{\beta_{1}^{2}P_{\tilde{A}_{p}}}{\alpha_{1}^{2}P_{S}+\beta_{1}^{2}P_{\tilde{A}_{p}}}\frac{\beta_{2}^{2}P_{\tilde{B}_{p}}}{\alpha_{2}^{2}P_{U}+\beta_{2}^{2}P_{\tilde{B}_{p}}}\right)\\\
\quad+\frac{R}{2}\log\left(\frac{P_{N}}{P_{\tilde{Q}_{p}+P_{\tilde{X}}+P_{N}}}\left(1+\frac{P_{\tilde{Q}_{p}}}{P_{\tilde{W}_{c}}+P_{\tilde{X}}+P_{N_{1}}}\right)\left(1+\frac{P_{\tilde{X}}}{P_{N_{1}}}\right)\right).\end{cases}$

The inner bound is derived from Theorem 2 by choosing the following auxiliary
r.v.s. The source encoder variables: $A_{c}=\emptyset,\quad B_{c}=\emptyset$,
$A_{p}=\alpha_{1}S+\beta_{1}\tilde{A}_{p}$, where
$\tilde{A}_{p}\sim\operatorname{\mathcal{N}}(0,P_{\tilde{A}_{p}})$ and
$B_{p}=\alpha_{2}U+\beta_{2}\tilde{B}_{p}+\gamma S$, given
$\tilde{B}_{p}\sim\operatorname{\mathcal{N}}(0,P_{\tilde{B}_{p}})$.

The channel encoder variables:

$\displaystyle Q_{c}\sim\operatorname{\mathcal{N}}(0,P_{Qc}),$ $\displaystyle
Q_{p}=Q_{c}+\tilde{Q_{p}},\quad\tilde{Q}_{p}\sim\operatorname{\mathcal{N}}(0,P_{\tilde{Q}_{p}}),$
$\displaystyle
W_{c}=Q_{c}+\tilde{W_{c}},\quad\tilde{W}_{c}\sim\operatorname{\mathcal{N}}(0,P_{\tilde{W}_{c}}),$
$\displaystyle
X=Q_{c}+\tilde{W_{c}}+\tilde{Q_{p}}+\tilde{X},\quad\tilde{X}\sim\operatorname{\mathcal{N}}(0,P_{\tilde{X}}),$

where $P=P_{\tilde{X}}+P_{Qc}+P_{\tilde{W}_{c}}+P_{\tilde{Q}_{p}}$.

Function $\tilde{S}$ defined as minimum mean square error (MMSE) estimator of
$S$ from $A_{c}$ and $A_{p}$, and $\tilde{U}$ is the MMSE estimator of $U$
from $B_{c},B_{p}$.

###  V-C Numerical evaluation

Fig. 2 shows outer bound obtained numerically from Proposition 1 for the case
of $\Delta_{S}=H(S)$ and $\Delta_{U}=H(U)$. Fig. 3 shows the inner bound
(Proposition 2) for the same parameters but for case 1 encoder input. The
other parameters are: $P_{S}=0.3$, $P_{U}=1$, $\sigma_{SU}=0.8$, $P=1$,
$P_{N_{1}}=0.10$, $P_{N_{2}}=0.40$.

Figure 2: Outer bound for Gaussian case and case 2 encoder input Figure 3:
Inner bound (every point above plane is achievable) for Gaussian case and case
1 encoder input

##  VI Conclusion

This paper presents an information-theoretic framework for the secure
transmission of semantic sources over wiretap channel. We characterize the
rate-distortion-equivocation region to see the theoretically possible trade-
off between distortion and secrecy requirements for semantic communications.
We present bounds on a general inner and outer rate-distortion-equivocation
region and reduce them to the Gaussian source and channel model to bring
closer a general model to a practical wireless communication setup.

[Proof of Theorem 1] The following set of inequalities will be used in our
proof:

$\displaystyle I(X^{n};Y^{n})\leq\sum_{i=1}^{n}I(X_{i};Y_{i})\leq nI(X;Y),$
(15)
$\displaystyle\frac{1}{k}I(U^{k};\hat{U}^{k})\geq\frac{1}{k}\sum_{i=1}^{k}I(U_{i};\hat{U}_{i})\geq
R_{U}(D_{U}+\epsilon),$ (16)
$\displaystyle\frac{1}{k}I(S^{k};\hat{S}^{k})\geq\frac{1}{k}\sum_{i=1}^{k}I(S_{i};\hat{S}_{i})\geq
R_{S}(D_{S}+\epsilon),$ (17)
$\displaystyle\sum_{i=1}^{k}I(S_{i};\hat{S}_{i})\leq\sum_{i=1}^{k}I(U_{i};\hat{U}_{i})\leq\sum_{i=1}^{n}I(X_{i};Y_{i}),$
(18)

where $\epsilon>0$.

First, we proof inequality in (9) as:

$\displaystyle R_{U}(D_{U}+\epsilon)$
$\displaystyle\leq^{(\ref{eq:bound_for_src_rate_distortion})}\frac{1}{k}\sum_{i=1}^{k}I(U_{i};\hat{U}_{i})\leq^{(\ref{eq:dpi})}\frac{1}{k}\sum_{i=1}^{n}I(X_{i};Y_{i})$
$\displaystyle\leq^{(\ref{eq:upper_bound_I_XY})}\frac{n}{k}I(X;Y)\leq^{(\ref{eq:condition_for_rate})}(R+\epsilon)I(X;Y).$
(19)

To obtain (10), we follow the same steps as in the proof of (9) with (17)
instead of (16).

The proof of (12) is as follows:

$\displaystyle
k(R_{k}+\epsilon)\geq^{(\ref{eq:condition_for_key_rate})}\log|\mathcal{K}|\geq
H(K)\geq H(K|Y^{n})$ $\displaystyle\geq H(K|Y^{n})-H(K|Y^{n}\hat{U}^{k})$
$\displaystyle=H(K,Y^{n})-H(Y^{n})$
$\displaystyle\quad-H(K,Y^{n},\hat{U}^{k})+H(Y^{n},\hat{U}^{k})$
$\displaystyle=H(\hat{U}^{k}|Y^{n})-H(\hat{U}^{k}|Y^{n}K)=^{(\ref{eq:dec_out_is_a_func_of_in_and_key})}H(\hat{U}^{k}|Y^{n})$
$\displaystyle\geq^{(\ref{eq:cond_for_src_equivocation})}H(\hat{U}^{k}|Y^{n})-[H(U^{k}|Z^{n})-k(\Delta_{U}-\epsilon)]$
$\displaystyle=H(\hat{U}^{k},Y^{n})-H(Y^{n})-H(U^{k},Z^{n})$
$\displaystyle\quad+H(Z^{n})+k(\Delta_{U}-\epsilon)$
$\displaystyle=H(Y^{n}|\hat{U}^{k})+H(\hat{U}^{k})-H(Y^{n})$
$\displaystyle\quad-H(Z^{n}|U^{k})-H(U^{k})+H(Z^{n})$
$\displaystyle\quad+k(\Delta_{U}-\epsilon)$
$\displaystyle=I(U^{k};Z^{n})-I(\hat{U}^{k};Y^{n})+H(\hat{U}^{k}|U^{k})$
$\displaystyle\quad-H(U^{k})+I(U^{k};\hat{U}^{k})+k(\Delta_{U}-\epsilon).$
(20)

The first three terms in (20) can be rewritten as follows:

$\displaystyle I(U^{k};Z^{n})-I(\hat{U}^{k};Y^{n})+H(\hat{U}^{k}|U^{k})=$
$\displaystyle=I(U^{k};Z^{n})-H(Y^{n})+H(Y^{n}|\hat{U}^{k})+H(\hat{U}^{k}|U^{k})$
$\displaystyle\geq I(U^{k};Z^{n})-H(Y^{n})+H(Y^{n}|\hat{U}^{k}U^{k})$
$\displaystyle\quad+H(\hat{U}^{k}|U^{k})$
$\displaystyle=I(U^{k};Z^{n})-H(Y^{n})+H(Y^{n},\hat{U}^{k}|U^{k})$
$\displaystyle\geq I(U^{k};Z^{n})-I(U^{k};Y^{n}).$

With the help of Lemma 1 (see Appendix), we have:

$I(U^{k};Z^{n})-I(U^{k};Y^{n})\geq n[I(X;Z)-I(X;Y)].$ (21)

Substituting (21) to (20) we obtain:

$\displaystyle
k(R_{k}+\epsilon)\geq^{(\ref{eq:converse_intermediate_result},\ref{eq:bound_for_src_rate_distortion})}n[I(X;Z)-I(X;Y)]-kH(U)+$
$\displaystyle\qquad+kR_{U}(D_{U}+\epsilon)+k(\Delta_{U}-\epsilon)$ (22)
$\displaystyle R_{k}+2\epsilon\geq\frac{n}{k}[I(X;Z)-I(X;Y)]-H(U)+$
$\displaystyle\qquad+R_{U}(D_{U}+\epsilon)+\Delta_{U}$
$\displaystyle\geq^{(\ref{eq:condition_for_rate})}(R+\epsilon)[I(X;Z)-I(X;Y)]-H(U)+$
$\displaystyle\qquad+R_{U}(D_{U}+\epsilon)+\Delta_{U}.$ (23)

To prove (13), we skip some steps due to its similarity with steps in proof of
(12) (we use $S^{k}$ instead of $U^{k}$).

$\displaystyle k(R_{k}+\epsilon)\geq...\geq
H(\hat{S}^{k}|Y^{n})-H(\hat{S}^{k}|Y^{n}K)$
$\displaystyle\geq^{(\ref{eq:cond_for_sem_equivocation})}H(\hat{S}^{k}|Y^{n})-[H(S^{k}|Z^{n})-k(\Delta_{S}-\epsilon)]$
$\displaystyle=I(S^{k};Z^{n})-I(\hat{S}^{k};Y^{n})-H(S^{k})+$
$\displaystyle\qquad+I(S^{k};\hat{S}^{k})+H(\hat{S}^{k}|S^{k})+k(\Delta_{S}-\epsilon)$
$\displaystyle\geq...\geq n[I(X;Z)-I(X;Y)]-kH(S)+$
$\displaystyle\qquad+I(S^{k};\hat{S}^{k})+k(\Delta_{S}-\epsilon)$ (24)
$\displaystyle\geq^{(\ref{eq:bound_for_sem_rate_distortion})}n[I(X;Z)-I(X;Y)]-kH(S)+$
$\displaystyle\qquad+kR_{S}(D_{S}+\epsilon)+k(\Delta_{S}-\epsilon).$ (25)

Now we proceed with the proof of (11). Our proof will rely on the following
inequality:

$\displaystyle\frac{1}{k}I(S^{k},U^{k};\hat{S}^{k},\hat{U}^{k})\geq\frac{1}{k}\sum_{i=1}^{k}I(S_{i},U_{i};\hat{S}_{i},\hat{U}_{i})\geq
R(D_{S},D_{U}).$ (26)

Thus, (11):

$\displaystyle
R(D_{S}+\epsilon,D_{U}+\epsilon)\leq^{(\ref{eq:inequality_indirect_rd})}\frac{1}{k}\sum_{i=1}^{k}I(S_{i},U_{i};\hat{S}_{i},\hat{U}_{i})$
$\displaystyle\leq^{(a)}\frac{1}{k}\sum_{i=1}^{n}I(X_{i};Y_{i})\leq^{(\ref{eq:upper_bound_I_XY})}\frac{n}{k}I(X;Y)$
$\displaystyle\leq^{(\ref{eq:condition_for_rate})}(R+\epsilon)I(X;Y),$ (27)

where (a) is due to the data processing inequality.

Finally, the proof of (14) is,

$\displaystyle k(R_{k}+\epsilon)\geq...\geq
H(K|Y^{n})-H(K|Y^{n},\hat{S}^{k},\hat{U}^{k})$
$\displaystyle=H(\hat{S}^{k},\hat{U}^{k}|Y^{n})-H(\hat{S}^{k},\hat{U}^{k}|Y^{n}K)=^{(\ref{eq:dec_out_is_a_func_of_in_and_key})}H(\hat{S}^{k},\hat{U}^{k}|Y^{n})$
$\displaystyle\geq^{(\ref{eq:cond_for_joint_equivocation})}H(\hat{S}^{k},\hat{U}^{k}|Y^{n})-[H(S^{k},U^{k}|Z^{n})-k(\Delta_{SU}-\epsilon)]$
$\displaystyle=H(\hat{S}^{k},\hat{U}^{k},Y^{n})-H(Y^{n})-H(S^{k},U^{k},Z^{n})$
$\displaystyle\quad+H(Z^{n})+k(\Delta_{SU}-\epsilon)$
$\displaystyle=H(Y^{n}|\hat{S}^{k},\hat{U}^{k})+H(\hat{S}^{k},\hat{U}^{k})-H(Y^{n})$
$\displaystyle\quad-H(Z^{n}|S^{k},U^{k})-H(S^{k},U^{k})+H(Z^{n})$
$\displaystyle\quad+k(\Delta_{SU}-\epsilon)$
$\displaystyle=I(S^{k},U^{k};Z^{n})-I(\hat{S}^{k},\hat{U}^{k};Y^{n})+H(\hat{S}^{k},\hat{U}^{k}|S^{k},U^{k})$
$\displaystyle\quad-H(S^{k},U^{k})+I(S^{k},U^{k};\hat{S}^{k},\hat{U}^{k})+k(\Delta_{SU}-\epsilon).$
(28)

The first three terms in (28) are,

$\displaystyle
I(S^{k},U^{k};Z^{n})-I(\hat{S}^{k},\hat{U}^{k};Y^{n})+H(\hat{S}^{k},\hat{U}^{k}|S^{k},U^{k})$
$\displaystyle\geq n[I(X;Z)-I(X;Y)].$

Substituting in (28), we have:

$\displaystyle
k(R_{k}+\epsilon)\geq^{(\ref{eq:inequality_indirect_rd})}n[I(X;Z)-I(X;Y)]-kH(S,U)$
$\displaystyle\qquad+kR(D_{S},D_{U})+k(\Delta_{SU}-\epsilon),$ $\displaystyle
R_{k}+2\epsilon\geq(R+\epsilon)[I(X;Z)-I(X;Y)]-H(S,U)$
$\displaystyle\qquad+R(D_{S},D_{U})+\Delta_{SU}.$

Letting $\epsilon\rightarrow 0$ completes the converse proof.

[Proof of Theorem 2]

Now we consider achievability proof for case 1 of encoder input (encoder has
access to both $u^{k}$ and $s^{k}$).

Source codebook:

We introduce 4 r.v.s $A_{c}$, $A_{p}$ and $B_{c}$, $B_{p}$ defined on
alphabets $\mathcal{A}_{c}$, $\mathcal{A}_{p}$ and $\mathcal{B}_{c}$,
$\mathcal{B}_{p}$. Random variables $A_{c}$ and $B_{c}$ correspond to a
codebook distribution of a common message part for semantics and source. While
$A_{p}$ and $B_{p}$ represent private a part of semantics and source. Let
$R_{ac}$, $R_{ap}$, $R_{bc}$, $R_{bp}$ be positive rates.

To construct a source codebook, we start by randomly and independently picking
$2^{kR_{ac}}$ typical $\mathcal{T}^{k}_{\delta}(A_{c})$ sequences from $A_{c}$
distribution. We call such sequence $a_{c}^{k}(s_{ac})$, where
$s_{ac}\in[1,2,...,2^{kR_{ac}}]$

For each sequence $a_{c}^{k}(s_{ac})$ we pick $2^{kR_{ap}}$ typical
$\mathcal{T}^{k}_{\delta}(A_{p}|a_{c}^{k}(s_{ac}))$ sequences from $A_{p}$
distribution and name it $a^{k}_{p}(s_{ac},s_{ap})$,
$s_{ap}\in[1,2,...,2^{kR_{ap}}]$.

For each $a_{c}^{k}(s_{ac})$ we pick $2^{kR_{bc}}$ sequences
$b^{k}_{c}(s_{ac},s_{bc})\in\mathcal{T}^{k}_{\delta}(B_{c}|a_{c}^{k}(s_{ac}))$,
$s_{bc}\in[1,2,...,2^{kR_{bc}}]$.

We finish codebook by picking for each previous sequences, $2^{kR_{bp}}$
sequences
$b^{k}_{p}(s_{ac},s_{ap},s_{bc},s_{bp})\in\mathcal{T}^{k}_{\delta}(B_{p}|a_{c}^{k}(s_{ac}),a^{k}_{p}(s_{ac},s_{ap}),b^{k}_{c}(s_{ac},s_{bc}))$,
$s_{bp}\in[1,2,...,2^{kR_{bp}}]$.

This codebook is revealed to Bob and Eve.

Channel codebook:

Let $Q_{c}$, $Q_{p}$ and $W_{c}$ be r.v. for channel codebook generation
defined on $\mathcal{Q}_{c}$, $\mathcal{Q}_{p}$, $\mathcal{W}_{c}$. Let
$R_{qc},R_{qp},R_{wc},R_{wp},R_{1},R_{2}$ be positive rates s.t:

$\displaystyle R_{1}<(R+\epsilon)I(Q_{p};Z|Q_{c}),$ $\displaystyle
R_{2}<(R+\epsilon)I(X;Z|W_{c}).$

From $\mathcal{T}^{n}_{\delta}(Q_{c})$ we pick $2^{kR_{qc}}$ sequences named
$q^{n}_{c}(r_{qc})$, where $r_{qc}\in[1,2,...,2^{kR_{qc}}]$ is a index of a
sequence.

For each $q^{n}_{c}(r_{qc})$ we pick $2^{k(R_{qp}+R_{1})}$ sequences
$q^{n}_{p}(r_{qc},r_{qp},r_{1})\in\mathcal{T}^{n}_{\delta}(Q_{p}|q^{n}_{c}(r_{qc}))$.

Also for each $q^{n}_{c}(r_{qc})$ we randomly pick $2^{kR_{wc}}$ sequences
$w^{n}_{c}(r_{qc},r_{wc})\in\mathcal{T}^{n}_{\delta}(W_{c}|q^{n}_{c}(r_{qc}))$.

And, finally, for each
$q^{n}_{c}(r_{qc}),q^{n}_{p}(r_{qc},r_{qp},r_{1}),w^{n}_{c}(r_{qc},r_{wc})$ we
pick $2^{k(R_{wp}+R_{2})}$ sequences
$x^{n}(r_{qc},r_{qp},r_{1},r_{wc},r_{wp},r_{2})\in\mathcal{T}^{n}_{\delta}(X|q^{n}_{c}(r_{qc}),q^{n}_{p}(r_{qc},r_{qp},r_{1}),w^{n}_{c}(r_{qc},r_{wc}))$.

This codebook is revealed to Bob and Eve. Further to shorten notations we will
skip indices in sequence names.

Source encoding:

We have $(s^{k},u^{k})$ as encoder input. We search for the first jointly
typical sequence $a^{k}_{c}$ s.t.
$(a^{k}_{c},s^{k})\in\mathcal{T}^{k}_{\delta}(A_{c},S)$.

Then, given $a^{k}_{c}$, we find first $a^{k}_{p}$ sequence s.t.
$(a^{k}_{p},s^{k})\in\mathcal{T}^{k}_{\delta}(A_{p},S|a^{k}_{c})$.

Also, given codeword $a^{k}_{c}$, we proceed by finding first codeword
$b^{k}_{c}$ s.t.
$(b^{k}_{c},u^{k})\in\mathcal{T}^{k}_{\delta}(B_{c},U|S,a^{k}_{c})$.

And, given all previous codewords $a^{k}_{c}$, $a^{k}_{p}$ and $b^{k}_{c}$, we
finish source encoding by finding first
$(b^{k}_{p},u^{k})\in\mathcal{T}^{k}_{\delta}(B_{p},U|S,a^{k}_{c},a^{k}_{p},b^{k}_{c})$.

Channel encoding:

We choose arbitrary one-to-one mapping
$(r_{qc},r_{qp},r_{wc},r_{wp})=g(s_{ac},s_{ap},s_{bc},s_{bp})$ which is used
to map source indices to channel indices. We assume that there exist mappings
$(r_{qc},r_{qp})=g_{1}(s_{ac},s_{ap})$ and
$(r_{wc},r_{wp})=g_{2}(s_{bc},s_{bp})$.

Now, given channel indexes $(r_{qc},r_{qp},r_{wc},r_{wp})$ as a result of
mapping $g$, we sequentially select $q^{n}_{c},q^{n}_{p},w^{n}_{c},x^{n}$,
from channel codebook. Alice transmit sequence $x^{n}\doteq
x^{n}(r_{qc},r_{qp},r_{1},r_{wc},r_{wp},r_{2})$, where $r_{1}$ and $r_{2}$
selected at random with uniform distribution.

Decoding:

Bob receives $y^{n}$. He sequentially searches in his codebook for a codewords
s.t.:

  1. 1.

$(q^{n}_{c},y^{n})\in\mathcal{T}^{n}_{\delta}(Q_{c},Y)$,

  2. 2.

$(q^{n}_{p},y^{n})\in\mathcal{T}^{n}_{\delta}(Q_{p},Y|q^{n}_{c})$,

  3. 3.

$(w^{n}_{c},y^{n})\in\mathcal{T}^{n}_{\delta}(W_{c},Y|q^{n}_{c})$,

  4. 4.

$(x^{n},y^{n})\in\mathcal{T}^{n}_{\delta}(X,Y|q^{n}_{c},q^{n}_{p},w^{n}_{c})$.

Then, given channel indexes $(r_{qc},r_{qp},r_{wc},r_{wp})$, Bob using inverse
mapping $g^{-1}$ gets source decoder indexes $(s_{ac},s_{ap},s_{bc},s_{bp})$
and decodes:

$\displaystyle\hat{s}^{k}$ $\displaystyle=\tilde{S}(a^{k}_{c},a^{k}_{p}),$
$\displaystyle\hat{u}^{k}$
$\displaystyle=\tilde{U}(a^{k}_{c},a^{k}_{p},b^{k}_{c},b^{k}_{p}),$

where $\tilde{S}:A^{k}_{c}\times A^{k}_{p}\rightarrow\hat{S}^{k}$ and
$\tilde{U}:A^{k}_{c}\times A^{k}_{p}\times B^{k}_{c}\times
B^{k}_{p}\rightarrow\hat{U}^{k}$ are functions.

Errors at encoding and decoding:

We consider the following events which correspond to errors at the encoding or
decoding stages.

Encoder errors:

$\displaystyle\mathcal{E}_{1}\doteq\\{\not\exists
a^{k}_{c}:(a^{k}_{c},s^{k})\in\mathcal{T}^{k}_{\delta}(A_{c},S)\\},$
$\displaystyle\mathcal{E}_{2}\doteq\\{\not\exists
a^{k}_{p}:(a^{k}_{p},s^{k})\in\mathcal{T}^{k}_{\delta}(A_{p},S|a^{k}_{c})\\},$
$\displaystyle\mathcal{E}_{3}\doteq\\{\not\exists
b^{k}_{c}:(b^{k}_{c},u^{k})\in\mathcal{T}^{k}_{\delta}(B_{c},U|S,a^{k}_{c})\\},$
$\displaystyle\mathcal{E}_{4}\doteq\\{\not\exists
b^{k}_{p}:(b^{k}_{p},u^{k})\in\mathcal{T}^{k}_{\delta}(B_{p},U|S,a^{k}_{c},a^{k}_{p},b^{k}_{c})\\}.$

Decoder errors:

$\displaystyle\mathcal{E}_{5}\doteq\\{\exists\hat{q}^{n}_{c}\neq
q^{n}_{c}:\hat{q}_{c}^{n},q^{n}_{c}\in\mathcal{T}^{n}_{\delta}(Q_{c},Y)\\},$
$\displaystyle\mathcal{E}_{6}\doteq\\{\exists\hat{q}^{n}_{p}\neq
q^{n}_{p}:\hat{q}^{n}_{p},q^{n}_{p}\in\mathcal{T}^{n}_{\delta}(Q_{p},Y|q^{n}_{c})\\},$
$\displaystyle\mathcal{E}_{7}\doteq\\{\exists\hat{w}^{n}_{c}\neq
w^{n}_{c}:\hat{w}^{n}_{c},w^{n}_{c}\in\mathcal{T}^{n}_{\delta}(W_{c},Y|q^{n}_{c})\\},$
$\displaystyle\mathcal{E}_{8}\doteq\\{\exists\hat{x}^{n}\neq
x^{n}:\hat{x}^{n},x^{n}\in\mathcal{T}^{n}_{\delta}(X,Y|q^{n}_{c},q^{n}_{p},w^{n}_{c})\\}.$

We upper bound probability of an “error” event as follows:

$Pr\\{\mathcal{E}\\}=P_{\mathcal{E}}\leq\sum_{i=1}^{8}P_{\mathcal{E}_{i}},$

where $P_{\mathcal{E}_{i}}=Pr\\{\mathcal{E}_{i}\\}$.

Given $k\rightarrow\infty$, it can be shown that:

  1. 1.

if $R_{ac}>I(A_{c};S)$
then $P_{\mathcal{E}_{1}}\rightarrow 0$,

  2. 2.

if $R_{ap}>I(A_{p};S|A_{c})$
then $P_{\mathcal{E}_{2}}\rightarrow 0$,

  3. 3.

if $R_{bc}>I(B_{c};U|S,A_{c})$
then $P_{\mathcal{E}_{3}}\rightarrow 0$,

  4. 4.

if $R_{bp}>I(B_{p};U|S,A_{c},A_{p},B_{c})$
then $P_{\mathcal{E}_{4}}\rightarrow 0$,

  5. 5.

if $R_{qc}<(R+\epsilon)I(Q_{c};Y)$
then $P_{\mathcal{E}_{5}}\rightarrow 0$,

  6. 6.

if $R_{qp}+R_{1}<(R+\epsilon)I(Q_{p};Y|Q_{c})$
then $P_{\mathcal{E}_{6}}\rightarrow 0$,

  7. 7.

if $R_{wc}<(R+\epsilon)I(W_{c};Y|Q_{c})$
then $P_{\mathcal{E}_{7}}\rightarrow 0$,

  8. 8.

if $R_{wp}+R_{2}<(R+\epsilon)I(X;Y|Q_{c},Q_{p},W_{c})$
then $P_{\mathcal{E}_{8}}\rightarrow 0$.

Analysis of expected distortion for semantics:

$\displaystyle\operatorname{\mathbb{E}}d_{s}\left(S^{k},\hat{S}^{k}\right)=\operatorname{\mathbb{E}}d_{s}\left(S^{k},\hat{f}_{s}(Y^{n})\right)$
$\displaystyle=^{(a)}P_{\mathcal{E}}\operatorname{\mathbb{E}}\left\\{d_{s}\left(S^{k},\hat{f}_{s}(Y^{n})\right)|\mathcal{E}\right\\}$
$\displaystyle\qquad+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{s}\left(S^{k},\hat{f}_{s}(Y^{n})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\leq
P_{\mathcal{E}}d_{S,m}+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{s}\left(S^{k},\hat{f}_{s}(Y^{n})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle=^{(b)}P_{\mathcal{E}}d_{S,m}+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{s}\left(S^{k},\tilde{S}(A^{k}_{c},A^{k}_{p})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\leq^{(c)}P_{\mathcal{E}}d_{S,m}+P_{\bar{\mathcal{E}}}(1+\epsilon_{1})\operatorname{\mathbb{E}}\left\\{d_{s}\left(S,\tilde{S}(A_{c},A_{p})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\leq^{(d)}P_{\mathcal{E}}d_{S,m}+(1+\epsilon_{1})\operatorname{\mathbb{E}}d_{s}\left(S,\tilde{S}(A_{c},A_{p})\right),$

where
$d_{S,m}=\max_{\mathcal{C},S^{k},Y^{n}}d_{s}\left(S^{k},\hat{f}_{s}(Y^{n})\right)$,
(a) due to law of total expectation, (b) because
$\hat{f}_{s}(Y^{n})=\tilde{S}(A^{k}_{c},A^{k}_{p})$ given no error occurs
$\bar{\mathcal{E}}$, (c) due to typical average lemma and
$(s^{k},a^{k}_{c},a^{k}_{p})\in\mathcal{T}^{k}_{\delta}(S,A_{c},A_{p})$, (d)
due to
$\operatorname{\mathbb{E}}(X|A)\leq\frac{\operatorname{\mathbb{E}}X}{P_{A}}$.

We conclude that the following is sufficient to satisfy distortion condition
(4) for semantics:

$D_{S}\geq\operatorname{\mathbb{E}}d_{s}\left(S,\tilde{S}(A_{c},A_{p})\right),$

given $k\rightarrow\infty$

Analysis of expected distortion for observation:

$\displaystyle\operatorname{\mathbb{E}}d_{u}\left(U^{k},\hat{U}^{k}\right)=\operatorname{\mathbb{E}}d_{u}\left(U^{k},\hat{f}_{u}(Y^{n})\right)$
$\displaystyle=^{(a)}P_{\mathcal{E}}\operatorname{\mathbb{E}}\left\\{d_{u}\left(S^{k},\hat{f}_{s}(Y^{n})\right)|\mathcal{E}\right\\}$
$\displaystyle\qquad+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{u}\left(U^{k},\hat{f}_{u}(Y^{n})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\leq
P_{\mathcal{E}}d_{U,m}+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{u}\left(U^{k},\hat{f}_{u}(Y^{n})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle=^{(b)}P_{\mathcal{E}}d_{U,m}+P_{\bar{\mathcal{E}}}\operatorname{\mathbb{E}}\left\\{d_{u}\left(U^{k},\tilde{U}(A^{k}_{c},A^{k}_{p},B^{k}_{c},B^{k}_{p})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\leq^{(c)}P_{\bar{\mathcal{E}}}(1+\epsilon_{1})\operatorname{\mathbb{E}}\left\\{d_{u}\left(S,\tilde{U}(A^{k}_{c},A^{k}_{p},B^{k}_{c},B^{k}_{p})\right)|\bar{\mathcal{E}}\right\\}$
$\displaystyle\qquad+P_{\mathcal{E}}d_{U,m}$
$\displaystyle\leq^{(d)}P_{\mathcal{E}}d_{U,m}+(1+\epsilon_{1})\operatorname{\mathbb{E}}d_{u}\left(U,\tilde{U}(A_{c},A_{p},B_{c},B_{p})\right),$

where
$d_{U,m}=\max_{\mathcal{C},U^{k},Y^{n}}d_{u}\left(U^{k},\hat{f}_{u}(Y^{n})\right)$,
(a) due to law of total expectation, (b) because
$\hat{f}_{u}(Y^{n})=\tilde{U}(A^{k}_{c},A^{k}_{p},B^{k}_{c},B^{k}_{p})$ given
$\bar{\mathcal{E}}$, (c) due to typical average lemma and
$(u^{k},a^{k}_{c},a^{k}_{p},b^{k}_{c},b^{k}_{p})\in\mathcal{T}^{k}_{\delta}(U,A_{c},A_{p},B_{c},B_{p})$,
(d) due to
$\operatorname{\mathbb{E}}(X|A)\leq\frac{\operatorname{\mathbb{E}}X}{P_{A}}$.

Thus, to satisfy distortion condition (5) it is sufficient to have:

$D_{U}\geq\operatorname{\mathbb{E}}d_{u}\left(U,\tilde{U}(A_{c},A_{p},B_{c},B_{p})\right),$

given $k\rightarrow\infty$

Equivocation analysis for observation:

Here we treat sequence indices $s_{**}$ and $r_{**}$ as random variables. We
start by analyzing equivocation for observation $U^{k}$:

$\displaystyle H(U^{k}|Z^{n})=H(U^{k}|M_{u},Z^{n})+I(U^{k};M_{u}|Z^{n})$
$\displaystyle=H(U^{k}|M_{u})+H(M_{u}|Z^{n})-H(M_{u}|U^{k},Z^{n})$
$\displaystyle\quad-I(U^{k};Z^{n}|M_{u}),$ (29)

where $M_{u}\doteq(s_{bc},s_{bp})$ is an encoded (by source encoder) message
for observation.

First term of (29):

$\displaystyle H(U^{k}|M_{u})=H(U^{k}|s_{bc},s_{bp})=H(S^{k},U^{k})$
$\displaystyle\quad-I(s_{bc},s_{bp};S^{k},U^{k})-H(S^{k}|U^{k},s_{bc},s_{bp})$
$\displaystyle=^{(a)}H(S^{k},U^{k})-H(s_{bc},s_{bp})-H(S^{k}|U^{k},s_{bc},s_{bp})$
$\displaystyle=H(S^{k},U^{k})-H(s_{bc})-H(s_{bp})+I(s_{bc};s_{bp})$
$\displaystyle\quad-H(S^{k}|U^{k},s_{bc},s_{bp}),$ (30)

where (a) because $(s_{bc},s_{bp})$ is a function of $(s^{k},u^{k})$

For second term (29) we have:

$\displaystyle
H(M_{u}|Z^{n})=H(s_{bc},s_{bp}|Z^{n})=^{(a)}H(r_{wc},r_{wp}|Z^{n})$
$\displaystyle\geq^{(b)}H(X^{n}|r_{wc})+H(Z^{n}|X^{n})-H(X^{n}|r_{wc},r_{wp},Z^{n})$
$\displaystyle\quad-H(Z^{n}|r_{wc})$
$\displaystyle=H(X^{n}|r_{wc})+H(Z^{n}|X^{n})-H(X^{n}|r_{wc},r_{wp},Z^{n})$
$\displaystyle\quad-H(Z^{n}|r_{qc},r_{wc})-I(Z^{n};r_{qc}|r_{wc}),$ (31)

where (a) due to $g_{2}$ mapping, (b) for same reasons as in [18, eq. 2.38].
Substituting (30) and (31) into (29) we have:

$\displaystyle H(U^{k}|Z^{n})\geq
H(S^{k},U^{k})-H(s_{bc})-H(s_{bp})+I(s_{bc};s_{bp})$
$\displaystyle\quad-H(S^{k}|U^{k},s_{bc},s_{bp})+H(X^{n}|r_{wc})+H(Z^{n}|X^{n})$
$\displaystyle\quad-H(X^{n}|r_{wc},r_{wp},Z^{n})-H(Z^{n}|r_{qc},r_{wc})-I(Z^{n};r_{qc}|r_{wc})$
$\displaystyle\quad-H(s_{bc},s_{bp}|U^{k},Z^{n})-I(U^{k};Z^{n}|s_{bc},s_{bp}).$
(32)

Now we reduce some terms in (32).

$\displaystyle
H(S^{k},U^{k})-H(S^{k}|U^{k},s_{bc},s_{bp})-H(s_{bc},s_{bp}|U^{k},Z^{n})$
$\displaystyle=^{(a)}H(U^{k})+I(S^{k};s_{bc},s_{bp};Z^{n}|U^{k}),$ (33)

where (a) because $(s_{bc},s_{bp})$ is a function of $(s^{k},u^{k})$.

$\displaystyle H(X^{n}|r_{wc},r_{wp},Z^{n})+I(U^{k};Z^{n}|s_{bc},s_{bp})$
$\displaystyle\leq^{(a)}H(X^{n}|r_{wc},r_{wp},Z^{n})+I(X^{n};Z^{n}|r_{wc},r_{wp})$
$\displaystyle=H(X^{n}|r_{wc},r_{wp}),$ (34)

where (a) due to data processing inequality and $g_{2}$ mapping.

$\displaystyle I(Z^{n};r_{qc}|r_{wc})\leq
H(r_{qc}|r_{wc})=H(r_{qc})-I(r_{qc};r_{wc}).$ (35)

Returning to (32) with (33), (34) and (35):

$\displaystyle H(U^{k}|Z^{n})\geq
H(U^{k})-H(s_{bc})-H(s_{bp})+I(s_{bc};s_{bp})$
$\displaystyle\quad+H(X^{n}|r_{wc})+H(Z^{n}|X^{n})-H(Z^{n}|r_{qc},r_{wc})$
$\displaystyle\quad-H(r_{qc})+I(r_{qc};r_{wc})-H(X^{n}|r_{wc},r_{wp})$
$\displaystyle\quad+I(S^{k};s_{bc},s_{bp};Z^{n}|U^{k}).$ (36)

Now we bound each term of (36):

  1. 1.

$H(U^{k})=kH(U)$ ($U^{k}$ is i.i.d.),

  2. 2.

$H(s_{bc})\leq kR_{bc}$,

  3. 3.

$H(s_{bp})\leq kR_{bp}$,

  4. 4.

$H(X^{n}|r_{wc})\geq k(R_{qc}+R_{qp}+R_{1}+R_{wp}+R_{2})-1-\epsilon_{1}$

($X^{n}$ is nearly uniform and [18, Lemma 2.5]),

  5. 5.

$H(Z^{n}|X^{n})=nH(Z|X)$

(channel is memoryless),

  6. 6.

$H(Z^{n}|r_{qc},r_{wc})\leq nH(Z|Q_{c},W_{c})+n\epsilon_{2}$

(see [18, eq. 2.50-2.52]),

  7. 7.

$H(r_{qc})\leq kR_{qc}$,

  8. 8.

$H(X^{n}|r_{wc},r_{wp})\leq k(R_{qc}+R_{qp}+R_{1}+R_{2})$,

  9. 9.

$I(s_{bc};s_{bp})+I(r_{qc};r_{wc})+I(S^{k};s_{bc},s_{bp};Z^{n}|U^{k})\geq 0$.

Collecting all terms we have:

$\displaystyle H(U^{k}|Z^{n})\geq
kH(U)-kR_{bc}-kR_{bp}+kR_{qc}+kR_{qp}+kR_{1}$
$\displaystyle\quad+kR_{wp}+kR_{2}-1-\epsilon_{1}+nH(Z|X)-nH(Z|Q_{c},W_{c})$
$\displaystyle\quad-n\epsilon_{2}-kR_{qc}-k(R_{qc}+R_{qp}+R_{1}+R_{2})$
$\displaystyle=kH(U)-kR_{bc}-kR_{bp}+kR_{wp}-kR_{qc}+nH(Z|X)$
$\displaystyle\quad-nH(Z|Q_{c},W_{c})-n\epsilon_{3}-1.$

Thus, to satisfy equivocation for source, it is sufficient to have:

$\displaystyle\Delta_{U}\leq H(U)-R_{bc}-R_{bp}+R_{wp}-R_{qc}$
$\displaystyle\quad+(R+\epsilon)(H(Z|X)-H(Z|Q_{c},W_{c}))-\epsilon,$

given $k\rightarrow\infty$.

Equivocation analysis for semantics:

For semantic equivocation, we have:

$\displaystyle H(S^{k}|Z^{n})=H(S^{k}|M_{s},Z^{n})+I(S^{k};M_{s}|Z^{n})$
$\displaystyle=H(S^{k}|M_{s})+H(M_{s}|Z^{n})-H(M_{s}|S^{k},Z^{n})$
$\displaystyle\quad-I(S^{k};Z^{n}|M_{s}),$ (37)

where $M_{s}\doteq(s_{ac},s_{ap})$ is an encoded (by source encoder) message
for semantics. First term of (37):

$\displaystyle H(S^{k}|M_{s})=H(S^{k}|s_{ac},s_{ap})$
$\displaystyle=H(S^{k},U^{k})-I(s_{ac},s_{ap};S^{k},U^{k})-H(U^{k}|S^{k},s_{ac},s_{ap})$
$\displaystyle=^{(a)}H(S^{k},U^{k})-H(s_{ac},s_{ap})-H(U^{k}|S^{k},s_{ac},s_{ap})$
$\displaystyle=H(S^{k},U^{k})-H(s_{ac})-H(s_{ap})+I(s_{ac};s_{ap})$
$\displaystyle\quad-H(U^{k}|S^{k},s_{ac},s_{ap}),$ (38)

where (a) because $(s_{ac},s_{ap})$ is a function of $(s^{k},u^{k})$. Second
term of (37):

$\displaystyle
H(M_{s}|Z^{n})=H(s_{ac},s_{ap}|Z^{n})=^{(a)}H(r_{qc},r_{qp}|Z^{n})$
$\displaystyle\geq^{(b)}H(X^{n}|r_{qc})+H(Z^{n}|X^{n})$
$\displaystyle\quad-H(X^{n}|r_{qc},r_{qp},Z^{n})-H(Z^{n}|r_{qc}),$ (39)

where (a) due to $g_{1}$ mapping, (b) see [18, eq. 2.38]. Gathering all terms,
we have:

$\displaystyle H(S^{k}|Z^{n})\geq
H(S^{k},U^{k})-H(s_{ac})-H(s_{ap})+I(s_{ac};s_{ap})$
$\displaystyle\quad-H(U^{k}|S^{k},s_{ac},s_{ap})+H(X^{n}|r_{qc})+H(Z^{n}|X^{n})$
$\displaystyle\quad-H(X^{n}|r_{qc},r_{qp},Z^{n})-H(Z^{n}|r_{qc})$
$\displaystyle\quad-H(s_{ac},s_{ap}|S^{k},Z^{n})-I(S^{k};Z^{n}|s_{ac},s_{ap}).$
(40)

We reduce some terms in (40).

$\displaystyle
H(S^{k},U^{k})-H(U^{k}|S^{k},s_{ac},s_{ap})-H(s_{ac},s_{ap}|S^{k},Z^{n})$
$\displaystyle=^{(a)}H(S^{k})+I(U^{k};s_{ac},s_{ap};Z^{n}|S^{k}),$ (41)

where (a) because $(s_{ac},s_{ap})$ is a function of $(s^{k},u^{k})$.

$\displaystyle H(X^{n}|r_{qc},r_{qp},Z^{n})+I(S^{k};Z^{n}|s_{ac},s_{ap})$
$\displaystyle\leq^{(a)}H(X^{n}|r_{qc},r_{qp},Z^{n})+I(X^{n};Z^{n}|r_{qc},r_{qp})$
$\displaystyle=H(X^{n}|r_{qc},r_{qp}),$ (42)

where (a) due to data processing inequality and $g_{1}$ mapping. Substituting
(41) and (42) into (40) we obtain:

$\displaystyle H(S^{k}|Z^{n})\geq
H(S^{k})-H(s_{ac})-H(s_{ap})+I(s_{ac};s_{ap})$
$\displaystyle\quad+H(X^{n}|r_{qc})+H(Z^{n}|X^{n})-H(Z^{n}|r_{qc})$
$\displaystyle\quad-H(X^{n}|r_{qc},r_{qp})+I(U^{k};s_{ac},s_{ap};Z^{n}|S^{k}).$
(43)

Now we bound each term of (43):

  1. 1.

$H(S^{k})=kH(S)$ ($S^{k}$ is i.i.d.),

  2. 2.

$H(s_{ac})\leq kR_{ac}$,

  3. 3.

$H(s_{ap})\leq kR_{ap}$,

  4. 4.

$H(X^{n}|r_{qc})\geq k(R_{wc}+R_{qp}+R_{1}+R_{wp}+R_{2})-1-\epsilon_{1}$

($X^{n}$ is nearly uniform and [18, Lemma 2.5]),

  5. 5.

$H(Z^{n}|X^{n})=nH(Z|X)$

(channel is memoryless),

  6. 6.

$H(Z^{n}|r_{qc})\leq nH(Z|Q_{c})+n\epsilon_{2}$

(see [18, eq. (2.50-2.52)]),

  7. 7.

$H(X^{n}|r_{qc},r_{qp})\leq k(R_{wc}+R_{wp}+R_{1}+R_{2})$,

  8. 8.

$I(U^{k};s_{ac},s_{ap};Z^{n}|S^{k})\geq 0$,

  9. 9.

$I(s_{ac};s_{ap})\geq 0$.

Returning to (43) we have:

$\displaystyle H(S^{k}|Z^{n})\geq kH(S)-kR_{ac}-kR_{ap}$
$\displaystyle\quad+k(R_{wc}+R_{qp}+R_{1}+R_{wp}+R_{2})-1-\epsilon_{1}$
$\displaystyle\quad+nH(Z|X)-nH(Z|Q_{c})-n\epsilon_{2}$
$\displaystyle\quad-k(R_{wc}+R_{wp}+R_{1}+R_{2})$
$\displaystyle=kH(S)-kR_{ac}-kR_{ap}+kR_{qp}$
$\displaystyle\quad+nH(Z|X)-nH(Z|Q_{c})-n\epsilon_{3}-1.$

Thus, to satisfy equivocation for semantics, it is sufficient to have:

$\displaystyle\Delta_{S}\leq H(S)-R_{ac}-R_{ap}+R_{qp}$
$\displaystyle\quad+(R+\epsilon)(H(Z|X)-H(Z|Q_{c}))-\epsilon,$

given $k\rightarrow\infty$.

Joint equivocation analysis:

$\displaystyle H(S^{k},U^{k}|Z^{n})=H(S^{k},U^{k}|M)+H(M|Z^{n})$
$\displaystyle\quad-I(S^{k},U^{k};Z^{n}|M)$
$\displaystyle=^{(a)}H(S^{k},U^{k}|M)+H(M|Z^{n}),$ (44)

where $M=(s_{ac},s_{bc},s_{ap},s_{bp})$ and (a) due to
$(S^{k},U^{k})\rightarrow M\rightarrow Z^{n}$.

First term of (44):

$H(S^{k},U^{k}|M)=^{(a)}H(S^{k},U^{k})-H(M),$

where (a) because $M$ is a function of $(S^{k},U^{k})$.

Second term of (44):

$\displaystyle
H(M|Z^{n})\geq^{(a)}H(X^{n}|r_{c})+H(Z^{n}|X^{n})-H(X^{n}|r_{c},r_{p},Z^{n})$
$\displaystyle\quad-H(Z^{n}|r_{c}),$

where $r_{c}=(r_{qc},r_{wc})$, $r_{p}=(r_{qp},r_{wp})$, (a) due to $g$
mapping, the fact that $H(r_{c},r_{p}|X^{n})=0$ and $X^{n}\rightarrow
Y^{n}\rightarrow Z^{n}$.

Returning to (44) we have:

$\displaystyle H(S^{k},U^{k}|Z^{n})\geq H(S^{k},U^{k})-H(M)+H(X^{n}|r_{c})$
$\displaystyle\quad+H(Z^{n}|X^{n})-H(X^{n}|r_{c},r_{p},Z^{n})-H(Z^{n}|r_{c}).$
(45)

Now we bound each term of (45):

  1. 1.

$H(S^{k},U^{k})=kH(S,U)$ ($S^{k}$ and $U^{k}$ are i.i.d),

  2. 2.

$H(M)\leq k(R_{ac}+R_{bc}+R_{ap}+R_{bp})$,

  3. 3.

$H(X^{n}|r_{c})\geq k(R_{qp}+R_{wp}+R_{1}+R_{2})-1-\epsilon_{1}$

($X^{n}$ is nearly uniform, [18, Lemma 2.5]),

  4. 4.

$H(Z^{n}|X^{n})=nH(Z|X)$,

  5. 5.

$H(X^{n}|r_{c},r_{p},Z^{n})\leq\epsilon_{2}$

(due to Fano’s inequality, see [18, eq. (2.49)]),

  6. 6.

$H(Z^{n}|r_{c})\leq nH(Z|Q_{c},W_{c})+n\epsilon_{3}$

(see [18, eq. (2.50-2.52)]).

Finally, we obtain:

$\displaystyle\frac{1}{k}H(S^{k},U^{k}|Z^{n})\geq
H(S,U)-R_{ac}-R_{bc}-R_{ap}-R_{bp}$
$\displaystyle\quad+(R+\epsilon)\left(H(Z|X)-H(Z|Q_{c},W_{c})\right)+\epsilon$
$\displaystyle\quad+R_{qp}+R_{wp}+R_{1}+R_{2}.$

Thus, to satisfy joint equivocation, it is sufficient to have:

$\displaystyle\Delta_{SU}\leq
H(S,U)-R_{ac}-R_{bc}-R_{ap}-R_{bp}+R_{qp}+R_{wp}$
$\displaystyle\quad+R_{1}+R_{2}+(R+\epsilon)\left(H(Z|X)-H(Z|Q_{c},W_{c})\right)-\epsilon.$

Summary of conditions:

$\begin{cases}R_{**},R_{1},R_{2}>0\\\ R_{ac}+R_{ap}=R_{qc}+R_{qp}\\\
R_{bc}+R_{bp}=R_{wc}+R_{wp}\\\ R_{ac}<R_{qc}\\\ R_{bc}<R_{wc}\\\
R_{ac}>I(A_{c};S)\\\ R_{ap}>I(A_{p};S|A_{c})\\\ R_{bc}>I(B_{c};U|S,A_{c})\\\
R_{bp}>I(B_{p};U|S,A_{c},A_{p},B_{c})\\\ R_{qc}<RI(Q_{c};Y)\\\
R_{qp}+R_{1}<RI(Q_{p};Y|Q_{c})\\\ R_{wc}<RI(W_{c};Y|Q_{c})\\\
R_{wp}+R_{2}<RI(X;Y|Q_{c},Q_{p},W_{c})\\\ R_{1}<RI(Q_{p};Z|Q_{c})\\\
R_{2}<RI(X;Z|W_{c})\\\
D_{S}\geq\operatorname{\mathbb{E}}d_{S}(S,\tilde{S}(A_{c},A_{p}))\\\
D_{U}\geq\operatorname{\mathbb{E}}d_{U}(U,\tilde{U}(A_{c},A_{p},B_{c},B_{p}))\\\
\Delta_{S}\leq H(S)-R_{ac}-R_{ap}+R_{qp}\\\
\qquad+(R+\epsilon)(H(Z|X)-H(Z|Q_{c}))-\epsilon\\\ \Delta_{U}\leq
H(U)-R_{bc}-R_{bp}+R_{wp}-R_{qc}\\\
\qquad+(R+\epsilon)(H(Z|X)-H(Z|Q_{c},W_{c}))-\epsilon\\\ \Delta_{SU}\leq
H(S,U)-R_{ac}-R_{bc}-R_{ap}-R_{bp}+R_{qp}+R_{wp}\\\
\qquad+R_{1}+R_{2}+(R+\epsilon)\left(H(Z|X)-H(Z|Q_{c},W_{c})\right)-\epsilon.\end{cases}$

The above system of inequalities can be reduced to the following system:

$\begin{cases}I(S;A_{c})<RI(Q_{c};Y),\\\
I(S;A_{c},A_{p})<RI(Y;Q_{c},Q_{p}),\\\
I(U;B_{c}|S,A_{c})<RI(W_{c};Y|Q_{c}),\\\
I(U;B_{c}|S,A_{c})+I(U;B_{p}|S,A_{c},A_{p},B_{c})<\\\
\qquad<R[I(W_{c};Y|Q_{c})+I(X;Y|Q_{c},Q_{p},W_{c})]\\\
D_{S}\geq\operatorname{\mathbb{E}}d_{S}(S,\tilde{S}(A_{c},A_{p})),\\\
D_{U}\geq\operatorname{\mathbb{E}}d_{U}(U,\tilde{U}(A_{c},A_{p},B_{c},B_{p})),\\\
\Delta_{S}\leq H(S|Ac,Ap)+R(H(Z|X)-H(Z|Qc))\\\ \qquad+RI(Qp;Y|Qc),\\\
\Delta_{U}\leq H(U)-I(A_{c},A_{p};S)-I(B_{c};U|A_{c})\\\
\qquad-I(B_{p};U|S,A_{c},A_{p},B_{c})+RH(Z|X)\\\ \qquad-
RH(Z|Q_{c},W_{c})+RI(Q_{p};Y|Q_{c})\\\ \qquad+RI(X;Y|Q_{c},Q_{p},W_{c}),\\\
\Delta_{SU}\leq H(S,U)-I(S;A_{c},A_{p})-I(U;B_{p}|S,A_{c},A_{p},B_{c})\\\
\qquad-I(U;B_{c}|S,A_{c})+R(H(Z|X)-H(Z|Qc,Wc))\\\
\qquad+RI(Q_{p};Y|Q_{c})+RI(X;Y|Q_{c},Q_{p},W_{c}).\end{cases}$

[Lemma 4]

###### Lemma 4

For $U^{k}\rightarrow X^{n}\rightarrow Y^{n}\rightarrow Z^{n}$ Markov chain
the following holds [8]:

$I(U^{k};Z^{n})-I(U^{k};Y^{n})\geq n[I(X;Z)-I(X;Y)].$ (46)

Proof:

$\displaystyle I(U^{k};Y^{n})=\sum_{i=1}^{n}I(U^{k};Y_{i}|Y^{i-1}),$
$\displaystyle I(U^{k};Y_{i}|Y^{i-1})=I(U^{k},\tilde{Z}^{i+1};Y_{i}|Y^{i-1})$
$\displaystyle\quad-I(\tilde{Z}^{i+1};Y_{i}|Y^{i-1},U^{k})$
$\displaystyle=I(U^{k};Y_{i}|Y^{i-1},\tilde{Z}^{i+1})$
$\displaystyle\quad+I(\tilde{Z}^{i+1};Y_{i}|Y^{i-1})-I(\tilde{Z}^{i+1};Y_{i}|U^{k},Y^{i-1}),$

where $\tilde{Z}^{i}=(Z_{i},Z_{i+1},...,Z_{n})$. Following the same steps for
$I(U^{k};Z^{n})$ we have:

$\displaystyle
I(U^{k};Y^{n})=\sum_{i=1}^{n}A+\sum_{i=1}^{n}B-\sum_{i=1}^{n}C,$
$\displaystyle
I(U^{k};Z^{n})=\sum_{i=1}^{n}D+\sum_{i=1}^{n}F-\sum_{i=1}^{n}G,$

where:

$\displaystyle A$ $\displaystyle=I(U^{k};Y_{i}|Y^{i-1},\tilde{Z}^{i+1}),$
$\displaystyle B$ $\displaystyle=I(\tilde{Z}^{i+1};Y_{i}|Y^{i-1}),$
$\displaystyle C$ $\displaystyle=I(\tilde{Z}^{i+1};Y_{i}|U^{k},Y^{i-1}),$
$\displaystyle D$ $\displaystyle=I(U^{k};Z_{i}|Y^{i-1},\tilde{Z}^{i+1}),$
$\displaystyle F$ $\displaystyle=I(Y^{i-1};Z_{i}|\tilde{Z}^{i+1}),$
$\displaystyle G$ $\displaystyle=I(Y^{i-1};Z_{i}|\tilde{Z}^{i+1}U^{k}).$

Due to Csiszár Sum Identity [16] we have:

$\displaystyle\sum_{i=1}^{n}B=\sum_{i=1}^{n}F,$
$\displaystyle\sum_{i=1}^{n}C=\sum_{i=1}^{n}G.$

Then:

$\displaystyle
I(U^{k};Z^{n})-I(U^{k};Y^{n})=\sum_{i=1}^{n}I(U^{k};Z_{i}|Y^{i-1},\tilde{Z}^{i+1})-$
$\displaystyle\quad-\sum_{i=1}^{n}I(U^{k};Y_{i}|Y^{i-1},\tilde{Z}^{i+1})$
$\displaystyle=nI(U^{k};Z|V)-nI(U^{k};Y|V),$ (47)

where $J$ is uniform r.v. with alphabet $\mathcal{J}=\\{1,...,n\\}$,
$Z=Z_{J}$, and $V=(Y^{i-1},\tilde{Z}^{i+1},J)$.

Due to $V\rightarrow W\rightarrow X\rightarrow Y\rightarrow Z$ Markov chain,
where $W=(V,U^{k})$, we can rewrite (47) as,

$\displaystyle nI(W;Z|V)-nI(W;Y|V)=n[I(W;Z)-I(V;Z)-$
$\displaystyle\quad-I(W;Y)+I(V;Y)]$ $\displaystyle=n[I(X;Z)-I(X;Z|W)-I(V;Z)-$
$\displaystyle\quad-I(X;Y)+I(X;Y|W)+I(V;Y)]$
$\displaystyle\geq^{(\ref{eq:less_noisy})}n[I(X;Z)-I(X;Y)]$

where the last inequality holds since the channel is less noisy:

$I(V;Y)\geq I(W;Z)\text{ and }I(X;Y|U^{k})\geq I(X;Z|U^{k})$ (48)

The proof is complete.

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††thanks: These two authors contributed equally ††thanks: These two authors
contributed equally ††thanks: Corresponding author††thanks: Corresponding
author††thanks: Corresponding author

# Enhanced photovoltaic effect in graphene-silicon Schottky junction under
mechanical manipulation

Dong Pu  Muhammad Abid Anwar  Jiachao Zhou  School of Micro-Nano Electronics,
Zhejiang University, Hangzhou, China  ZJU Global Scientific and Technological
Innovation Center, Hangzhou, China  Renwei Mao  Xin Pan  ZJU-UIUC Institute
(ZJUI), Zhejiang University, Haining, China  Jian Chai  Feng Tian  Hua Wang
<EMAIL_ADDRESS>School of Micro-Nano Electronics, Zhejiang University,
Hangzhou, China  ZJU Global Scientific and Technological Innovation Center,
Hangzhou, China  Huan Hu<EMAIL_ADDRESS>School of Micro-Nano
Electronics, Zhejiang University, Hangzhou, China  ZJU-UIUC Institute (ZJUI),
Zhejiang University, Haining, China  Yang Xu<EMAIL_ADDRESS>School of
Micro-Nano Electronics, Zhejiang University, Hangzhou, China  ZJU Global
Scientific and Technological Innovation Center, Hangzhou, China  ZJU-UIUC
Institute (ZJUI), Zhejiang University, Haining, China

###### Abstract

Graphene-silicon Schottky junction (GSJ) which has the potential for large-
scale manufacturing and integration can bring new opportunities to Schottky
solar cells for photovoltaic (PV) power conversion. However, the essential
power conversion limitation for these devices lies in the small open-circuit
voltage ($V_{oc}$), which depends on the Schottky barrier height (SBH). In
this study, we introduce an electromechanical method based on the
flexoelectric effect to enhance the PV efficiency in GSJ. By atomic force
microscope (AFM) tip-based indentation and in situ current measurement, the
current-voltage (I-V) responses under flexoelectric strain gradient are
obtained. The $V_{oc}$ is observed to increase for up to 20$\%$, leading to an
evident improvement of the power conversion efficiency. Our studies suggest
that strain gradient may offer unprecedented opportunities for the development
of GSJ based flexo-photovoltaic applications.

Due to its special two-dimensional structural properties and excellent
performance [1], graphene has the potential to be integrated into existing
semiconductor technologies and used in next-generation electronics. Recent
research has shown that the formation of junctions between graphene and three-
dimensional or two-dimensional semiconductors [2] can produce the
rectification effect of a typical Schottky junction. Its tunable Schottky
barrier makes graphene junctions an excellent platform for studying the
transport properties of interfaces and has led to applications in scenarios
such as photodetection [3], light-speed communication [4], and chemical and
biological detection [5]. Solar energy collection and conversion have
attracted much attention in recent years. The Schottky junction devices can
naturally work as solar cell or photovoltaic cell [6], as the built-in
electrical field provides the voltage potential difference that drives the
current [7]. Conventional metal/semiconductor Schottky devices suffer from the
contact instability, high cost and high-temperature fabrication process.
Graphene, which has unique optical properties and excellent mechanical
properties, offers Schottky solar cells with low sheet resistance, high
optical transparency, large area growth, and low-cost transferring. Over the
last decade, many studies have been focused on Gr/Si Schottky junction [8, 9,
10, 11] (GSJ) for solar cell applications. An overall power conversion
efficiency (PCE) of 1-1.7$\%$ is achieved with open circuit voltage ($V_{oc}$)
and short circuit current ($J_{SC}$) linearly depending on the intensity of
incident light[6]. By chemical doping, PCE can be further increased for
$8.5\%$ [12]. With systematical optimization including the number of graphene
layers, PCE can surpass $3\%$ [13]. Furthermore, substrate metasurface [14,
15], such as Si nanowire, Si nanohole array, and additional antireflection
layer [16] are proposed for PCE enhancement.

Figure 1: Experimental characterization of the GSJ device. a. The Scanning
electron micrograph (SEM) of the device. The white bar is
$100\text{\,}\mu\mathrm{m}$. b. The surface topography of the device measured
by AFM. c. Current distribution of Gr/Si junction measured by AFM with bias
voltage $V_{bias}$$-1\text{\,}\mathrm{V}$ and applied force
$0.26\text{\,}\mu\mathrm{N}$. The white scale bar represents
$600\text{\,}\mathrm{p}\mathrm{m}$. d. The Raman mapping of the device. e. The
current-voltage response curve. The inset picture shows the logarithmic
current as a function of the bias voltage.

However, one primary limitation for the GSJ Schottky solar cell is the low
open circuit voltage $V_{oc}$, which relates to the small Schottky barrier
height (SBH). Here, we introduce an electromechanical coupling, called
flexoelectricity [17], to increase the SBH, thus enhancing the performance of
GSJ solar cell. The flexoelectric effect describes the linear coupling between
electric polarization and strain gradient in solid state materials [18, 19].
It suggests that the polarization can originate from the strain gradient even
in the centrosymmetric systems. Compared with the piezoelectric effect that
has been studied extensively, the response of the flexoelectric effect is very
weak, and remains underexplored [17]. The electric polarization induced by a
strain gradient is typical of the order of
${10}^{-9}\text{\,}\mathrm{C}\mathrm{/}\mathrm{m}^{2}$ [17, 20] at the macro
scale. As the geometric size scales down, the strain gradient is inversely
proportional to the spatial scale [21]. Thus micro-nano structure is able to
achieve a large strain gradient, the flexoelectric effect induced by a strain
gradient, and dominates over the piezoelectric effect at nanoscale[22, 23]. In
recent years, By using an atomic force microscope (AFM) to introduce large
strain gradients, a number of experimental studies concentrate on the
mechanism of flexoelectricity and its applications have emerged [24, 25, 26].
Conductive AFM (CAFM) probe coated with metal can introduce a strain gradient
and simultaneously monitor the current flow through the junctions. The strain
gradient breaks the inversion symmetry of centrosymmetric materials and
induces electric field polarization, named flexoelectric effect [20]. In
contrast, a uniform strain cannot induce dipoles in graphene [27], and it is
difficult to change the Schottky barrier. The barrier height of the Schottky
junction interface between the probe and silicon can be tuned by the
flexoelectric effect [20, 28]. The flexo-photovoltaic effect (FPV) was found
in perovskites [29, 30] and two-dimensional material system [31, 32], and
significantly improves the solar cell performance. These motivate us to
systematically investigate the flexo-photovoltaic in GSJ.

In this study, we introduce the flexo-photovoltaic effect in the Gr/Si
Schottky junction. We find the GSJ performance as a solar cell can be largely
enhanced through the flexoelectric effect by using AFM tip pressing. By in
situ adding mechanical force on the GSJ, the current flows through the
junction to the tip can be detected, then being read out based on the CAFM
module using AFM. The current-voltage curves under different applied forces
are analyzed, and we obtained the corresponding SBH variation as a function of
applied force. Under illumination, the GSJ device shows PV effect. $V_{oc}$
can be improved by this electromechanical effect. We finally demonstrate the
enhanced PV through the flexoelectric effect in GSJ.

The graphene/silicon junction Schottky devices are prepared with a single
layer of CVD graphene being wet-transferred to the lightly-doped n-type
silicon layer forming the two dimensional and three dimensional (2d-3d)
Schottky contact. An n-doped Si/SiO2
$500\text{\,}\mu\mathrm{m}$$/$$100\text{\,}\mu\mathrm{m}$ substrate with a
resistivity of 1-$10\text{\,}\Omega\cdot\mathrm{c}\mathrm{m}$ corresponding to
a doping concentration of
$4.5\text{\times}{10}^{14}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ to
$4.94\text{\times}{10}^{15}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ is used.

The bottom of the silicon substrate is mechanically scraped to remove the thin
oxide layer and coated with GaIn and copper to form an ohmic contact [33]. The
scanning electron micrograph (SEM) of the device is shown in Fig.1a. The
graphene covers the silicon window (dark area) forming the Schottky contact,
connecting with the Au electrode (surrounding gray part), and forming an ohmic
contact with Au electrodes. The graphene is etched into ribbons. The surface
morphology of the Gr ribbon edge tested by AFM is shown in Fig.1b. The left-
hand side flatten area is silicon covered by graphene, while the other is bare
silicon after the graphene is etched. With the tip contacting the GSJ area,
the current can be read out with the CAFM module. Figure.1c shows the high
spatial resolution current distribution of the junction with $V_{bias}=$
$-1\text{\,}\mathrm{V}$. Figure.1d shows the corresponding Raman mapping on
the graphene ribbon device.

Figure 2: a. Schematic of the CAFM experimental setup. b. Finite element
analysis (FEA) of strain distribution in the substrate under pressing. c.
Energy bandgap structure of the GSJ with laser on. d. The bandgap structure
bending by tip-pressing induced flexoelectricity modulation.

The intensity of the 2D peak versus the G peak and the intensity of the D peak
versus the G peak are shown to illustrate the clear graphene ribbon areas.
According to the contour scale bar, the 2D peak has a larger peak, while the D
peak has a smaller peak, indicating the high quality of the graphene in the
sample. With the probe connecting the bottom copper and the top-surface
electrode separately, we obtain the basic current-voltage (I-V) response of
the device using a semiconductor analyzer (Agilent B1500), shown in Fig.1e.
The red line shows I-V response with the light turned off while the blue line
represents the light-on case. In the light-off case, the I-V curves show the
typical Schottky diode properties of GSJ. The center wavelength is
$532\text{\,}\mathrm{n}\mathrm{m}$ and the approximate power density used is
$50\text{\,}\mu\mathrm{W}$. The reverse current is on the order of
${10}^{-10}\text{\,}\mathrm{A}$, showing a good diode characteristic. When the
light turns on, the reverse photocurrent exceeds $10\text{\,}\mu\mathrm{A}$.
The device performs a typical photoresponse of GSJ, and the inset picture
shows the half-logarithmic curve of the I-V results. In the reverse bias
region, the photocurrent reaches nearly 5 orders of magnitude under the effect
of the photoresponse. In the following experiments, we use AFM to do further
tests to study the electromechanical effects on the responses by in situ
exerting mechanical stress. Note that due to the current-limiting protection
of the instruments ($20\text{\,}\mathrm{n}\mathrm{A}$), we focus on the
results in the range of $\pm 20\text{\,}\mathrm{n}\mathrm{A}$ in the following
tests.

Considering the graphene massless carrier property and the inhomogeneity of
the Gr/Si contact, the modified equation for GSJ based on thermal emission
theory can be expressed as the following formula[7, 34],

$J=A^{*}T^{3}\exp\left(-\frac{\bar{\phi}_{B}-\frac{\delta_{P}^{2}}{2k_{B}T}}{k_{B}T}\right)[\exp\left(\frac{qV}{\eta
k_{B}T}-1\right)]$ (1)

where $A^{*}=$
$0.011\,58\text{\,}\mathrm{A}\mathrm{/}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{K}^{3}$
and $\delta_{P}=$ $135\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$, and $\eta$,
$k_{B}$, $T$ and $q$ are the ideal factor, the Boltzmann constant, temperature
and elementary charge, respectively. The effective working area $A$ of GSJ for
our device here is
$1.25\text{\times}{10}^{-3}\text{\,}\mathrm{c}\mathrm{m}^{2}$. Fitting Fig.1e
with Eq.1, we then obtain the Schottky barrier height (SBH) $\bar{\phi}_{B}=$
$0.6\text{\,}\mathrm{e}\mathrm{V}$ before applying mechanical stress. To find
the effect of electrical polarization on the Schottky junction induced by
strain gradient, we use the Conductive-AFM tip to exert stress on the graphene
surface and in situ read-out the current ($I_{gsj}$) by sweeping the bias
voltage $V_{bias}$, the schematic of the experimental setup is shown in
Fig.2a.

Figure 3:  GSJ I-V responses under tip-pressing obtained by AFM. Photo-
response effects on the GSJ performance with fixed applied force (a. Linear
ordinate coordinates; b. Logarithmic ordinate coordinates). c. GSJ I-V
responses of increased applied forces with laser turned off. d. GSJ I-V
responses of increased applied forces with laser turned on.

The tip coated with conductive diamond (Adama-AD40) is directly pressed onto
the CVD graphene layer. Note that, in these AFM based experiments, $V_{bias}$
is applied to the bottom ohmic electrode and the current reversely flows
through the Gr/Si Schottky junction to the probe for current detection. Thus
the forward bias response is under the condition of negative bias voltage
($V_{bias}<0$). Directly experimental calibration of the tip-induced strain
distribution is still challenging [24]. In the following, we estimate the
strain distribution based on finite element simulation via COMSOL. As shown in
Fig.1, the graphene is etched into ribbons. Due to the atomic thickness of
graphene and the free-standing edge of the ribbon, the graphene layer has
negligible effects on the out-of-plane deformation of the junction. The finite
element analysis on the strain distribution under pressing thus only takes the
tip and the silicon substrate into consideration for simplification. We set
the tip radius to $10\text{\,}\mathrm{n}\mathrm{m}$ according to the SEM
imaging measurement. The bottom of the substrate is set to be fixed according
to the actual situation of the experiment. The tip is set using diamond
material with density
$3515\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$, Young’s modulus
$105\text{\times}{10}^{10}\text{\,}\mathrm{P}\mathrm{a}$ and Poisson’s ratio
0.1 while the silicon substrate with density
$2329\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$, Young’s modulus
$170\text{\times}{10}^{9}\text{\,}\mathrm{P}\mathrm{a}$ and Poisson’s ratio
0.28. The half-cross-section of the substrate is shown in Fig.2b, as the
contact area has rotation symmetry of z-axis. While at an applied force on the
order of $\sim$$5\text{\,}\mu\mathrm{N}$, the corresponding strain is about
$\pm 0.3\text{\,}$. For the vertical direction (z-axis), the strain gradient
is estimated to be on the order of ${10}^{7}\text{\,}\mathrm{m}^{-1}$.

For a typical graphene low-doped n-type silicon junction, the electronic band
structure and energy-band diagram are shown in Fig.2c. While under
illumination, part of the photo-generated carriers moves to the graphene thus
the barrier height slightly changes. Note that in our following experiments,
the laser central wavelength from AFM is $840\text{\,}\mathrm{n}\mathrm{m}$.
In Fig.2d, we show the further changes (dashed blue lines) of the barrier
height. While under tip pressing, the strain gradient in the substrate silicon
breaks the inverse symmetry of the centrosymmetric material, which results in
a flexoelectric effect [29, 20] (the blue arrow represents the polarization).
The corresponding built-in electric field thus increases the SBH. This leads
to the flexoelectricity induced Schottky barrier height tuning of the device,
which will consequently affect the photovoltaic effect in this junction.

Figure 4:  a. The GSJ current as a function of bias voltage under different
applied forces. The orange curves show the laser-off cases while the blue
curves show the laser-on cases. b. The open-circuit voltage as a function of
applied forces. The blue points represent the laser-on cases while the orange
points represent the laser-off cases. c. The extracted Schottky barrier height
as a function of applied forces.

Figure 3 shows the GSJ I-V responses obtained directly by AFM while applying
forces. While the laser is turned off, the GSJ shows a typical Schottky diode
feature (the orange line in Fig.3a). As aforementioned, the forward bias range
is when $V_{bias}<0$, while the reverse bias area $V_{bias}>0$. While
$V_{bias}>$$-0.3\text{\,}\mathrm{V}$, the response current reaches the upper
testing limit of our AFM instrument. The open-circuit voltage ($V_{oc}$)
increases to $\sim$$0.4\text{\,}\mathrm{V}$, shown clearly in Fig.3b. The
short-circuit current ($J_{sc}$) can not be directly obtained, because of the
instrument current limits. Nevertheless, the photoresponse (in Fig.3a and
Fig.3b) implies that this GSJ device has an evident photovoltaic effect. To
further investigate the mechanical manipulation of the PV effect, we applied
forces by AFM nanoindentation on the GSJ. We first turn the laser off to study
the force induced effects on the GSJ. The response current (absolute value) in
the forward bias area can be enlarged by increasing the applied force (Fig.3c)
from $2.07\text{\,}\mu\mathrm{N}$ to $7.58\text{\,}\mu\mathrm{N}$, while the
response current in reverse bias area maintains being cut off. Similarly, for
each applied force, we then collect the current data by sweeping the bias
voltage $V_{bias}$ when turning the laser on, shown in Fig.3d. In the forward
bias area, the GSJ current ($I_{GSJ}$) maintained being enlarged (absolute
value) when the applied force is increased in contrast to the Fig.3c. Due to
the laser-induced photoresponse, the current dramatically changed. There
exists an evident photovoltaic area. By directly contrasting the I-V responses
under different applied forces, the half-logarithmic graph of the absolute GSJ
current as a function of the bias voltage is shown in Fig.4a. For the laser-
off cases (orange lines), the open-circuit voltage $V_{oc}$ remains stable
under different applied forces. Remarkably, for the laser-on cases (blue
lines), due to the existing photovoltaic effects, $V_{oc}$ varies from near
zero to around $-0.5\text{\,}\mathrm{V}$ in contrast to the laser-off cases.
The absolute value of $V_{oc}$ is increased from $0.38\text{\,}\mathrm{V}$ to
$0.46\text{\,}\mathrm{V}$ for more than 20$\%$, as shown in Fig.4b (the blue
circles). Fitting the laser-off cases with Eq.1, the Schottky barrier height
as a function of applied forces is obtained, as shown in Fig.4c. The strain
gradient in the substrate originates from the contacts between the tip and the
substrate surface [24], which breaks the inversion symmetry in the silicon
[20, 28] and causes an extra flexoelectricity induced built-in potential in
addition to the native depletion layer in the GSJ. The SBH can be additionally
increased by the forces.

For this GSJ, the open-circuit voltage can be obtained for $I=0$ while short-
circuiting current for $V=0$. When $V=0$, the simple expression can be
obtained [7] $I_{sc}\approx-I_{ph}$, where $I_{ph}$ is the photogenerated
current. The $V_{oc}$ can be expressed as,

$V_{oc}\approx\frac{\eta
k_{B}T}{q}\ln(\frac{I_{ph}}{I_{0}})\approx\frac{\eta}{q}\phi_{B}+Const.$ (2)

The $V_{oc}$ has approximately a linear relation with the Schottky barrier
height, and agrees well with experiments.

Considering the photoresponse of Gr/Si Schottky junction, we need the short-
circuit current $I_{sc}$ and fill factor $FF$ for estimation. However, because
of the limitation of our instrument, the short-circuit current can not be
precisely measured experimentally. Here, our laser power has not been changed
as an AFM detection light source for the laser-on cases, i.e. photogenerated
current $I_{ph}$ is fixed. We then assume that the short-circuit current for
different applied forces are equal, i.e. for each case, when $V_{bias}=0$, the
GSJ reaches the reverse saturated state. In addition, when $V_{bias}>V_{oc}$,
the current rises sharply to the upper limit in a series of parabolic shapes.
Considering the actual $I_{sc}\gg$$20\text{\,}\mathrm{n}\mathrm{A}$, the fill
factors $FF$ are assumed to be approximately equal for the cases under
different forces. Consequently, the power conversion efficiency (PCE) can be
expressed as $\displaystyle PCE=\frac{I_{sc}V_{oc}FF}{P_{in}}$, which can be
improved for about at least 20$\%$ attributed to the enhancement of $V_{oc}$
under strain gradient.

In this article, we investigate the flexoelectricity-enhanced photovoltaic
effect in GSJ. Using the AFM, we apply different forces to generate the strain
gradient in the junction and in situ obtain the response current of GSJ under
these conditions. By experimental validation, we find that the open-circuit
voltage ($V_{oc}$) can be enhanced from $0.38\text{\,}\mathrm{V}$ to
$0.46\text{\,}\mathrm{V}$ by the applied forces when the laser is turned on
and the Schottky barrier height can be increased from
$0.65\text{\,}\mathrm{e}\mathrm{V}$ to $0.7\text{\,}\mathrm{e}\mathrm{V}$,
respectively. As a consequence, the power conversion efficiency can be
enhanced for more than 20$\%$ with the assumption that the short-circuit
current $I_{sc}$ is identical and similar fill factor $FF$. Our work shed
light on the GSJ devices for PV effect enhancement with mechanical
manipulation approach and pave the way for the PV enhancements 2D
heterostructure based on the similar mechanism. At present, our research is
still in the early stage of exploration, and combining this technology with
integration is still a challenge for future applications.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under Grants 92164106 and 61874094, China Postdoctoral Science
Foundation (2021M692789), and in part by the Fundamental Research Funds for
the Central Universities under Grants K20200060 and 2021FZZX001-17. We also
thank Prof. Chunli Zhang for his valuable discussions and comments.

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