# The Model Inversion Eavesdropping Attack in Semantic Communication Systems

Yuhao Chen, Qianqian Yang†, Zhiguo Shi, Jiming Chen  College of Control
Science and Engineering, Zhejiang University, Hangzhou 310007, China
College of Information Science and Electronic Engineering, Zhejiang
University, Hangzhou 310007, China
The State Key Laboratory of Industrial Control Technology, Hangzhou 310007,
China
{csechenyh, qianqianyang20†, shizg<EMAIL_ADDRESS>
###### Abstract

In recent years, semantic communication has been a popular research topic for
its superiority in communication efficiency. As semantic communication relies
on deep learning to extract meaning from raw messages, it is vulnerable to
attacks targeting deep learning models. In this paper, we introduce the model
inversion eavesdropping attack (MIEA) to reveal the risk of privacy leaks in
the semantic communication system. In MIEA, the attacker first eavesdrops the
signal being transmitted by the semantic communication system and then
performs model inversion attack to reconstruct the raw message, where both the
white-box and black-box settings are considered. Evaluation results show that
MIEA can successfully reconstruct the raw message with good quality under
different channel conditions. We then propose a defense method based on random
permutation and substitution to defend against MIEA in order to achieve secure
semantic communication. Our experimental results demonstrate the effectiveness
of the proposed defense method in preventing MIEA.

##  I Introduction

Recently, semantic communication has been widely believed to be one of the
core technologies for the sixth generation (6G) of wireless networks because
of its high communication efficiency[1]. Compared with the current research on
communication which focuses on transmitting mapped bit sequences of the raw
message[2, 3, 4], semantic communication systems transmit compacted semantic
features. Existing literature in semantic communication mainly exploits the
deep learning (DL) techniques to extract the semantic features from the raw
message. For instance, Han _et al_. [5] proposed to extract the text-related
features from the speech signal as the semantic features and remove the
redundant content. On the receiver’s side, the semantic features can be
reconstructed by a deep learning model into the original message or directly
applied for downstream tasks such as image classification and speech
recognition.

Although many works have been proposed for semantic communication considering
different aspects, few studies have taken into account the security
problems[6, 7, 8]. Tung _et al_.[6] proposed to encrypt the transmitted signal
in semantic communication, but the encryption algorithm incurs a large
computation overhead. Security is crucial in semantic communication for two
main reasons. Firstly, semantic communication is more prone to privacy leakage
compared to traditional communication. In traditional communication systems,
the bit sequences being transmitted contain redundant bits to ensure reliable
transmission, which can be used to provide a certain level of privacy
protection. However, the semantic communication systems transmit compact and
more semantic-related symbols which may reveal more private information.
Secondly, deep-learning-based semantic communication may be vulnerable to
attacks targeting DL models. Extensive studies have been conducted on attacks
on the DL model, a review of which can be referred to [9]. If the semantic
features being transmitted are eavesdropped by a malicious attacker, the
attacker can reconstruct the raw message by utilizing the DL-based attack
techniques. The attacker can also add perturbation to the transmitted data,
causing the semantic communication system to make incorrect decisions on
downstream tasks. For example, Sagduyu _et al_. [7] proposed a multi-domain
evasion attack to cause the semantic communication system to make incorrect
classifications, which is achieved by introducing noises to input images or
the semantic features. Du _et al_.[8] proposed a semantic data poisoning
attack, which causes the receiver to receive irrelevant messages from the
transmitter. For example, the receiver wants to receive an image with a pear
but gets an image with an apple instead. This attack is performed by
minimizing the difference between the semantic features of the targeted
message and the irrelevant message.

In this paper, we consider the security issue in semantic communication
systems and introduce the model inversion eavesdropping attack (MIEA) for
semantic communication, where an attacker eavesdrops the transmitted symbols
and attempts to reconstruct the original message from them by inverting the DL
model used at the transmitter. We perform MIEA under both the white-box and
the black-box settings. The attacker has knowledge of the DL model in the
white-box setting while not in the black-box setting. To defend against MIEA,
we also propose a defense method based on random permutation and substitution.
Evaluations demonstrate that the MIEA attack works under different channel
conditions, i.e., different values of the signal-to-noise ratio (SNR), which
reveals the risk of privacy leaks in semantic transmission. Numerical results
also validate the effectiveness of our proposed defense method.

This paper is organized as follows: Section II introduces the basic ideas of
semantic communications. In section III, we present the proposed MIEA under
both the white-box and black-box setting, and propose our defense method. In
section IV, we evaluate the effectiveness of the proposed MIEA and the
proposed defense method. Section V concludes our work.

##  II Fundamentals

In this section, we provide the fundamentals of semantic communication and the
eavesdropping performed by the attacker. We consider a semantic communication
system which transmits images over wireless channels. As shown in Fig. 1, the
transmitter of the semantic communication system consists of a semantic
encoder and a channel encoder. The semantic encoder extracts the semantic
features $\boldsymbol{z}$ from the raw image $\boldsymbol{x}$, while the
channel encoder maps $\boldsymbol{z}$ into the transmitted features
$\boldsymbol{y}_{\mathrm{f}}\in\mathbb{R}^{h\times w\times c}$, where $h,w,c$
denote the height, the width and the channel of the transmitted features
respectively. Before transmission, $\boldsymbol{y}_{\mathrm{f}}$ is reshaped
into the transmitted symbols $\boldsymbol{y}\in\mathbb{R}^{N\times 2}$, where
$N=\frac{h\times w\times c}{2}$ and the two channels are the real parts and
imaginary parts of the signal to be transmitted, respectively.
$\boldsymbol{y}$ is then transmitted over a wireless channel, which we denote
as the main channel to distinguish from the channel used by the attacker. The
received signal $\hat{\boldsymbol{y}}$ at the receiver side can be
characterized by

$\hat{\boldsymbol{y}}=\boldsymbol{H}_{\mathrm{m}}\boldsymbol{y}+\boldsymbol{n}_{\mathrm{m}},$
(1)

where $\boldsymbol{H}_{\mathrm{m}}$ is a matrix which reflects the main
channel effect such as multi-path propagation, fading and interference, while
$\boldsymbol{n}_{\mathrm{m}}$ is a zero-mean additive white Gaussian noise.
The receiver of the semantic communication system consists of a channel
decoder and a semantic decoder. The receiver first reshapes
$\hat{\boldsymbol{y}}$ back to the transmitted features
$\hat{\boldsymbol{y}}_{\mathrm{f}}$. Then the channel decoder maps
$\hat{\boldsymbol{y}}_{\mathrm{f}}$ back to the semantic features
$\hat{\boldsymbol{z}}$. The semantic decoder then reconstructs the image
$\hat{\boldsymbol{x}}$ from $\hat{\boldsymbol{z}}$. We jointly train the
semantic encoder, channel encoder, semantic decoder and channel decoder using
the following loss function:

$\mathcal{L}=\frac{1}{N}\sum\limits_{i=1}^{N}\|\boldsymbol{x}-\hat{\boldsymbol{x}}\|^{2}+\lambda
T(\hat{\boldsymbol{x}}),$ (2)

where $N$ is the number of the training data batch and

$T(\hat{\boldsymbol{x}})=\sum\limits_{i,j}(|\hat{\boldsymbol{x}}_{i+1,j}-\hat{\boldsymbol{x}}_{i,j}|^{2}+|\hat{\boldsymbol{x}}_{i,j+1}-\hat{\boldsymbol{x}}_{i,j}|^{2})^{\beta/2}.$
(3)

The first term in (2) computes the mean square error (MSE) between
$\boldsymbol{x}$ and $\hat{\boldsymbol{x}}$. The second term
$T(\hat{\boldsymbol{x}})$ is the total variation [10] that measures the
smoothness of the reconstructed image $\hat{\boldsymbol{x}}$, where
$\hat{\boldsymbol{x}}_{i,j}$ denotes the pixel value at the position $(i,j)$
and $\beta$ controls the smoothness of the image, with larger $\beta$ being
more piecewise-smooth. The hyper-parameter $\lambda$ balances the two terms.
In our work, we choose $\beta=1$ and $\lambda=1$.

Figure 1: Illustration of the semantic communication with MIEA.

Next, we introduce how an attacker eavesdrops the transmitted signal under the
semantic communication system. We follow the naming convention in the security
research, with Alice, Bob and Eve representing the sender, receiver and
attacker respectively. Suppose Alice wants to send an image to Bob. As shown
in the lower part of Fig. 1, since the transmitted symbols $\boldsymbol{y}$ is
transmitted over the wireless channel, it can easily be captured by any
unauthorized receiver. Assume that there exists an attacker Eve who intercepts
$\boldsymbol{y}$ and attempts to reconstruct the raw image from it. The
wireless channel between Alice and Eve is referred to as the eavesdropper
channel[11]. The received signal at Eve is given by

$\hat{\boldsymbol{y}}_{\mathrm{e}}=\boldsymbol{H}_{\mathrm{e}}\boldsymbol{y}+\boldsymbol{n}_{\mathrm{e}},$
(4)

Similarly, $\boldsymbol{H}_{\mathrm{e}}(\boldsymbol{\cdot})$ represents the
eavesdropper channel matrix and $\boldsymbol{n}_{\mathrm{e}}$ is a zero-mean
additive white Gaussian noise. After eavesdropping
$\hat{\boldsymbol{y}}_{\mathrm{e}}$, Eve is able to reconstruct the image,
denoted as $\hat{\boldsymbol{x}}_{\mathrm{e}}$, which will be detailed section
III. Note that to avoid confusion, we use the received image to denote
$\hat{\boldsymbol{x}}$ received by Bob and the eavesdropped image to denote
$\hat{\boldsymbol{x}}_{\mathrm{e}}$ eavesdropped by Eve.

##  III The Proposed MIEA and its Defense

In this section, we first elaborate the idea of MIEA. To reconstruct
$\hat{\boldsymbol{x}}_{\mathrm{e}}$, Eve performs MIA[12] using either the
white-box attack or the black-box attack, which depends on the knowledge of
the semantic encoder and channel encoder that Eve has. Then we propose an
effective defense method that defends against both types of attack.

###  III-A White-box Attack

In the white-box attack, Eve knows the parameters and structure of the
semantic encoder and channel encoder. For example, the semantic communication
system is publicly available or available through purchase, such as JPEG. In
this case, Eve can directly use the semantic encoder and channel encoder to
reconstruct the image. We denote the two encoders as a single function
$f(\cdot)$ which maps a given image $\boldsymbol{x}$ to the transmitted symbol
$\boldsymbol{y}$, that is, $\boldsymbol{y}=f(\boldsymbol{x})$. The
reconstructed image $\hat{\boldsymbol{x}}_{\mathrm{e}}$ can be obtained by
solving the following optimization problem:

$\hat{\boldsymbol{x}}_{\mathrm{e}}=\mathop{\arg\min}\limits_{\boldsymbol{x}}\|\hat{\boldsymbol{y}}_{\mathrm{e}}-f(\boldsymbol{x})\|^{2}+\lambda
T(\boldsymbol{x}).$ (5)

The first term in (5) is the MSE between $\hat{\boldsymbol{y}}_{\mathrm{e}}$
and $f(\boldsymbol{x})$, while the second term $T(\boldsymbol{x})$ is the
total variation defined in (3) that guarantees the smoothness of
$\boldsymbol{x}$. Similar to (2), we set $\beta=1$ and $\lambda=1$ here. The
optimization problem (5) can be solved by performing the gradient descent,
which iteratively updates the input $\boldsymbol{x}$ so that (5) can be
minimized.

###  III-B Black-box Attack

In the black-box attack, Eve lacks knowledge of the parameters and structures
of both encoders. In this case, Eve uses an inverse network of the two
encoders, denoted as $f^{-1}(\cdot)$, to inverse
$\hat{\boldsymbol{y}}_{\mathrm{e}}$ back to
$\hat{\boldsymbol{x}}_{\mathrm{e}}$. Specifically, $f^{-1}$ takes
$\hat{\boldsymbol{y}}_{\mathrm{e}}$ as input and outputs $\boldsymbol{x}$,
i.e., $f^{-1}(\hat{\boldsymbol{y}}_{\mathrm{e}})=\boldsymbol{x}$. To train
$f^{-1}$, we assume that Eve can feed a batch of samples
$\mathbb{X}=\\{\boldsymbol{x}_{1},\boldsymbol{x}_{2},...,\boldsymbol{x}_{m}\\}$
into the encoder and capture the corresponding transmitted symbols
$\mathbb{Y}_{\mathrm{e}}=\\{\hat{\boldsymbol{y}}_{1},\hat{\boldsymbol{y}}_{2},...,\hat{\boldsymbol{y}}_{m}\\}$,
where $m$ is the number of the samples. Eve then trains $f^{-1}$ using
$\mathbb{Y}_{\mathrm{e}}$ as the input and $\mathbb{X}$ as the ground truth
output. We use the $l_{2}$ norm as the loss function and employ stochastic
gradient descent to train the inverse network:

$f^{-1}=\mathop{\arg\min}\limits_{g}\frac{1}{m}\sum\limits_{i=1}^{m}\|g(\hat{\boldsymbol{y}}_{i})-\boldsymbol{x}_{i}\|^{2},$
(6)

where we use $g$ to represent the inverse network being optimized. Once the
inverse network is trained, Eve is able to reconstruct the image from any
newly eavesdropped signal.

###  III-C Defense Method against MIEA

To defend against MIA in deep learning, researchers have proposed various
techniques such as differential privacy (DP)[12] or attacker-aware training
[13]. The DP technique involves adding an additive Laplacian noise to the raw
image, while the attacker-aware training adds a regularization term to the
loss function during training, which maximizes the MSE between the
reconstructed image and the raw image. Both methods prevent Eve from
reconstructing high-quality images while maintaining the performance of
downstream tasks. However, in the transmission task considered in this paper,
both Bob and Eve attempt to reconstruct the image from $\hat{\boldsymbol{y}}$
and $\hat{\boldsymbol{y}}_{\mathrm{e}}$ respectively. Furthermore, we assume
that both the main channel and eavesdropper channel are AWGN channel, i.e.,
$H_{\mathrm{m}}=H_{\mathrm{e}}$. Then the difference between
$\hat{\boldsymbol{y}}$ and $\hat{\boldsymbol{y}}_{\mathrm{e}}$ is
$|\boldsymbol{n}_{\mathrm{m}}-\boldsymbol{n}_{\mathrm{n}}|$, which is
relatively small. Therefore, if Eve fails to reconstruct high-quality images
under the defense methods above, Bob will also fail, which contradicts the
goal of the transmission task. To prevent Eve from reconstructing images as
well as maintaining the image quality received by Bob, an intuitive solution
is to encrypt $\boldsymbol{y}$ using common cryptography algorithms, but this
will incur a large computation overhead. To reduce the computation overhead,
we propose a defense method based on random permutation and substitution of
the transmitted features $\hat{\boldsymbol{y}_{\mathrm{f}}}$, which can
simultaneously defend against both types of attack.

_Random Permutation and Substitution_. We first introduce the random
permutation operation. For the transmitted features
$\boldsymbol{y}_{\mathrm{f}}$, we randomly permute the tensor along the first
dimension $h$. We define the permutation scheme $P$ as a random permutation of
the array $[0,1,...,h-1]$, where each element represents the index (0-indexed)
of $\boldsymbol{y}_{\mathrm{f}}$. After applying $P$, we obtain the permuted
transmitted features $\boldsymbol{y}^{\mathrm{p}}_{\mathrm{f}}$.

Then we perform the substitution operation on
$\boldsymbol{y}^{\mathrm{p}}_{\mathrm{f}}$ by swapping some of the $y_{i}$
with $y^{\prime}_{i}$ from another transmitted features
$\boldsymbol{y}^{\prime}_{\mathrm{f}}$, where $i$ in $y_{i}$ and
$y^{\prime}_{i}$ indicates that substitution is performed in the same position
of both transmitted features. Note that we remove the subscript f in $y_{i}$
to avoid complex notations. If Alice sends several images to Bob,
$\boldsymbol{y}^{\prime}_{\mathrm{f}}$ can be the transmitted features of the
next image. If Alice only sends one image to Bob,
$\boldsymbol{y}^{\prime}_{\mathrm{f}}$ can be a random-noise tensor, which
will also be sent to Bob. Similarly, we define the substitution scheme $S$ as
a sub-array of the array $[0,1,...,h-1]$, where every $i$ in $S$ indicates the
$y_{\mathrm{f},i}$ that should be substituted. After substitution,
$\boldsymbol{y}^{\mathrm{p}}_{\mathrm{f}}$ becomes
$\boldsymbol{y}^{\mathrm{s}}_{\mathrm{f}}$.

Figure 2: An example of the proposed defense method.

We give an example to explain the idea of random permutation and substitution.
As shown in Fig. 2, assume that $h=5$, then
$\boldsymbol{y}_{\mathrm{f}}=[y_{0},y_{1},...,y_{4}]$, where
$y_{i}\in\mathbb{R}^{1\times w\times c}$, $i=0,1,...,4$. We also assume that
there is another transmitted features
$\boldsymbol{y}^{\prime}_{\mathrm{f}}=[y^{\prime}_{0},y^{\prime}_{1},...,y^{\prime}_{4}]$
If $P=[4,2,1,0,3]$ for $\boldsymbol{y}_{\mathrm{f}}$ and
$P^{\prime}=[0,3,4,1,2]$ for $\boldsymbol{y}^{\prime}_{\mathrm{f}}$, then
$\boldsymbol{y}^{\mathrm{p}}_{\mathrm{f}}=[y_{4},y_{2},y_{1},y_{0},y_{3}]$ and
${\boldsymbol{y}^{\prime}}^{\mathrm{p}}_{\mathrm{f}}=[y^{\prime}_{0},y^{\prime}_{3},y^{\prime}_{4},y^{\prime}_{1},y^{\prime}_{2}]$.
If $\boldsymbol{y}_{\mathrm{f}}$ and $\boldsymbol{y}^{\prime}_{\mathrm{f}}$
share a common $S=[0,2]$, then
$\boldsymbol{y}^{\mathrm{s}}_{\mathrm{f}}=[y^{\prime}_{0},y_{2},y^{\prime}_{4},y_{0},y_{3}]$
and
${\boldsymbol{y}^{\prime}}^{\mathrm{s}}_{\mathrm{f}}=[y_{4},y^{\prime}_{3},y_{1},y^{\prime}_{1},y^{\prime}_{2}]$.
Suppose that Bob knows the $P$ and $S$ before transmission. After reshaping
$\hat{\boldsymbol{y}}$, Bob will first recover
$\hat{\boldsymbol{y}_{\mathrm{f}}}$ from
$\hat{\boldsymbol{y}}^{\mathrm{s}}_{\mathrm{f}}$ and then feed
$\hat{\boldsymbol{y}_{\mathrm{f}}}$ into the channel decoder. However, since
Eve does not know $P$ and $S$, Eve will try to reconstruct
$\hat{\boldsymbol{x}}_{\mathrm{e}}$ directly from
$\hat{\boldsymbol{y}}^{\mathrm{s}}_{\mathrm{f}}$, which is shown to be
infeasible in IV-C. Moreover, since different $P$ and $S$ are used for each
transmission, it would be difficult for Eve to determine the correct $P$ and
$S$ for each eavesdropped signal.

_Scheme Selection_. The proposed method is dependent on Bob having knowledge
of $P$ and $S$ before transmission. Hence it is necessary for Alice and Bob to
share two common sets of schemes, namely the permutation scheme set
$\mathbb{P}$ and the substitution set $\mathbb{S}$, which are kept secret from
Eve. Both sets comprise multiple schemes that can be employed for permutation
and substitution. Before each image transmission, Alice generates a value pair
$V=\\{p,s\\}$, which is used to select the corresponding $P$ and $S$ from
$\mathbb{P}$ and $\mathbb{S}$, respectively. $V$ is first encrypted using a
secret key $K$ shared between Alice and Bob. Then the encrypted $V$ is
transmitted to Bob, which cannot be modified by the main channel. Hence error-
free techniques such as error correction and retransmission are utilized to
transmit $V$. After receiving the $\hat{\boldsymbol{y}_{\mathrm{s}}}$ and the
encrypted $V$, Bob decrypts $V$ using $K$ and determines $P$ and $S$, from
which $\hat{\boldsymbol{y}}$ can be recovered.

Figure 3: Visualization of MIEA for the white-box attack and the black-box
attack under different channel conditions. For each channel condition, the
eavesdropped image by the white-box attack is displayed on the left and the
one obtained by the black-box attack is on the right.

##  IV Evaluations

In this section, we present our experiments to evaluate MIEA and the proposed
defense method. We first evaluate MIEA’s performance for the white-box attack
and black-box attack. We then show the effectiveness of the proposed defense
method. We use the semantic communication model DeepJSCC[14] to transmit
images from the CelebA dataset[15], which we crop and resize to $180\times
180$ in the evaluation. The semantic encoder and decoder each have four
convolutional layers, while the channel encoder and decoder have one
convolutional layer. We assume both the main channel and eavesdropper channel
to be AWGN channel and denote the channel condition as the combination of the
main channel’s SNR and the eavesdropper channel’s SNR. Although the main
channel is not considered in MIEA, we still perform evaluation under different
SNRs of the main channel, as we use different DeepJSCC models for each SNR
value. For each evaluation, we consider the SNR of both channels to be 0dB,
10dB and 20dB, resulting in nine different channel conditions.

###  IV-A Evaluation Setup

Before evaluating the performance of both attacks, we train the DeepJSCC model
on the CelebA dataset using three different SNR values for the main channel
(0dB, 10dB, and 20dB), resulting in three distinct DeepJSCC models. As stated
in [14], the SNR value determines the standard deviation of
$\boldsymbol{n}_{\mathrm{m}}$ when the transmission power is normalized to 1.
We train the CelebA dataset with a batch size of 128, using Adam [16] as the
optimizer with a learning rate of $10^{-3}$.

To measure the image quality, we use two metrics, i.e., the structural
similarity index measure (SSIM) and the peak signal-to-noise ratio (PSNR)[14],
where higher values of SSIM and PSNR indicate better quality.

###  IV-B Evaluation of MIEA

We first evaluate MIEA for the two types of attack. For the white-box attack,
where Eve reconstructs the image by minimizing (5), we employ Adam[16] as the
optimizer with a learning rate of $10^{-3}$ and we initialize
$\hat{\boldsymbol{x}}_{\mathrm{e}}$ to an all-zero tensor. For the black-box
attack, we use an inverse network $f^{-1}(\cdot)$ consisting of an upsampling
layer and two convolution layers. Then we train $f^{-1}(\cdot)$ by solving the
optimization problem in (6), where we choose the CelebA test dataset as
$\mathbb{X}$ and obtain its corresponding transmitted symbols $\mathbb{Y}$.
Similarly, we use Adam as the optimizer and set the learning rate to
$10^{-3}$.

Fig. 3 shows the performance of MIEA on the DeepJSCC model under different
channel conditions, with the SSIM and PSNR given below each image. The first
two columns in Fig. 3 are baselines for comparisons with the images
eavesdropped by MIEA, where the first column displays the original images
$\boldsymbol{x}$ transmitted by Alice, and the second column shows the images
received by Bob under different main channel’s SNRs. As shown in the first two
columns, increasing the SNR improves image quality, as indicated by higher
average SSIM and PSNR values. Additionally, higher SNRs reveal more details in
the images, such as the female’s hair.

The remaining columns in Fig. 3 display the eavesdropped images obtained by
MIEA. In the following evaluations in this paper, for each channel condition,
we show the eavesdropped image by the white-box attack on the left and the one
obtained by the black-box attack on the right. Additionally, TableI lists the
average SSIM and PSNR of the eavesdropped images of individuals selected from
the CelebA training set. The first row in the table shows the quality of the
images received by Bob, and for each channel condition, the values on the top
show the quality of eavesdropped images obtained by the white-box attack, and
the values on the bottom show the quality of the eavesdropped images obtained
by the black-box attack. We note that we choose CelebA training set for
evaluation because the CelebA test set is used for training $f^{-1}(\cdot)$ in
the black-box attack. It can be seen from Fig. 3 and Table I that the quality
of eavesdropped images improves as the SNR of the eavesdropper channel
increases for a given SNR of the main channel. Moreover, for a given SNR of
the eavesdropper channel, the quality of eavesdropped images is similar under
different SNRs of the main channel. It also can be observed that the SSIM and
PSNR values in the black-box attack are generally larger than those in the
white-box attack. This is because the black-box attack requires training
$f^{-1}(\cdot)$ before reconstructing any image from the eavesdropped signal,
which needs many samples from $\mathbb{X}$ and $\mathbb{Y}$. In contrast, the
white-box attack directly reconstructs the image from the eavesdropped signal
without any training in advance. Although the SSIM and the PSNR of the
eavesdropped images in both attacks are lower than those of the images
received by Bob, the eavesdropped images are visually recognizable and their
privacy is compromised, which confirms the effectiveness of MIEA and reveals
the risk of privacy leaks in current semantic communication.

TABLE I: The average SSIM and PSNR of the eavesdropped images under different
channel conditions

| Main Channel SNR
---|---
| 0dB | 10dB | 20dB
| Reconstructed
---
Images by Bob
30.02dB / 0.70 | 32.28dB / 0.79 | 33.28dB / 0.80
EC1 SNR 0dB | 17.44dB / 0.31 | 16.69dB / 0.30 | 18.58dB / 0.32
21.65dB / 0.46 | 22.24dB / 0.46 | 22.85dB / 0.46
EC SNR 10dB | 18.58dB / 0.40 | 17.80dB / 0.40 | 20.56dB / 0.43
23.50dB / 0.53 | 23.76dB / 0.54 | 24.33dB / 0.56
EC SNR 20dB | 18.74dB / 0.43 | 17.94dB / 0.42 | 20.84dB / 0.46
23.88dB / 0.57 | 23.98dB / 0.59 | 24.41dB / 0.61

  * 1

EC refers to the eavesdropper channel.

###  IV-C Evaluation of the Proposed Defense Method

Next, we evaluate the proposed defense method by repeating the evaluation of
MIEA in section IV-B with the defense method implemented on
$\boldsymbol{y}_{\mathrm{f}}$.

Fig. 4 visualizes the eavesdropped images by the two attack types after
applying the proposed defense method, using the same individual as in Fig. 3.
It can be observed that the eavesdropped images are visually unrecognizable,
demonstrating the effectiveness of the proposed defense method in preventing
Eve from eavesdropping on raw images. We can also see that the contour of the
female in the white-box attack is less obvious than that in the black-box
attack, suggesting that the defense against the white-box attack is superior
to that against the black-box attack. This is because that Eve has no prior
knowledge of the defense method when performing the white-box attack, whereas
$f^{-1}(\cdot)$ used in the black-box attack has learned some knowledge of the
defense method from the training samples.

Figure 4: Visualization of MIEA for both attacks under different channel
conditions after applying the proposed method. TABLE II: The average SSIM and
PSNR of the eavesdropped images by MIEA after defense

EC MC1 | 0dB | 10dB | 20dB
---|---|---|---
0dB | 8.03dB / 0.02 | 8.74dB / 0.02 | 6.94dB / 0.00
11.36dB / 0.11 | 11.41dB / 0.07 | 12.51dB / 0.07
10dB | 8.55dB / 0.04 | 7.70dB / -0.01 | 9.07dB / 0.05
11.72dB / 0.16 | 11.34dB / 0.11 | 13.22dB / 0.13
20dB | 8.02dB / 0.03 | 13.31dB / 0.09 | 8.39dB / 0.02
12.59dB / 0.21 | 11.55dB / 0.10 | 11.54dB / 0.07

  * 1

MC refers to the main channel.

In addition, we provide the average SSIM and PSNR of the eavesdropped images
for both attacks in Table. II. For a given SNR of the main channel, the SSIM
and PSNR do not increase as the SNR of the eavesdropper channel increases
because different $P$ and $S$ are used for different transmitted features. The
average SSIM and PSNR of the eavesdropped images by the black-box attack are
larger than those by the white-box attack, which is consistent with the
observation from Fig. 3. Overall, the average SSIM and PSNR are relatively
small, which indicates the effectiveness of the proposed defense method in
preventing Eve from obtaining meaningful information from the eavesdropped
signal.

Figure 5: Visualization of MIEA for both attacks under different channel conditions after applying only the random permutation. TABLE III: The average SSIM and PSNR of the eavesdropped images when applying only random permutation EC MC | 0dB | 10dB | 20dB
---|---|---|---
0dB | 8.05dB / 0.02 | 8.73dB / 0.06 | 8.50dB / 0.02
14.00dB / 0.22 | 11.43dB / 0.10 | 13.25dB / 0.13
10dB | 8.32dB / 0.06 | 8.02dB / 0.03 | 8.87dB / 0.01
14.55dB / 0.26 | 12.97dB / 0.19 | 13.19dB / 0.12
20dB | 8.36dB / 0.04 | 8.05dB / 0.02 | 8.77dB / 0.07
14.97dB / 0.28 | 12.83dB / 0.18 | 12.83dB / 0.13

Next, we conduct an ablation study to further validate our proposed method.
Fig. 5 and Table. III demonstrate the eavesdropped images and the average SSIM
and PSNR for both attacks by applying only the random permutation. As shown in
Fig. 5, when only the random substitution is applied, the white-box attack can
be effectively defended, while the black-box attack can still reconstruct
visually recognizable images for some $P$ and $S$. Moreover, the average SSIM
and PSNR for the black-box attack in Table. III are larger than those in
Table. II, which demonstrates that only random permutation is insufficient for
defending against MIEA.

Fig. 6 and Table. IV show the related results for both attacks by applying
only the random substitution. For both attacks, most of the eavesdropped
images are visually recognizable, indicating that the attacker can still
obtain sensitive information from the transmitted symbols, even though some of
the semantic features have been substituted. The average SSIM and PSNR in
Table. IV are larger than those in Table. III, which means that the random
permutation is more effective than random substitution in defending against
MIEA. From the ablation study, we can observe that the proposed defense method
outperforms both the random-permutation-based and random-substitution-based
defense methods, demonstrating that both permutation and substitution are
essential for the effectiveness of the proposed defense method.

Figure 6: Visualization of MIEA for both attacks under different channel conditions after applying only the random substitution. TABLE IV: The average SSIM and PSNR of the eavesdropped images when applying only random substitution EC MC | 0dB | 10dB | 20dB
---|---|---|---
0dB | 8.99dB / 0.14 | 8.91dB / 0.10 | 15.80dB / 0.16
16.06dB / 0.28 | 15.00dB / 0.25 | 14.70dB / 0.20
10dB | 15.62dB / 0.19 | 9.16dB / 0.14 | 10.43dB / 0.16
14.68dB / 0.26 | 14.49dB / 0.26 | 15.80dB / 0.27
20dB | 12.68dB / 0.18 | 15.35dB / 0.18 | 16.13dB / 0.21
15.78dB / 0.30 | 14.53dB / 0.27 | 17.29dB / 0.28

##  V Conclusion

In this paper, we propose MIEA to expose privacy risks in semantic
communication. MIEA enables an attacker to eavesdrop on the transmitted
symbols through an eavesdropper channel and reconstruct the raw message by
inverting the DL model employed in the semantic communication system. We
consider MIEA under the white-box attack and the black-box attack and propose
a novel defense method based on random permutation and substitution to defend
against both types of attack. In our evaluation, we first examine MIEA for
both attacks under various channel conditions. We then conduct experiments and
an ablation study to demonstrate the effectiveness of our proposed defense
method.

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# Complexity of non-trivial sound speed in inflation

Lei-Hua Liu<EMAIL_ADDRESS>Department of Physics, College of
Physics, Mechanical and Electrical Engineering, Jishou University, Jishou
416000, China  Ai-Chen Li<EMAIL_ADDRESS>Institut de Ciències del
Cosmos, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat
de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain

###### Abstract

In this paper, we study the impact of non-trivial sound on the evolution of
cosmological complexity in inflationary period. The vacuum state of curvature
perturbation could be treated as squeezed states with two modes, characterized
by the two most essential parameters: angle parameter $\phi_{k}$ and squeezing
parameter $r_{k}$. Through $Schr\ddot{o}dinger$ equation, one can obtain the
corresponding evolution equation of $\phi_{k}$ and $r_{k}$. Subsequently, the
quantum circuit complexity between a squeezed vacuum state and squeezed states
are evaluated in scalar curvature perturbation with a type of non-trivial
sound speed. Our results reveal that the evolution of complexity at early
times shows the rapid solution comparing with $c_{S}=1$, in which we implement
the resonant sound speed with various values of $\xi$. In these cases, it
shows that the scrambling time will be lagged with non-vanishing $\xi$.
Further, our methodology sheds a new way of distinguishing various
cosmological models.

##  I Introduction

With the development of $\rm AdS/CFT$ Maldacena:1997re , the understanding of
the nature of spacetime has been made significant progress, in which a bulk
gravitational theory is equivalent to a CFT at its corresponding boundary.
This conjecture can be dubbed as a particular realization of the holographic
principle, consequently one can consider that gravity is an emergent
phenomenon not as a fundamental force in nature. For the further development
of this methodology, another well-known conjecture was proposed that the
spactime comes via quantum entanglement VanRaamsdonk:2010pw , especially from
a realizion called $\rm ER=EPR$ Maldacena:2013xja . In light of this logic, it
motivates us for exploring the nature of spactime from the perspective of
quantum entanglement Swingle:2014uza ; Faulkner:2013ica ; Bianchi:2012ev ;
Lashkari:2013koa ; Balasubramanian:2013lsa . Thus, holographic entanglement
has become a key step for investigating the essence of spacetime. However,
Ref. Stanford:2014jda  found that a boundary QFT will reach a thermal
equilibrium within a short time, the evolution of its corresponding wormhole
will cost much longer time comparing with QFT. Therefore, it motives that
there is another quantum quantity named by complexity for describing the
evolution of wormhole Hartman:2013qma ; Liu:2013iza . More precisely, this
evolution can be depicted by holographic complexity. For calculating
complexity in light of holographic principle, Susskind and his collaborators
have proposed two conjectures: CV conjecture (complexity=volum) and CA
conjecture (complexity=action), in which the first one is translated into the
computation of maximum volume of the wormhole in bulk space and the second is
referred as the computation of action evaluated on a bulk subregion called
Wheeler-DeWitt patch. After that, extensive studies of such theories have
produced many profound results, see for example Cai:2016xho ; Carmi:2017jqz ;
Carmi:2016wjl ; Reynolds:2016rvl ; Chapman:2018dem ; Chapman:2018lsv ;
Reynolds:2017lwq ; Li:2020ark .

Motivated by the AdS/CFT, a hotspot research direction in the field of high
energy physics is to investigate the physics of circuit complexity from
viewpoint of quantum field theory. Currently, there are mainly two methods for
calculating the complexity: a). Nielsen $\rm et~{}al.$ are pioneering to
propose a geometrical method for computing the complexity in phase space of
quantum gates Nielsen2006AGA ; Nielsen2006QuantumCA ; Dowling2008TheGO ; b).
Under the framework of information geometry, one can use the ”Fubini-Study
distance” to proceed Chapman:2017rqy . According to these two methods, one can
use the wave functions in position space to explicitly obtain the complexity
Jefferson:2017sdb ; Bhattacharyya:2018bbv ; Guo:2018kzl . One can also use the
covariance matrix to calculate the complexity Chapman:2018hou ;
Hackl2018CircuitCF ; Alves2018EvolutionOC ; Camargo2019ComplexityAA ;
Ali2020PostquenchEO ; Khan:2018rzm . Although the implication of complexity in
high energy physics is still elementary, there are some attempts are
investigated, such as the definition of complexity in QFT
Jefferson2017CircuitCI . Even the Hawking radiation can be treated by this
kind of quantum circuit complexity Zhao2019AQC ; Brown2019ThePL . More
striking, the spacetime in some sense can be interpreted as the quantum
circuit complexity Chandra2021SpacetimeAA .

In light of Nielson’s geometrical method Nielsen2006AGA , it can be
implemented into the early Universe, namely for the inflationary epoch. The
essence of curvature perturbation is quantum perturbation, however, we are not
capable of distinguishing that it is quantum or classical from observations.
The standard procedure to deal with the circuit complexity is utilizing
squeezed state satisfied with the minimal uncertain relation, thus it
naturally utilizes this squeezed state as a ground state of curvature
perturbation. Ref. Bhargava2020QuantumAO  investigates the relation between
the bounce Universe and complexity, in which they found that the post bounce
was most signigicant and also gave a rough estimation of scrambling time
scale. Similar work has been studied in Lehners2020QuantumCC , in which more
inflationary models are studied and they found that the complexity in the
matter dominant period (MD) is growing fastest. Once taking effects of
background expansion into account Bhattacharyya:2020rpy , it is possible to
study the evolution of circuit complexity for curvature perturbation where
they found that the inflationary period has the most simple linear relation of
complexity evolution and its value will be enhanced in later Universe. From
another aspect, the complexity could be dubbed as an integral part of web of
diagnostics for quantum chaos Roberts2016ChaosAC ;
Balasubramanian2019QuantumCO ; Ali2020ChaosAC ; Yang2019TimeEO ;
Bhattacharyya2019TowardsTW ; Barbon2019OnTE  leading to the essential
information about the scrambling time and Lyapunov exponent $\rm etc$
Ali2020ChaosAC , in which it will provide the new perspective for exploring
the cosmology.

For the cosmological field, it also motives us to explore the quantum
complexity of quantum cosmology, especially for the very early universe
Bhattacharyya:2020rpy . In Bhattacharyya:2020rpy ; Bhargava2020QuantumAO , the
quantum circuit complexity has been calculated for the scalar curvature
perturbations in simple inflation model with constant sound speed, i.e.
$c^{2}_{S}=1$. Alternatively, it is valuable to evaluate the deviation from
the trivial should speed, we will consider the effects from non-trivial sound
speed into the quantum circuit complexity, especially for the different
partial differential equations of parameters of squeezed state also comparing
with Ref. Bhattacharyya:2020rpy ; Bhargava2020QuantumAO . Thus, our purpose is
to study the effects of varying sound speed on the evolution of quantum
circuit complexity in background of inflationary de-Sitter spacetime. In
particular, we are interested in a type of sound speed resonance (SSR)
Cai:2018tuh . This mechanism is of significant importantce since it can be
realized in various aspects for enhancing the power spectrum for curvature
perturbation and primordial gravitational waves Cai:2020ovp ; Zhou:2020kkf ;
Cai:2021yvq . On the other hand, the very recent Refs. Li:2021kfq ; Li:2021sro
has investigated the quantum curcuit complexity in k-essence inflation and BPS
states, it shows the very different evolution of curcuit complexity. In order
to capture the various chaotic features of SSR, the investigations of its
evolution of complexity will lead to distinguish various cosmological models.

This paper is organized as follows. In section II, we will review the non-
trivial sound speed $c_{S}^{2}$ from the perspective of resonant production of
PBH. In section III, the squeezed state of cosmological perturvations will be
given, in which there exists two parameters for this squeezed states. In
section IV, the complexity of these cosmological perturbation will be obtained
under the framework of geometrical method via Nielson’s work. In section V, we
give our main conclusions and discussions.

##  II The effects of non-trivial sound speed

Before recalling the non-trivial sound speed, we will consider the Friedmann-
Lemaitre-Robertson-Walker background metric considered as our working metric,

$ds^{2}=a(\eta)^{2}(-d\eta^{2}+d\vec{x}^{2}),$ (1)

where $a(\eta)$ is scale factor in conformal time and $\vec{x}$ denotes the
spatial vector. Under this background, we could define the perturbation of
some scalar field in inflationary period
$\phi(x_{\mu})=\phi_{0}(\eta)+\delta\phi(x_{\mu})$ with its corresponding
metric defined by

$ds^{2}=a(\eta)^{2}\bigg{(}-(1+\psi(\eta,x))d\eta^{2}+(1-\psi(\eta,x))dx^{2}\bigg{)},$
(2)

where $\psi(\eta,x)$ is the perturbation of metric. Being armed these two
metrics and the perturbations of some scalar field, the perturbed action can
be denoted by the curvature perturbation

$S=\frac{1}{2}\int
dtd^{3}xa^{3}\frac{\dot{\phi}^{2}}{H^{2}}\bigg{[}\dot{\mathcal{R}}^{2}-\frac{1}{a^{2}}(\partial_{i}\mathcal{R})^{2}\bigg{]},$
(3)

where $H=\frac{\dot{a}}{a}$,
$\mathcal{R}=\psi+\frac{H}{\dot{\phi}_{0}}\delta\phi$. Action (3) can be
transferred into canonical normalized scalar field in light of the Mukhanov
variable $v=z\mathcal{R}$ where $z=a\sqrt{2\epsilon}$ with
$\epsilon=-\frac{\dot{H}}{H^{2}}=1-\frac{\mathcal{\mathcal{H}^{\prime}}}{\mathcal{H}^{2}}$,

$S=\frac{1}{2}\int d\eta d^{3}x\bigg{[}v^{\prime
2}-(\partial_{i}v)^{2}+\frac{z^{\prime}}{z}v^{2}-2\frac{z^{\prime}}{z}v^{\prime}v\bigg{]},$
(4)

where prime implies that the derivative with respect to the conformal time
$\eta$ even for $\mathcal{H}$. In action (4), it clearly indicates the
perturbation of curvature is of trivial sound speed namely, $c_{S}=1$. To
evaluate the influence of non-trivial sound speed for the complexity, we will
introduce a simple mechanism of production of PBH from a resonant sound speed.
In this new mechanism, the key ingredient for producing the enhanced value of
curvature perturbation is the non-trivial sound speed $c_{S}$. For a general
non-trivial sound speed ArmendarizPicon:1999rj ; Garriga:1999vw , it is
defined by a canonical variable $v=z\zeta$ with $z=\sqrt{2\epsilon}a/c_{S}$
where $\epsilon$ has the same definition as the previous discussions. For the
canonical variable, its corresponding Fourier mode $v_{k}$ satisfied with the
following equation of motion,

$v_{k}^{\prime\prime}+(c_{S}^{2}k^{2}-\frac{z^{\prime\prime}}{z})v_{k}=0,$ (5)

This equation of motion is our starting point for constructing the model. If
the sound speed $c_{S}$ is one, it will nicely recover eom of curvature
perturbation for slow-roll single-field inflation. As for the deviation from
one of $c_{S}$, it experiences the non-canonical kinetic term, in which the
inflation embedded into string theory Alishahiha:2004eh ; Silverstein:2003hf .
Meanwhile, this mechanism can be realized in DBI inflation Chen:2020uhe . From
the perspective of effective field theory, when integrating out the heavy
modes, it could also lead to the non-trivial sound speed $c_{S}$
Achucarro:2012sm .

Since the amplification of curvature perturbation is highly relevant with this
non-trivial $c_{S}$. In order to realize the formation of PBH, the ehanced
power spectrum at some certain scales is necesseary dubbed as one kind of
mechanism. To characterize this enhanced power spectrum, Ref. Cai:2018tuh  has
proposed the sound speed is of the following form,

$c_{S}^{2}=1-2\xi[1-\cos(k_{*}\tau)],$ (6)

where $\xi$ is the amplitude of oscillation and $k_{*}$ is the frequency of
oscillation. The motivation for exploring this specific sound speed is that it
could reveal rich phenomelogical implications, in which this mechanism can be
realized in various theoretical models, such DBI inflation, even for the
multifield inflation, $\it e.t.c$. Due to its extensive implication, we dub it
as an important example of non-trivial sound speed to explore the evolution of
curcuit complexity. Combining eq. (6) and eq. (5) and keeping the first order
of $\xi$ for $\frac{z^{\prime\prime}}{z}$ since the value of $\xi$ is tiny
compared with one. Subsequently, changing variable as $x=k_{*}\tau$, then eq.
(5) will become Mathieu equation,

$\frac{d^{2}v_{k}}{dx^{2}}+(A_{k}-2q\cos 2x)v_{k}=0$ (7)

where $A_{k}=k^{2}/k_{*}^{2}+2q-4\xi$ and $q=2\xi-(k^{2}/k_{*}^{2})\xi$. There
is a key feature of the Mathieu equation, since it exists an instability
region of eq. (7) depicting its exponential growth, in which it can be read by

$v_{k}\propto\exp(\xi k_{*}\tau/2).$ (8)

This exponential growth could lead to the amplification of curvature
perturbation. Consequently, it will realize the generation of PBH during
inflation. Observing that eq. (7) belongs to the narrow resonance due to the
small value of $\xi$. And the resonance only occurs as $k=nk_{*}$ with $n$ is
an integer. Lastly, we will evaluate the deviation from the standard sound
speed, particularly the information from $\xi$.

Figure 1: It shows the resonant sound speed varying with scale factor
$a(\tau)$ with various values of $\xi$. For a resonable input of $\xi$, we set
its value are $0,~{}0.1,~{}0.04,~{}0.08$, respectively.

In figure (1), it clearly indicates that the deviation from the standard sound
speed ($c_{S}=1$) varying with scale factor $a(\tau)$, in which it tells that
there exists an oscillation at the very beginning of inflation ($a\approx
0.012$) and meanwhile this oscillation will be larger as enhancing the value
of $\xi$, where we have used $\eta=-\frac{1}{Ha}$. From figure (1), we could
also know that the parameter $\xi$ cannot be large due to the constraints of
deviation from standard sound speed. Moreover, we will utilize this parameter
to investigate its effects to the cosmological complexity.

##  III The squeezed quantum states for cosmological perturbations

In this paper, we will proceed with the complexity in inflationary period,
namely for the quantum fluctuations. The complexity are usually depicting the
chaotic and unstable feature of a statistical system. In the quantum level,
the most simple and applicable system is the so-called inverted harmonic
oscillator, in which its Hamiltonian is
$H=\frac{1}{2}p^{2}-\frac{1}{2}k^{2}x^{2}$ where $p$ is the momentum and $k$
is the frequency of oscillator. This inverted oscillator is an appropriate
approximation for describing one chaotic and unstable system. Meanwhile, one
could observe that the Hamiltonian operator of an cosmological system in
momentum space is of the same mathematical structure as this inverted
oscillator. This the reason we could use the methodology in quantum system
into the cosmological system. The details of calculating the cosmological
complexity are showing in Section IV. From another aspect, the wave function
of the inverted quantum harmonic oscillator is the Gaussian wave function,
thus the squeezed state is a natural way for realizing the evolution of the
inflationary quantum curcuit.

For convenience, we will use the Mukhanov variable for depicting the action.
To be more precise, EOM (5) is our starting point for calculating the
complexity. To obtain EOM (5), its corresponding action can be explicitly
written as

$\displaystyle S=\int d\eta L=\frac{1}{2}\int d\eta d^{3}x\bigg{(}v^{\prime
2}-c_{S}^{2}(\partial_{i}v)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\big{(}\frac{z^{\prime}}{z}\big{)}^{2}v^{2}-2\frac{z^{\prime}}{z}v^{\prime}v\bigg{)}$
(9)

Here, the expression of $c^{2}_{s}$ is given by $\eqref{non trivial sound
speed}$. The canonical momentum is obtained from $\eqref{PerCano}$

$\displaystyle\pi(\eta,\vec{x})=\frac{\delta L}{\delta
v^{\prime}(\eta,\vec{x})}=v^{\prime}-\frac{z^{\prime}}{z}v$ (10)

And then the Hamiltonian $H=\int d^{3}x(\pi v^{\prime}-\mathcal{L})$ is

$\displaystyle H=\frac{1}{2}\int
d^{3}x\big{[}\pi^{2}+c_{S}^{2}(\partial_{i}v)^{2}+\frac{z^{\prime}}{z}(v\pi+\pi
v)\big{)}\big{]}$ (11)

Promoting the Mukhanov variable $v$ to the quantum field and decomposing it
into Fourier modes

$\displaystyle\hat{v}(\eta,\vec{x})=\int\frac{d^{3}k}{(2\pi)^{3/2}}\frac{1}{\sqrt{2k}}\big{(}\hat{c}_{-\vec{k}}^{\dagger}v_{k}^{\star}(\eta)+\hat{c}_{\vec{k}}v_{k}(\eta)\big{)}e^{i\vec{k}\cdot\vec{x}}$
(12)

Meanwhile, the corresponding $\hat{\pi}(\eta,\vec{x})$ is

$\displaystyle\hat{\pi}(\eta,\vec{x})=i\int\frac{d^{3}k}{(2\pi)^{3/2}}\sqrt{\frac{k}{2}}\big{(}\hat{c}_{-\vec{k}}^{\dagger}u_{k}^{\star}(\eta)-\hat{c}_{\vec{k}}u_{k}(\eta)\big{)}e^{i\vec{k}\cdot\vec{x}}$
(13)

where $\hat{c}_{-\vec{k}}^{\dagger}$ and $\hat{c}_{\vec{k}}$ represent the
creation and annihilation operators respectively. By choosing an appropriate
normalization condition for mode functions $u_{k}(\eta),~{}v_{k}(\eta)$, one
can give the following Hamiltonian

$\displaystyle\hat{H}=\int d^{3}k\hat{\mathcal{H}}_{k}=\int
d^{3}k\big{\\{}\frac{k}{2}(c_{S}^{2}+1)\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}$
$\displaystyle\quad~{}~{}+\frac{k}{2}(c_{S}^{2}+1)\hat{c}_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\big{(}\frac{k}{2}(c_{S}^{2}-1)+\frac{iz^{\prime}}{z}\big{)}\hat{c}_{\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}^{\dagger}$
$\displaystyle\quad~{}~{}+\big{(}\frac{k}{2}(c_{S}^{2}-1)-\frac{iz^{\prime}}{z}\big{)}\hat{c}_{\vec{k}}\hat{c}_{-\vec{k}}\big{\\}}$
(14)

In case of $c^{2}_{S}=1$, the $\eqref{HamWithCs}$ reduces to a Hamiltonian
which is similar to the form of the inverted harmonic oscillator
Barton1986QuantumMO . The unitary evolution operator acting on a state can be
parameterized in the factorized form Grishchuk:1990bj ;
Albrecht1994InflationAS

$\displaystyle\hat{\mathcal{U}}_{\vec{k}}(\eta,\eta_{0})=\hat{\mathcal{S}}_{\vec{k}}(r_{k},\phi_{k})\hat{\mathcal{R}}_{\vec{k}}(\theta_{k})$
(15)

In $\eqref{UniOperator}$, the $\hat{\mathcal{R}}_{\vec{k}}$ is the two-mode
rotation operator written in term of the rotation angle $\theta_{k}(\eta)$

$\displaystyle\hat{\mathcal{R}}_{\vec{k}}(\theta_{k})=\exp\big{[}-i\theta_{k}(\eta)\big{(}\hat{c}_{\vec{k}}\hat{c}_{\vec{k}}^{\dagger}+\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{-\vec{k}}\big{)}\big{]}$
(16)

Meanwhile, $\hat{\mathcal{S}}$ is the two-mode squeeze operator written in
term of the squeezing parameter $r_{k}(\eta)$ and the squeezing angle
$\phi_{k}(\eta)$ respectively.

$\displaystyle\hat{S}_{\vec{k}}(r_{k},\phi_{k})=\exp\big{[}r_{k}(\eta)\big{(}e^{-2i\phi_{k}(\eta)}\hat{c}_{\vec{k}}\hat{c}_{-\vec{k}}-e^{2i\phi_{k}(\eta)}\hat{c}_{-\vec{k}}^{\dagger}\hat{c}_{\vec{k}}^{\dagger}\big{)}\big{]}$
(17)

Since the rotation operators only give an irrelevant phase factor when acting
on the initial vacuum state, we will not include it in subsequent analysis. By
acting the squeeze operator on the two-mode vacuum state
$|0;0\rangle_{\vec{k},-\vec{k}}$, a two-mode squeezed state is obtained

$\displaystyle|\Psi\rangle_{sq}=\frac{1}{\cosh
r_{k}}\sum_{n=0}^{\infty}(-1)^{n}e^{2in\phi_{k}}\tanh^{n}r_{k}|n;n\rangle_{\vec{k},-\vec{k}}$
(18)

in which $|n;n\rangle_{\vec{k},-\vec{k}}$ represents the two-mode excited
state

$\displaystyle|n;n\rangle_{\vec{k},-\vec{k}}=\frac{1}{n!}\big{(}\hat{c}_{\vec{k}}^{\dagger}\big{)}^{n}\big{(}\hat{c}_{-\vec{k}}^{\dagger}\big{)}^{n}|0;0\rangle_{\vec{k},-\vec{k}}$
(19)

Together $\eqref{HamWithCs},\eqref{squeeState}$ with $Schr\ddot{o}dinger$
equation

$\displaystyle i\frac{d}{d\eta}|\Psi\rangle_{sq}=\hat{H}|\Psi\rangle_{sq}$
(20)

We give the differeitial equations which control the time evolution of the
squeezing paremeters $r_{k}(\eta)~{},\phi_{k}(\eta)$

$\displaystyle-\frac{dr_{k}}{d\eta}=\frac{k}{2}(c_{S}^{2}-1)\sin(2\phi_{k})+\frac{z^{\prime}}{z}\cos(2\phi_{k})$
(21)
$\displaystyle\frac{d\phi_{k}}{d\eta}=\frac{k(c_{S}^{2}+1)}{2}-\frac{k}{2}(c_{S}^{2}-1)\cos
2\phi_{k}\coth 2r_{k}$ $\displaystyle\quad\quad+\frac{z^{\prime}}{z}\sin
2\phi_{k}\coth 2r_{k}$ (22)

These two equations are difficult to solve analytically, thus we have to
consider the numerical solutions. For the sake of simplicity for calculation,
the variable $\log_{10}a$ are utilized to take place of the conformal time
$\eta$. Under this variable transformation, together with the expression
$\eqref{non trivial sound speed}$ and $z=\frac{\sqrt{2}\epsilon a}{c_{S}}$,
equations $\eqref{parameter eq}$ are expanded as

$\displaystyle\frac{10^{y}H_{0}}{\ln[10]}\cdot\frac{dr}{dy}=\big{(}\frac{2k_{\star}\xi\sin(\frac{2k_{\star}}{10^{y}H_{0}})}{1-2\xi+2\xi\cos(\frac{2k_{\star}}{10^{y}H_{0}})}-10^{y}H_{0}\big{)}\cdot\cos(2\phi_{k})$
$\displaystyle\quad\quad\quad~{}~{}+k\xi\big{(}1-\cos(\frac{2k_{\star}}{10^{y}H_{0}})\big{)}\cdot\sin(2\phi_{k})$
(23)
$\displaystyle\frac{10^{y}H_{0}}{\ln[10]}\cdot\frac{d\phi_{k}}{dy}=k\big{(}1-\xi\big{(}1-\cos(\frac{2k_{\star}}{H_{0}\cdot
10^{y}})\big{)}\big{)}$
$\displaystyle\quad\quad+\big{(}10^{y}H_{0}-\frac{2k_{\star}\xi\sin(\frac{2k_{\star}}{10^{y}H_{0}})}{1-2\xi+2\xi\cos(\frac{2k_{\star}}{10^{y}H_{0}})}\big{)}\cdot\sin
2\phi_{k}\coth 2r_{k}$
$\displaystyle\quad\quad+k\xi\big{(}1-\cos(\frac{2k_{\star}}{H_{0}\cdot
10^{y}})\big{)}\cdot\cos 2\phi_{k}\coth 2r_{k}$ (24)

in which we have denoted $y=\log_{10}a$. Being armed with this variable
transformation, one can obtain the numerical solution of $\phi_{k}$ and
$r_{k}$ depicted by figure (2).

Figure 2: The numerical solutions of $\phi_{k}$ and $r_{k}$ in terms of
$\log_{10}(a)$ with $\xi=0$, $\xi=0.01$, $\xi=0.04$ and $\xi=0.08$. Our plots
adopts $H_{0}=1$.

Figure (2) shows the evolution of $\phi_{k}$ and $r_{k}$ with various values
of parameter $\xi$. In the upper panel of figure (2), the trend of numerical
solutions of $r_{k}$ with various $\xi$ is almost the same, particularly for
the case of $\xi=0$ which will be recovered the solution of Ref.
Bhargava2020QuantumAO . $r_{k}$ plays an essential role in determining the
trend of complexity showing in figure (3). Combining with the analysis for
figure (3), our result indicates that linear growth for $\xi$ will be lagged
with non-vanishing $\xi$, precisely speaking that the linear growing part will
appear after horizon exit comparing with the case of $\xi=0$ in Ali2020ChaosAC
, in which the linear growing part of $\xi=0$ occurs at the horizon exit. This
is a huge difference for resulting in the evolution of complexity highly
relevant with the property of a quantum chaotic system. The second difference
comes via the oscillation behavior at the very early universe, it shows an
enhanced oscillation with increasing the value of $\xi$. Finally, all of these
cases will be approached a nearly the same slope leading to the same evolution
after all these cases exit the horizon. Generically, we recall that the
definition of resonant sound speed $c_{S}$, in which the frequency of
oscillation for resonant sound speed will also be enlarged as improving the
value of $\xi$ indicating in figure (1). In some sense, the fast oscillation
of non-trivial sound speed leads to a more rapid oscillation of $r_{k}$. As
for $\phi_{k}$, the only difference occurs that in the case of $\xi=0.8$,
$\phi_{k}$ will be faster approaching a constant comparing with other cases.
For a careful reader, it will be easily found that the constant will be
smaller as decreasing the value of $\xi$ until $\xi=0.8$ whose evolution will
show an obvious difference. Being armed with these two parameters, we will
analyze the evolution of complexity.

##  IV The complexity of cosmological squeezed states

In this paper, we will evaluate the circuit complexity by using Nielsen’s
method Nielsen2006AGA ; Nielsen2006QuantumCA ; Dowling2008TheGO . First, a
reference state $|\psi^{R}\rangle$ is given at $\tau=0$. And then, we suppose
that a target state $|\psi^{T}\rangle$ could be obtained at $\tau=1$ by acting
a unitary operator on $|\psi^{R}\rangle$, namely

$\displaystyle|\psi^{T}\rangle_{\tau=1}=U(\tau=1)|\psi^{R}\rangle_{\tau=0}$
(25)

As usual, $\tau$ parametrizes a path in the Hilbert space. Generally, the
unitary operator is constructed from a path-ordered exponential of a
Hamiltonian operator

$\displaystyle
U(\tau)=\overleftarrow{\mathcal{P}}\exp\bigg{(}-i\int^{\tau}_{0}dsH(s)\bigg{)}$
(26)

where the $\overleftarrow{\mathcal{P}}$ indicates right-to-left path ordering.
The Hamiltonian operator $H(s)$ can be expanded in terms of a basis of
Hermitian operators $M_{I}$, which are the generators for elementary gates

$\displaystyle H(s)=Y(s)^{I}M_{I}$ (27)

The coefficients $Y(s)^{I}$ are identified as the control functions that
determine which gate should be switched on or switched off at a given
parameter. Meanwhile, the $Y(s)^{I}$ satisfy the $Schr\ddot{o}dinger$ equation

$\displaystyle\frac{dU(s)}{ds}=-iY(s)^{I}M_{I}U(s)$ (28)

Then a $cost~{}functional$ is defined as follows

$\displaystyle\mathcal{C}(U)=\int_{0}^{1}\mathcal{F(U,\dot{U})}d\tau$ (29)

The complexity is obtained by minimizing the functional
$\eqref{costfunctional}$ and finding the shortest geodesic distance between
the reference and target states. Here, we restrict our attentions on the
$quadratic$ cost functional

$\displaystyle\mathcal{F}(U,Y)=\sqrt{\sum_{I}(Y^{I})^{2}}$ (30)

In this work, the target state is the two-mode squeezed vacuum state
$\eqref{squeeState}$. After projecting $|\Psi\rangle$ into the position space,
the following wavefunction is implied Martin:2019wta ; Lvovsky:2014sxa

$\displaystyle\Psi_{sq}(q_{\vec{k}},q_{-\vec{k}})=\sum_{n=0}^{\infty}(-1)^{n}e^{2in\phi_{k}}\frac{\tanh^{n}r_{k}}{\cosh
r_{k}}\langle q_{\vec{k}};q_{-\vec{k}}|n;n\rangle_{\vec{k},-\vec{k}}$
$\displaystyle~{}=\frac{\exp[A(r_{k},\phi_{k})\cdot(q_{\vec{k}}^{2}+q_{-\vec{k}}^{2})-B(r_{k},\phi_{k})\cdot
q_{\vec{k}}q_{-\vec{k}}]}{\cosh
r_{k}\sqrt{\pi}\sqrt{1-e^{-4i\phi_{k}}\tanh^{2}r_{k}}}$ (31)

in which the coefficients $A(r_{k},\phi_{k})$ and $B(r_{k},\phi_{k})$ are

$\displaystyle
A(r_{k},\phi_{k})=\frac{k}{2}\bigg{(}\frac{e^{-4i\phi_{k}}\tanh^{2}r_{k}+1}{e^{-4i\phi_{k}}\tanh^{2}r_{k}-1}\bigg{)}$
(32) $\displaystyle B(r_{k},\phi_{k})=2k\bigg{(}\frac{e^{-2i\phi_{k}}\tanh
r_{k}}{e^{-4i\phi_{k}}\tanh^{2}r_{k}-1}\bigg{)}$ (33)

By using a suitable rotation in vector space $(q_{\vec{k}},q_{-\vec{k}})$, the
exponent in $\eqref{squeeposition}$ could be rewritten by a form of diagonal
matrix

$\displaystyle\Psi_{sq}(q_{\vec{k}},q_{-\vec{k}})=\frac{\exp[-\frac{1}{2}\tilde{M}^{ab}q_{a}q_{b}]}{\cosh
r_{k}\sqrt{\pi}\sqrt{1-e^{-4i\phi_{k}}\tanh^{2}r_{k}}}$ (34)
$\displaystyle\tilde{M}=\left(\begin{array}[]{cc}\Omega_{\vec{k}^{\prime}}&0\\\
0&\Omega_{-\vec{k}^{\prime}}\end{array}\right)=\left(\begin{array}[]{cc}-2A+B&0\\\
0&-2A-B\end{array}\right)$ (39)

Meanwhile, the reference state is the unsqueezed vacuum state,

$\displaystyle\Psi_{00}(q_{\vec{k}},q_{-\vec{k}})=\langle
q_{\vec{k}};q_{-\vec{k}}|0;0\rangle_{\vec{k},-\vec{k}}$
$\displaystyle\quad\quad\quad\quad\quad~{}=\frac{\exp[-\frac{1}{2}(\omega_{\vec{k}}q_{\vec{k}}^{2}+\omega_{-\vec{k}}q_{-\vec{k}}^{2})]}{\pi^{1/2}}$
$\displaystyle\quad\quad\quad\quad\quad~{}=\frac{\exp[-\frac{1}{2}\tilde{m}^{ab}q_{a}q_{b}]}{\pi^{1/2}}$
(40) $\displaystyle\tilde{m}=\left(\begin{array}[]{cc}\omega_{\vec{k}}&0\\\
0&\omega_{-\vec{k}}\end{array}\right)$ (43)

Notice that (40) is the Guassian state related to the target state
corresponding to the squeezed state via Hamilton operator (26), in which this
operator will not change structure of the wavefunctions. Meanwhile, the
squeezed state is one kind of Gaussian state, thus the reference state should
also be a Gaussian state. According to the definition $\eqref{ComWaveFunc}$,
one can associate the target state $\eqref{targerstate}$ with the reference
state $\eqref{referstate}$ through a unitary transformation

$\displaystyle\Psi_{\tau}(q_{\vec{k}},q_{-\vec{k}})=\tilde{U}(\tau)\Psi_{00}(q_{\vec{k}},q_{-\vec{k}})\tilde{U}^{\dagger}(\tau)$
(44)
$\displaystyle\Psi_{\tau=0}(q_{\vec{k}},q_{-\vec{k}})=\Psi_{00}(q_{\vec{k}},q_{-\vec{k}})$
(45)
$\displaystyle\Psi_{\tau=1}(q_{\vec{k}},q_{-\vec{k}})=\Psi_{sq}(q_{\vec{k}},q_{-\vec{k}})$
(46)

where $\tilde{U}(\tau)$ is a $GL(2,C)$ unitary matrix which give the shortest
geodesic distance between the target state and the reference state in operator
space. As considered by the work Jefferson:2017sdb , $\tilde{U}(\tau)$ will
take the form

$\displaystyle\tilde{U}(\tau)=\exp[\sum_{k=1}^{2}Y^{k}(\tau)M_{k}^{diag}]$
(47)

where the $\\{M_{k}^{diag}\\}$ represent the 2 diagonal generators of
$GL(2,C)$

$\displaystyle M_{1}^{diag}=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)~{},~{}M_{2}^{diag}=\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)$ (52)

Note that the off-diagonal components are set to zero since they will increase
the distance between two states. The complex variables $\\{Y^{I}(\tau)\\}$ are
constructed as Ali:2018fcz

$\displaystyle Y^{I}(\tau)=Y^{I}(\tau=1)\cdot\tau+Y^{I}(\tau=0)$ (53)

From the boundary conditions $\eqref{boun1}$ and $\eqref{boun2}$, one obtains

$\displaystyle\text{Im}(Y^{1,2})\big{|}_{\tau=0}=\text{Re}(Y^{I})\big{|}_{\tau=0}=0$
(54)
$\displaystyle\text{Im}(Y^{1,2})\big{|}_{\tau=1}=\frac{1}{2}\ln\frac{|\Omega_{\vec{k},\vec{-k}}|}{\omega_{\vec{k},\vec{-k}}}$
(55)
$\displaystyle\text{Re}(Y^{1,2})\big{|}_{\tau=1}=\frac{1}{2}\arctan\frac{\text{Im}(\Omega_{\vec{k},\vec{-k}})}{\text{Re}(\Omega_{\vec{k},\vec{-k}})}$
(56)

And then the complexity could be rewritten as the following line length

$\displaystyle
C(\tilde{U})=\int^{1}_{0}d\tau\sqrt{G_{IJ}\dot{Y}^{I}(\tau)({\dot{Y}}^{J}(\tau))^{\star}}$
(57)

in which $G_{ij}$ is an induced metric for the group manifold. As indicated by
the work Jefferson:2017sdb , there exists an arbitrariness in term of the
choice of $G_{IJ}$. In the previous illustration, it clearly indicates that
$\tilde{U}(\tau)$ is of $GL(2,C)$ group structure, its corresponding geometry
in operator space is a flat geometry, namely, $G_{IJ}=\delta_{IJ}$ which means
that various group structures will lead to the different topology for its
induced metric. Here, we should emphasize that this induced metric is not the
metric for the spacetime but for the operator parameter minifold. Thus, the
complexity of eq. (57) can be applied into various cosmological background
when $\tilde{U}(\tau)$ has the $GL(2,C)$ group structure. Substitute
$\eqref{StraLineYtau}$ into $\eqref{CompleLineLeng}$, the expression for the
complexity is calculated as

$\displaystyle\mathcal{C}(k)=\frac{1}{2}\bigg{(}\big{(}\ln\frac{|\Omega_{\vec{k}}|}{\omega_{\vec{k}}}\big{)}^{2}+\big{(}\arctan\frac{\text{Im}(\Omega_{\vec{k}})}{\text{Re}(\Omega_{\vec{k}})}\big{)}^{2}$
$\displaystyle\quad\quad+\big{(}\ln\frac{|\Omega_{-\vec{k}}|}{\omega_{-\vec{k}}}\big{)}^{2}+\big{(}\arctan\frac{\text{Im}(\Omega_{\vec{-k}})}{\text{Re}(\Omega_{\vec{-k}})}\big{)}^{2}\bigg{)}^{1/2}$
(58)

Figure 3: Complexity is based on the numeical solution of $r_{k}$ and
$\phi_{k}$. These colorful lines are corresponding to the previous figure (2).
Our plots adopt $H_{0}=1$ and red curves for $\xi=0.08$, green curve for
$\xi=0.04$, dark blue for $\xi=0.01$, the light blue for $\xi=0$
(corresponding to $c_{S}^{2}=1$). The larger value leads to the faster
oscillation for complexity and will result in lagging the scrambling time. The
scrambling time is considered as the complexity begins to increase and the
Lyapunov index is identifield with the growing of complexity becomes linear as
mentioned in Ref Ali2020ChaosAC .

From figure (3), the complexity is explicitly obtained via the numerical
solution of $r_{k}$ and $\phi_{k}$, which indicates that the generic evolution
of complexity. Combining with the analysis for figure (2), the huge difference
comes via the linear growing part of the complexity. As Ali2020ChaosAC
indicated, a quantum chaotic system is characterizing by the circuit
complexity, especially for the time scale as the complexity begins to increase
can be identified with scrambling time, also Ref. Ali2020ChaosAC ;
Bhattacharyya:2020kgu  shows that the Lyapunov index can be approximately
equaled to he slope of the linear growth. In light of their investigations and
the above analysis showing in figure (2), the scrambling time will be lagged
behind the time of horizon exit when increasing the value of $\xi$. Another
difference is also for oscillation in the very early universe. Recalling that
the formation of PBH in Cai:2018tuh , it is generating at $k=nk_{*}$ in an
inflationary period, at late times we cannot observe the formation of PBH. As
a physical consequence, the corresponding evolution of complexity will not be
changed agreed to our numerical simulation in figure (2). As for the
inflationary period, we only consider the resonant sound speed whose dynamical
behavior will result in the oscillation of $c_{S}$ with the specific peak in
the very early universe. Also as a physical consequence, the evolution of
complexity will be oscillating faster in the very early universe. From the
perspective of a chaotic system, the scrambling time (a time scale of a
chaotic system) will be enhanced by adding the chaos in a chaotic system (the
formation of PBH in our model) which is also agreed with our numerical
simulation. Thus, it is reasonable to expect that our method could reveal a
more fruitful evolution of complexity with various non-trivial sound speed
inflationary models.

##  V Summary and discussion

The motivation to investigate the cosmological complexity is that it sheds a
new way for distinguishing various inflationary models. As its practical
realization, we evaluate the impact of non-trivial sound speed on the
evolution of cosmological complexity in light of Nielson’s geometrical method.
Under the framework of standard cosmological perturbation theory and combining
with the non-trivial sound speed, we utilize the squeezed state for
investigations of curvature perturbation in the inflationary period. To
achieve this goal, the most essential squeezed parameters are $r_{k}$ and
$\phi_{k}$ in the phase space, respectively. Following the procedure, we
obtain the evolution equation of $r_{k}$ and $\phi_{k}$ in terms of conformal
time depicted by Eq. (22), in which there is a non-trivial sound speed term,
in which we consider resonant sound speed as a special case. In a quantum
chaotic system, the chaos is characterizing by the scrambling time and
Lyapunov index. Meanwhile, Ref Ali2020ChaosAC  has indicated that the time
scale as the complexity begins to increase can be identified with scrambling
time, particularly, the Lyapunov index is dubbed as the slope as the growing
part of complexity becomes linear. In figure (3), our numerical simulation of
complexity has clearly shown that the scrambling time will be lagged as
enhancing the value of $\xi$, in which it was agreed with the reality since
the formation of PBH will span a time scale of non-linear evolution of
complexity. Another difference comes via the faster oscillation of complexity
caused by this resonant sound speed.

Here, we emphasize the importance of investigations for the SSR belonging to
the non-trivial sound speed. Firstly, the resonant sound speed (6) has
extensive phenomelogical implications, as the previous discussions, this
mechansim can be realized in DBI inflation, even for exploring the
gravitatioal waves Cai:2020ovp ; Zhou:2020kkf  and analyzing the ehanced power
spectrum of gravtational waves Cai:2021yvq  under this mechanism. Since this
mechanism can be realized in many models, the evolution of cosmological
complexity for the resonant sound speed (6) will manifest the chaotic feature
of these extensive models. In Li:2021kfq ; Li:2021sro , it studied the
evolution of cosmological complexity in k-essence inflationary model and BPS
states, in which they show the various evolution for the cosmological
complexity. Consequently, it sheds a new light for disguishing various
cosmological models.

Next, we will utilize our method for investigating the more inflationary
models with various non-trivial sound speed according to Eqs. 22, in which
these two equations are valid for any non-trivial sound speed in inflation. In
light of their solutions, we can simulate the evolution of complexity in
different inflationary models. For analyzing the late time evolution of
complexity, we need to take the potential into account. If translating into
our method, there must exist extra terms in action (9) left to the future
work. Furtherly, we could investigate the complexity from the modified
gravity, especially for $f(R)$ gravity Liu:2018hno ; Liu:2018htf  or the
multifield inflation Liu:2020zzv ; Liu:2019xhn , transformed into Einstein
frame, there are non-trivial sound speed and possible extra terms of affecting
the evolution of complexity. Finally, we could also work the complexity for a
black hole Information loss problem solver Choudhury:2020hil  and the
connection with Out-of-Time Ordered Correlators Choudhury:2020yaa  (OTOC).

## Acknowledgements

LH is funded by NSFC grant NO. 12165009 and Hunan Provincial Department of
Education, NO. 19B464. AC is supported by NSFC grant no.11875082 and the
University of Barcelona (UB) / China Scholarship Council (CSC) joint
scholarship. AC is also supported by National Natural Science Foundation of
China under Grant Nos.11947086. We are grateful for the numerical calculation
via Prof. Ding-Fang Zeng from Institute of Theoretical Physics In Beijing
University of Technology

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