# Examining the Implications of Deepfakes for Election Integrity

Hriday Ranka1, Mokshit Surana111footnotemark: 1, Neel Kothari111footnotemark:
1, Veer Pariawala111footnotemark: 1, Pratyay Banerjee111footnotemark: 1,
Aditya Surve111footnotemark: 1, Sainath Reddy Sankepally111footnotemark: 1,
Raghav Jain1, Jhagrut Lalwani1, Swapneel Mehta1 Equal contribution.

###### Abstract

It is becoming cheaper to launch disinformation operations at scale using AI-
generated content, in particular ’deepfake’ technology. We have observed
instances of deepfakes in political campaigns, where generated content is
employed to both bolster the credibility of certain narratives (reinforcing
outcomes) and manipulate public perception to the detriment of targeted
candidates or causes (adversarial outcomes). We discuss the threats from
deepfakes in politics, highlight model specifications underlying different
types of deepfake generation methods, and contribute an accessible evaluation
of the efficacy of existing detection methods. We provide this as a summary
for lawmakers and civil society actors to understand how the technology may be
applied in light of existing policies regulating its use. We highlight the
limitations of existing detection mechanisms and discuss the areas where
policies and regulations are required to address the challenges of deepfakes.

## Introduction

In recent years, the digital world has undergone rapid advancements, resulting
in the emergence of sophisticated technologies that blur the boundaries
between reality and fiction. During these events, deepfake technology has been
a cause for concern due to its potential implications and dangerous
consequences. Coined from the terms ”deep learning” and ”fake,” deepfakes
utilize advanced algorithms to generate hyper-realistic multimedia content,
often indistinguishable from authentic material (Mirsky and Lee 2021).

While AI-generated images and deepfakes both make use of artificial
intelligence, they serve vastly different purposes primarily arising from the
intent behind content creation and distribution. The former involves the
creation of realistic images through algorithms that learn from real data,
whereas the latter–technologically a subset thereof–typically aims to deceive
viewers into believing they are accessing authentic content. This article
(Becker and Laycock 2023) discusses the utilization of advanced technologies
to produce lifelike and personalized dynamic facial visuals, as well as
developing and adjusting various high-caliber static content. However, (Wang
et al. 2022) underscores how intent matters, showcasing the potential abuse of
deepfakes in producing fake images for scientific publications.

There are a variety of methods used to generate deepfakes; with state-of-the-
art approaches including diffusion-based models and generative adversarial
networks (GANs). Stable diffusion models, such as DALL-E 2, Midjourney, and
Stable Diffusion are neural networks trained on a large dataset of images and
captions to generate convincing images from text descriptions (Chen et al.
2023). On the other hand, GANs such as Cycle-GAN, DCGAN, and WGAN are deep
learning systems commonly used for image generation, data augmentation, music
generation, and deepfake creation (Remya Revi, Vidya, and Wilscy 2021). The
quality of deepfakes generated using GANs depends on the quantity and variety
of training data, and the use of GANs to synthesize minimum training data for
deepfake generation has been an area of active research (Singh, Sharma, and
Smeaton 2020).

## The Dangers of Deepfakes for Democratic Elections

The use of deepfake technology to spread disinformation poses a significant
threat to free and fair democratic elections. Deepfakes can serve as a potent
tool for malicious actors to manipulate voters and influence election outcomes
(McKenzie 2023; Appel and Prietzel 2022). There are several ways deepfakes
endanger democratic processes:

  1. 1.

Deepfakes can directly alter voter preferences and spread disinformation about
candidates by making them appear to take policy positions they do not hold or
engage in illegal behavior (Ray 2021; Appel and Prietzel 2022). This could
undermine trust in the electoral process.

  2. 2.

Coordinated disinformation campaigns utilizing deepfake videos could prevent
citizens from voting by spreading false information about election procedures
or intimidating voters through blackmail (Pawelec 2022). This form of voter
suppression damages participation.

  3. 3.

Deepfakes amplified through social media and messaging platforms can rapidly
reach millions of viewers (Christopher 2020; Jee 2020), confusing them about
candidates and issues. Widespread false or misleading information shaped by
deepfakes harms informed civic discourse.

  4. 4.

There are already instances of political deepfakes circulating globally, such
as the manipulated videos of Nancy Pelosi and Volodymyr Zelenskyy (News 2019;
Miller 2022). In the 2023 Argentina elections, both leading candidates, Javier
Milei and Sergio Massa, created and spread deepfake images and videos of each
other to portray their opponent negatively (David Feliba 2023). This
demonstrates the real-world vulnerability of elections.

## DeepFake Creation and Identification

### Generative Models

Deepfakes are fueled by technological advancements in generative models,
broadly including autoencoders, GANs, transformer-based models, and diffusion-
based models.

The historical developments in deepfake technology are as follows (Masood et
al. 2023):

  * •

Pre-2014: Traditional Techniques and Early Autoencoders: Before 2014, the
field of manipulated multimedia predominantly utilized conventional methods
such as splicing and copy-move, with an early occurrence dating as far back as
1860. Autoencoders, a generative model originating in the 1980s, garnered
interest in the early 2000s and made significant contributions to the
development of early generative models. Nevertheless, their influence on the
progression of deepfake technology was diminished by more sophisticated
models.

  * •

2014-2017: Emergence of GANs: In 2014, the introduction of Generative
Adversarial Networks (GANs) by Ian Goodfellow (Goodfellow et al. 2020) brought
about a significant change in deepfake technology. During this time, GANs
emerged as a highly influential and transformative factor, with academic
initiatives such as Face2Face and Synthesizing Obama playing a significant
role in the initial advancements. In September 2017, a significant event took
place on Reddit when a user named ”deepfake” shared the initial authentic
deepfake. This entailed the utilization of computer-generated videos
showcasing renowned actresses with their faces effectively replaced with
explicit content. This occurrence garnered public interest, indicating a
significant turning point in the advancement of deepfake technology. The event
brought to light the fact that complex generative models could be used for bad
and dishonest purposes, which raised awareness and led to more thought about
ethical issues and regulatory actions.

  * •

2018 Onwards: Integration of Transformers and Diffusion Models: Over the
following years, the deepfake technology landscape continued to progress. The
advent of open-source projects such as DeepFaceLab in 2018 has played a
significant role in making deepfake creation tools more accessible.
Furthermore, there was a significant change in the investigation of
transformers beyond their original utilization in natural language processing.
Researchers have acknowledged the adaptability of transformers, expanding
their application to include image synthesis and other tasks unrelated to
text.

Currently, the advancements in deepfake technology revolve around the
integration of transformers and diffusion models. The objective of this
collaborative approach is to attain generative outcomes of superior quality,
with a focus on enhancing the authenticity and capabilities of the produced
content. This technology advances with a focus on enhancing security. The
progress is crucial, especially in the context of elections, where the threat
of deepfakes contributing to misinformation campaigns continues to be a
significant concern.

### GAN-based Architectures

Very broadly, existing deepfake detection techniques can be divided into 2
categories, based on the consideration of change of characteristic/genuine
attributes in space (spatial consideration) and the ones that consider changes
in space as well as time (spatio-temporal). For the first class of techniques,
researchers aim to capture spatial features and perform the classification of
visual data (video/image) as real or fake (Bonettini et al. 2020; Raza, Munir,
and Almutairi 2022; Ismail et al. 2021; Afchar et al. 2018; Coccomini et al.
2022a). On the other hand, models utilizing time-series and image features in
combination have proven to be more accurate in identifying deepfakes (de Lima
et al. 2020; Zhang et al. 2022; Coccomini et al. 2022b; Cai et al. 2023; Jung,
Kim, and Kim 2020; Tariq, Lee, and Woo 2020; Cozzolino et al. 2021; Wodajo,
Atnafu, and Akhtar 2023).

#### Evaluating GAN-based DeepFake Generation

Accuracy Ratings: Prominent tools like FaceSwap-GAN (An 2022) exhibit superior
accuracy since perceptual loss improves the direction of eyeballs to be more
realistic and consistent with the input face. It also smoothes out artifacts
in the segmentation mask, resulting in higher output quality and rendering
dependable options for a range of uses. Similarly, tools like Simswap, Fewshot
FT Gan, and Faceshifter boast high accuracy. But it’s important to recognize
that some tools—like FaceApp and StyleGAN—have poorer accuracy levels.
StyleGAN (Brownlee 2020) which relies on traditional GAN generators, inherits
the interpretability and control issues associated with typical GAN models.
Due to the generators’ limited knowledge of latent space qualities and image
synthesis techniques, it may be difficult to comprehend and manipulate
components inside the StyleGAN framework.

Usability Analysis: CycleGAN(Zhu et al. 2017) stands out for its exceptional
user-friendliness and versatility. Its simplicity lies in the elimination of
the requirement for paired data (Paired training data consists of training
examples x i, y i N i=1 , where the correspondence between x i and y i
exists), making it accessible to a broader user base. The capability to
seamlessly work with unpaired data not only simplifies the process but also
proves cost-effective, addressing challenges associated with obtaining
extensive and reliable paired datasets. This user-friendly approach, coupled
with high accuracy in image-to-image translation, positions CycleGAN as a
valuable tool. The usability of face swapping and attribute manipulation tools
like Faceapp, SimSwap, Fewshot FT GAN, and FaceShifter depends on factors such
as user interface, documentation, and the level of technical expertise
required.

Security Assessment: By avoiding information loss, improving representation
ability, and rejecting an attribute-independent constraint, AttGAN (He et al.
2019) prioritizes a secure facial attribute editing approach. The framework’s
security is strengthened by the attribute classification constraint on
generated images, which guarantees accurate attribute manipulations. Utilizing
adversarial learning and reconstruction provides additional resilience to
maintain original facial features and produce realistic images. However, these
implementation strategies have varied levels of defense against adversarial
attacks, so they must be carefully considered. Overall, compared to models
with more stringent constraints, AttGAN’s combination of these features
improves security

Computational Efficiency: CycleGAN is recognized for its computational
efficiency, especially in handling unpaired data. Usability varies, with
CycleGAN being user-friendly, while others, like Style-GAN variants, may
demand deep learning expertise and substantial computational resources.

### Challenges faced by existing tools and techniques

Challenges arise in the performance of deepfake detection algorithms when
faced with low-quality films compared to high-resolution videos. Videos may
undergo various transformations, including reshaping, rotations, and
compression, necessitating flexible detection algorithms to maintain efficacy.

Time consumption emerged as a significant concern for real-world applications
of deepfake-detection techniques. Despite their potential impact on social
security, existing detection methods still face limitations in terms of
extensive time requirements, hindering widespread adoption in practical
scenarios.

The challenge of insufficient data for specific characters during the creation
of deepfake models was highlighted. While models are often trained on specific
datasets, they may struggle to produce accurate outputs when faced with
limited data for a particular character. Retraining models for each distinct
target character is a time-consuming process.

Dataset quality was identified as another challenging area, with most datasets
created under ideal conditions that differ from real-world testing scenarios.
This misalignment in dataset quality adds complexity to the development and
evaluation of deepfake-detection algorithms.

Despite the availability of various deepfake-generation tools, inherent flaws
and limitations persist. These tools are often tailored to specific traits,
emphasizing the need for additional research to enhance their efficiency.
Consequently, the creation of general-purpose deepfake-generation tools
remains a complex and challenging process that warrants further investigation.

## Reviewing Platform Policies against Deepfakes

The regulation of deepfake content on social media platforms has become a
critical issue in recent years.

With the rise of AI-generated manipulated media, platforms like Meta, X
(formerly Twitter), Reddit, Tiktok, and YouTube have implemented various
policies to address the spread of deepfakes (Center 2023).

### Meta

(Monika Bickert 2020)

  * •

Removal of Manipulated Media: Meta will remove audio, photos, or videos,
including deepfakes, if they violate any of their Community Standards, such as
those related to graphic violence, voter suppression, and hate speech.

  * •

Detection Efforts: Meta has launched the Deep Fake Detection Challenge and is
collaborating with experts to address deepfakes and manipulated media.

### X (formerly Twitter)

(X 2023)

  * •

Prohibition of Misleading Media: X prohibits the sharing of synthetic,
manipulated, or out-of-context media that may deceive or confuse people and
lead to harm.

However, memes, satire, animations, and cartoons are generally not in
violation of this policy.

  * •

Labeling and Consequences: In some cases, X may label posts containing
misleading media and take action to reduce the visibility of the post on the
platform.

### YouTube

(PTI-News 2023)

  * •

Disclosure Requirement: YouTube will require creators to disclose altered or
synthetic content that is realistic, including using AI tools. The platform
will inform viewers about such content through labels in the description panel
and video player.

  * •

Removal and Labeling: YouTube may remove AI-generated or manipulated content
that simulates an identifiable individual, and it will work with creators to
ensure they understand the new requirements.

### Reddit

(Peters 2020)

  * •

Reddit does not allow content that impersonates individuals or entities in a
misleading or deceptive manner, including deepfakes or other manipulated
content presented to mislead, or falsely attributed to an individual or
entity.

### Tiktok

(NBC-News 2023; Vincent 2023)

  * •

TikTok bans deepfakes of private figures and young people, and all realistic
AI deepfakes must be ”clearly disclosed”.

### Monitoring the Implementation of Current Deepfake Regulations

Monitoring the enforcement of current deepfake regulations involves assessing
the adherence of various stakeholders, evaluating the efficiency of detection
tools, and addressing emerging challenges. We explore the strategies and
considerations involved in overseeing the implementation of existing deepfake
regulations:

  1. 1.

Stakeholder Compliance: Regulatory success hinges on the compliance of key
stakeholders, including social media platforms, technology companies,
political campaigns, and content creators. Continuous monitoring of these
entities is essential to ensuring they are actively adopting measures to
prevent the creation and dissemination of malicious deepfakes. Collaborative
efforts between regulatory bodies and stakeholders can facilitate the exchange
of best practices, ensuring a unified approach to tackling deepfake-related
threats.

  2. 2.

Effectiveness of Detection Tools: The efficacy of deepfake detection tools
plays a pivotal role in enforcing regulations. Regular assessments of the
performance of existing detection mechanisms are crucial to identifying
strengths, weaknesses, and areas for improvement. This involves evaluating the
accuracy, speed, and adaptability of tools for detecting evolving deepfake
techniques. Ongoing research and development are necessary to enhance the
capabilities of detection tools and address emerging challenges in real time.

  3. 3.

Educational Initiatives: Monitoring the implementation of regulations extends
beyond enforcement measures to include educational initiatives. Informing the
public, political candidates, and election officials about the existence of
deepfake threats, the regulatory framework in place, and preventive measures
is vital. Periodic assessments of the effectiveness of educational campaigns
can guide adjustments and refinements to ensure they remain relevant and
impactful.

  4. 4.

Adaptability to Evolving Threats: The landscape of deepfake technology is
dynamic, with new advancements and variations emerging regularly. Monitoring
the implementation of regulations requires a proactive approach to stay ahead
of evolving threats. Regulatory bodies should establish mechanisms for
continuous threat assessment, allowing timely modification of regulations to
address emerging challenges effectively.

  5. 5.

International Collaboration: Given the global nature of information
dissemination and potential cross-border impact, international collaboration
is essential for effective regulation. Monitoring the implementation of
deepfake regulations involves fostering partnerships between countries,
sharing intelligence, and collectively addressing challenges. Regular forums
for collaboration can facilitate the exchange of insights and strategies to
combat the transnational aspects of deepfake threats.

### Areas where interventions/policies and regulations are required

The emergence of deepfakes poses significant challenges to the integrity of
elections. To address these challenges, various areas require targeted
interventions, policies, and regulations.

  1. 1.

Authentication Protocols: Implementing protocols to authenticate digital
content can help distinguish genuine media from deepfakes. This includes the
development of digital watermarks or certification systems (Westerlund 2019).

  2. 2.

Transparency Requirements: Legislation mandating the disclosure of AI-
manipulated content can increase transparency. Any altered media should be
clearly labeled to inform the public about its modified nature (Langa 2021).

  3. 3.

Media Literacy Programs: Educating the public, especially voters, about the
existence and nature of deepfakes is crucial. Media literacy programs can
teach people how to critically assess and verify the authenticity of the
information they receive (El Mokadem 2023).

  4. 4.

Legal Frameworks Against Misuse: There should be clear legal consequences for
maliciously creating or distributing deepfakes (Feeney 2021).

## Conclusion

This paper urgently calls for societal, policymaker, and regulatory action to
safeguard elections from the pervasive threat of deepfake technology. It
meticulously explores the dissemination of deepfakes and their implications
for election security, advocating for comprehensive regulatory frameworks to
counter the growing accessibility of AI-generated disinformation. The paper
delves into the intricacies of the deepfake generation, emphasizing its
profound impact on politics, trust, and democratic processes. While detection
tools show promise, challenges persist, necessitating research and
adaptability. Varied deepfake policies among technology and social media
companies underscore the importance of vigilant monitoring for effective
implementation. Recognizing the dynamic nature of these challenges, the paper
urges continuous proactive efforts from policymakers, technologists, and the
public to enhance detection capabilities and safeguard democracy against the
evolving threats of deepfake technology. Ongoing commitment to innovation,
collaboration, and democratic principles is crucial for ensuring the
resilience of electoral processes against AI-generated disinformation. We also
express our sincere gratitude to Mr. Eric Davis for his invaluable feedback
and insightful comments on this paper.

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## Appendix

## From Theory to Practice: Deepfake Detection Examples

Analyzing the effectiveness of deepfake-detection tools.

1\. Sensity AI(SensityAI 2023): Sensity AI presents a method for identifying
GAN-generated images, especially those that belong to the StyleGAN2 model.
Sensity’s model was used to analyze 975 profile pictures from LinkedIn
accounts. The research study focused on fictitious accounts on the platform
with GAN-generated profile photos (Goldstein and DiResta 2022). As evidenced
by the results, there was over 90% confidence in the ability to identify GAN-
generated images for 968 profile pictures. Surprisingly, 900 of these images
had a confidence score higher than 99.9%, demonstrating the resilience of
Sensity’s model. The majority of the identified images were linked to the
StyleGAN2 model, highlighting the efficacy of Sensity AI in detecting specific
types of GAN-generated content.

2\. Truepic: A startup company based in the United States has created a system
that utilizes mobile apps to allow regular users and freelancers to capture
images and store them on the company’s servers. The purpose of saving the
images is to maintain their integrity. As a result, comparing any fraudulent
attempt with the image kept on the servers makes it easy to spot. Truepic
utilizes blockchain technology (Hasan and Salah 2019) to securely store
metadata associated with saved images, guaranteeing their immutability. This
method is highly dependent on placing a significant amount of trust in Truepic
about the authenticity and integrity of the uploaded images. The operational
details of incorporating logos, text tickers, subtitles, or closed captions
into images or video frames are not readily apparent.

3\. Deeptrace(DeepTraceTechnologies 2023): Deeptrace employs machine learning
algorithms for detecting deepfake videos, showcasing high accuracy rates
achieved through comprehensive audio-based, visual-based, and text-based
analysis. Its real-time detection capabilities, scalability for analyzing
large datasets, and continuous learning to adapt to new deepfake techniques
position it as a versatile solution. Deeptrace effectively addresses the
challenges posed by deepfakes, making it a valuable asset for organizations
seeking reliable detection tools.


††thanks: They contribute equally to this work. ††thanks: They contribute
equally to this work.

# Krylov Complexity in Open Quantum Systems

Chang Liu  Institute for Advanced Study, Tsinghua University, Beijing, 100084,
China  Haifeng Tang  Institute for Advanced Study, Tsinghua University,
Beijing, 100084, China  Department of Physics, Tsinghua University, Beijing,
100084, China  Hui Zhai<EMAIL_ADDRESS>Institute for Advanced Study,
Tsinghua University, Beijing, 100084, China

###### Abstract

Krylov complexity is a novel measure of operator complexity that exhibits
universal behavior and bounds a large class of other measures. In this letter,
we generalize Krylov complexity from a closed system to an open system coupled
to a Markovian bath, where Lindbladian evolution replaces Hamiltonian
evolution. We show that Krylov complexity in open systems can be mapped to a
non-hermitian tight-binding model in a half-infinite chain. We discuss the
properties of the non-hermitian terms and show that the strengths of the non-
hermitian terms increase linearly with the increase of the Krylov basis index
$n$. Such a non-hermitian tight-binding model can exhibit localized edge modes
that determine the long-time behavior of Krylov complexity. Hence, the growth
of Krylov complexity is suppressed by dissipation, and at long-time, Krylov
complexity saturates at a finite value much smaller than that of a closed
system with the same Hamitonian. Our conclusions are supported by numerical
results on several models, such as the Sachdev-Ye-Kitaev model and the
interacting fermion model. Our work provides insights for discussing
complexity, chaos, and holography for open quantum systems.

Operator complexity describes how an operator becomes increasingly complicated
under the Heisenberg time evolution. The concept of operator complexity has
emerged as a novel tool in studying quantum matters B. Yoshida ; R.C. Myers ;
A. Streicher ; R.-Q. Yang ; S. Sharma ; K.-Y. Kim ; A. Streicher-2 ; A. Lucas
; O. Parrikar ; O. Parrikar-2 . It can characterize the chaotic behavior and
integrability of a quantum many-body Hamiltonian, and it is correlated with
the dynamics of quantum information processes. Through holography, it also
becomes a new entity to study black hole physics. A mathematically rigorous
definition of operator complexity depends on the choice of a pre-defined
basis. Previously, various measures of operator complexity have been proposed
and studied in different contexts B. Yoshida ; R.C. Myers ; A. Streicher ;
R.-Q. Yang ; S. Sharma ; K.-Y. Kim ; A. Streicher-2 ; A. Lucas ; O. Parrikar ;
O. Parrikar-2 .

Recently, the Krylov recursion method has been applied to investigate operator
complexity Altman . It is proposed that the operator complexity in the Krylov
basis, called Krylov complexity, exhibits universal behaviors and can bound a
large class of other measures Altman . Thanks to its advantages, Krylov
complexity has attracted considerable attention from various communities R.
Sinha ; A. Gorsky ; X. Cao ; A. Dymarsky ; A. Gorsky-2 ; M. Sasieta ; J. Simon
; Z. Y. Xian ; J. Sonner ; A. Lucas ; M. Smolkin ; J. D. Noh ; C. J. Lin ; D
Shouvik ; D. Patramanis ; D. Patramanis-2 ; J. Sully ; D. Rosa ; A. del Campo
; J. Sonner-2 ; T. Pathak ; Q. Wu ; J. Gemmer ; B. Swingle ; S Choudhury ; A.
Roy ; S Liu ; Y. Yang ; S Pawar ; Z. Y. Fan ; Z. Y. Fan-2 ; J. Sonner-3 .
Nevertheless, research on Krylov complexity has so far been limited to closed
systems. In this letter, we generalize Krylov complexity from a closed system
to an open system. In a closed system, operator growth is governed by
Hamiltonian, while for an open system coupled to a Markovian bath, operator
growth is governed by Lindbladian. Here we will discuss how this change from
Hamiltonian to Lindbladian affects the behavior of Krylov complexity.

Figure 1: (a) Schematic the mapping between Krylov complexity in open systems
and a non-hermitian tight-binding model in a half-infinite chain. (b) Krylov
complexity $\mathcal{K}(t)$ in open systems (red solid line), compared with
$\mathcal{K}(t)$ in a closed system with the same Hamiltonian (blue dashed
line). Krylov complexity is suppressed by dissipation.

Review of Krylov Complexity. Before starting the generalization, let us
briefly review Krylov complexity in a Hamiltonian system Altman . First of
all, for a system with its Hilbert space spanned by $\\{\ket{i}\\}$, an
operator $\hat{X}=\sum_{ij}X_{ij}\ket{i}\bra{j}$ can be mapped to a state in
the double space, denoted by
$\ket{\hat{X}}=\sum_{ij}X_{ij}\ket{i}\otimes\ket{j}$. We introduce a super-
operator $\hat{\mathcal{L}}$ acting on an operator $\hat{X}$ as
$\hat{\mathcal{L}}\hat{X}=[\hat{H},\hat{X}]$, and
$\ket{\hat{\mathcal{L}}\hat{X}}$ is the state corresponding to the operator
$\hat{\mathcal{L}}\hat{X}$. Hence, using the Baker-Campbell-Hausdorff formula,
the Heisenberg evolution of a reference operator
$\hat{O}(t)=e^{i\hat{H}t}\hat{O}e^{-i\hat{H}t}$ can be expressed as expanding
the state $\ket{\hat{O}(t)}$ in a set of basis
$\ket{\hat{\mathcal{L}}^{n}\hat{O}}$ as

$\ket{\hat{O}(t)}=\sum_{n}\frac{(it)^{n}}{n!}\ket{\hat{\mathcal{L}}^{n}\hat{O}}.$
(1)

However, this set of basis $\ket{\hat{\mathcal{L}}^{n}\hat{O}}$ is neither
normalized nor orthogonal. Hence, we first need to apply the Gram-Schmidt
procedure with the infinite-temperature inner product $\langle
O_{1}|O_{2}\rangle=\text{Tr}[O_{1}^{\dagger}O_{2}]$ to orthogonalize this set
of basis. This results in the Krylov basis $\\{\ket{\hat{\mathcal{W}}_{n}}\\}$
as

$\displaystyle\ket{\hat{\mathcal{W}}_{0}}=\frac{1}{b_{0}}\ket{\hat{O}},$ (2)
$\displaystyle\ket{\hat{\mathcal{W}}_{1}}=\frac{1}{b_{1}}\ket{\hat{\mathcal{L}}\hat{\mathcal{W}_{0}}},$
(3)
$\displaystyle\ket{\hat{\mathcal{W}}_{n}}=\frac{1}{b_{n}}\left(\ket{\hat{\mathcal{L}}\hat{\mathcal{W}}_{n-1}}-b_{n-1}\ket{\hat{\mathcal{W}}_{n-2}}\right)\
\ \text{for}\ \ n\geqslant 2.$ (4)

Here $\\{b_{n}\\}$ are called the Lanczos coefficients introduced to normalize
these states. It is discussed that $b_{n}$ increases linearly in $n$ for a
generic chaotic Hamiltonian Altman , until it saturates at large enough $n$
for a finite system R. Sinha ; Z. Y. Xian ; J. Sonner ; J. Sonner-3 .
Therefore, we can expand the state $\ket{\hat{O}(t)}$ under the Krylov basis
as

$\ket{\hat{O}(t)}=\sum_{n}\varphi_{n}(t)\ket{\hat{\mathcal{W}}_{n}}.$ (5)

$\varphi_{n}(t)$ satisfy the following tight-binding model that describes
single-particle hopping in a half-infinite chain Altman

$i\partial_{t}\varphi_{n}=-b_{n+1}\varphi_{n+1}-b_{n}\varphi_{n-1}.$ (6)

This particle sits at $n=0$ with only $\varphi_{0}$ being non-zero at $t=0$,
and hops away from $n=0$ at finite $t$. Krylov complexity $\mathcal{K}(t)$ is
defined as the mean distance measured from $n=0$ as

$\mathcal{K}(t)=\sum_{n}n|\varphi_{n}(t)|^{2}.$ (7)

Summary of Results. Our generalization of Krylov complexity to open systems
keeps using the Krylov basis defined in Eq. 2-4. However, the operator
dynamics is changed from the Heisenberg evolution to the Lindblad evolution. A
Lindbladian contains both the Hamiltonian part and the dissipative part with
the dissipation operator $\hat{M}$. The main results are schematically shown
in Fig. 1 and summarized as follows. We emphasize that these results are also
universal for chaotic Hamiltonians with generic dissipations.

1) Krylov complexity in an open system can also be mapped to a particle
hopping in a half-infinite chain, but described by a non-hermitian tight-
binding model as

$i\partial_{t}\varphi_{n}=-b_{n+1}\varphi_{n+1}-b_{n}\varphi_{n-1}-i\gamma\sum_{m}d_{nm}\varphi_{m},$
(8)

where $\gamma$ represents the dissipation strength. $d_{nm}$ are dominated by
their diagnoal terms $d_{nn}$.

2) For hermitian dissipation operator $\hat{M}$, $d_{n}$ is always positive.
And for a generic chaotic Hamiltonian, $d_{n}$ grows linearly in $n$ until it
saturates at $n>n_{\text{s}}$. $n_{\text{s}}$ increases linearly with the
increasing of the system size.

3) When $\gamma>\gamma_{\text{c}}$, this imaginary part of the spectrum of
this non-hermitian tight-binding model exhibits a gap. The wave functions of
the modes below the gap are localized at the edge. $\gamma_{\text{c}}$
decreases toward vanishing when $n_{\text{s}}$ increases.

4) The growth of Krylov complexity is suppressed by dissipation. For
$\gamma>\gamma_{\text{c}}$, the localized modes below the gap dominates the
long-time evolution of Krylov complexity, therefore, at long-time, Krylov
complexity saturates to a value much smaller the fully scrambled case.

Figure 2: Coefficients of the non-hermitian terms $|d_{nm}|$ for the SYK model
(a) and for the spinless fermion model (b). For (a), we have chosen
$\hat{M}_{i}=\hat{\psi}_{i}$ and $\hat{O}=i\hat{\psi}_{1}\hat{\psi}_{2}$, and
we have set $J=1$. The system contains total number of Majorana fermion
$N=24$. For (b), we have chosen
$\hat{M}_{i}=\hat{c}^{\dagger}_{i}+\hat{c}_{i}$ and $\hat{O}=\hat{n}_{1}$, and
we have set $J_{1}=1$, $J_{2}=0.2$, $V_{1}=0.6$ and $V_{2}=0.1$. The model
contains total number of sites $N=13$.

Non-hermitian Tight-Binding Model. Now we illustrate these results in detail.
First of all, we consider the dynamical equation of an operator $\hat{O}$
under the Lindblad evolution as

$\frac{d\hat{O}(t)}{dt}=i[\hat{H},\hat{O}]+\gamma\sum\limits_{i}(\pm
2\hat{M}^{\dagger}_{i}\hat{O}\hat{M}_{i}-\\{\hat{M}^{\dagger}_{i}\hat{M}_{i},\hat{O}\\}),$
(9)

where $\hat{M}_{i}$ are dissipation operators. Here we note that the minus
sign should be taken when both $\hat{O}$ and $\hat{M}_{i}$ are fermionic
operators F. Schwarz ; supple . This is crucial for following discussion and
we present the detailed derivation and explain the origin of this minus sign
in the supplementary material supple . Substituting Eq. 5 into both sides of
Eq. 22, we arrive at Eq. 8 and $d_{nm}$ is given by

$d_{nm}=\sum\limits_{i}\text{Tr}[\hat{W}^{\dagger}_{n}\\{\hat{M}^{\dagger}_{i}\hat{M}_{i},\hat{W}_{m}\\}\pm
2\hat{W}^{\dagger}_{n}\hat{M}^{\dagger}_{i}\hat{W}_{m}\hat{M}_{i}].$ (10)

Note that the total weight $\mathcal{Z}=\sum_{n}|\varphi_{n}|^{2}$ is
conserved in the hermitian case but is not conserved in the non-hermitian
case. Nevertheless, we still define the Krylov complexity as the mean distance
measured from $n=0$ in the half-infinite chain, and the definition now needs
to be modified as

$\mathcal{K}(t)=\frac{1}{\mathcal{Z}}\sum_{n}n|\varphi_{n}(t)|^{2}.$ (11)

Illustrating Examples. To illustrate the physics concretely, we consider two
representative models. The first model is the Sachdev-Ye-Kitaev (SYK) model A.
Kitaev ; S. Sachdev ; J. Maldacena . The Hamiltonian reads

$\hat{H}_{\text{S}}=\sum\limits_{i<j<k<l}J_{ijkl}\hat{\psi}_{i}\hat{\psi}_{j}\hat{\psi}_{k}\hat{\psi}_{l},$
(12)

where $\hat{\psi}_{i}$ $(i=1,\dots,N)$ denotes $N$ Majorana fermions in the
system. $J_{ijkl}$ are independent random Gaussian variables with variances
given by $\overline{J^{2}_{ijkl}}=3!J^{2}/N^{3}$. The second model is a one-
dimensional lattice model of interacting spinless fermions. The Hamiltonian
reads

$\displaystyle\hat{H}_{\text{F}}=$
$\displaystyle-\sum\limits_{i}\left(J_{1}\hat{c}^{\dagger}_{i}\hat{c}_{i+1}+J_{2}\hat{c}^{\dagger}_{i}\hat{c}_{i+2}+\text{h.c.}\right)$
$\displaystyle+\sum\limits_{i}\left(V_{1}\hat{n}_{i}\hat{n}_{i+1}+V_{2}\hat{n}_{i}\hat{n}_{i+2}\right),$
(13)

where $J_{1}$ and $J_{2}$ are the nearest and next nearest hopping strengths,
and $V_{1}$ and $V_{2}$ are the nearest and the next nearest interaction
strengths. Here we include the next nearest hopping and interaction to break
the integrability. The SYK model contains random and all-to-all interactions,
while the spinless fermion model has locality. These two models represent two
different types of chaotic Hamiltonians. Aside from these two models, we have
also numerically studied other models, such as the Ising model with transverse
and longitudinal fields and the spin-$1/2$ Hubbard model. The results are
similar.

Behavior of $d_{nm}$. In Fig. 10, we plot typical $d_{nm}$ for these two
models code . It is clear that the diagnoal matrix elements $d_{nn}$ (short-
noted as $d_{n}$) are much larger than all off-diagonal matrix elements. For
hermitian operators $\hat{M}_{i}$ and using the fact
$\hat{W}^{\dagger}_{n}=(-1)^{n}\hat{W}_{n}$, it is straightforward to show
that

$d_{n}=\sum\limits_{i}\text{Tr}[[\hat{W}_{n},\hat{M}_{i}]^{\dagger}[\hat{W}_{n},\hat{M}_{i}]],$
(14)

if plus sign is taken in Eq. 10. And

$d_{n}=\sum\limits_{i}\text{Tr}[\\{\hat{W}_{n},\hat{M}_{i}\\}^{\dagger}\\{\hat{W}_{n},\hat{M}_{i}\\}],$
(15)

if minus sign is taken in Eq. 10, and anti-commutators replace commutators. In
both cases, $d_{n}$ are always non-negative real numbers.

Figure 3: Coefficients $d_{n}$ for the diagonal components of the non-
hermitian terms for the SYK model (a) and for the spinless fermion model (b).
For (a), four different curves cover four different cases: $\hat{O}$ and
$\hat{M}_{i}$ are either fermionic operator $\hat{\psi}_{i}$ or bosonic
operator $i\hat{\psi}_{i}\hat{\psi}_{j}$. We have set $J=1$. The system
contains total number of Majorana fermion $N=24$. For (b), four different
curves cover four different cases: $\hat{O}$ and $\hat{M}_{i}$ are either
fermionic operator $\hat{\psi}_{i}$ or bosonic operator $\hat{n}_{i}$, where
$\hat{\psi}_{i}$ denotes $\hat{c}^{\dagger}_{i}+\hat{c}_{i}$. We have set
$J_{1}=1$, $J_{2}=0.2$, $V_{1}=0.6$ and $V_{2}=0.1$. The model contains total
number of sites $N=13$. $n_{\text{s}}$ marks the places where $d_{n}$
saturates.

Since Ref. Altman  has shown that $b_{n}$ increases linearly in $n$ for a
generic chaotic Hamiltonian, here we focus on the behavior of $d_{n}$, as
shown in Fig. 3. We find that $d_{n}$ also increases linearly with the
increasing of $n$ and saturates when $n>n_{\text{s}}$.

This behavior of $d_{n}$ can be understood as follows. Suppose $\hat{M}_{i}$
is a local operator at site-$i$, and if $\hat{W}_{n}$ acts trivially at
site-$i$, then these two operators commute with each other and this commutator
does not contribute to $d_{n}$. Hence, when operator size of $\hat{W}_{n}$
increases with the increasing of $n$, $d_{n}$ increases. Inspired by this
argument, let us then assume $d_{n}$ increases as $\sim n^{\delta}$. Below we
argue $\delta=1$. To this end, we ultilize the result of Ref. Altman  that in
a closed system, Krylov complexity is a proper bound of the out-of-time-
ordered commutator (OTOC). Here we consider the OTOC
$\langle|[\hat{O}(t),\hat{M}_{i}]|^{2}\rangle$ of the closed system at
infinite temperature, and we have

$\displaystyle\sum_{i}\langle|[\hat{O}(t),\hat{M}_{i}]|^{2}\rangle=\sum_{i}\langle|[\sum_{n}\varphi^{\text{c}}_{n}(t)\hat{W}_{n},\hat{M}_{i}]|^{2}\rangle$
$\displaystyle\approx\sum_{i}\sum_{n}|\varphi^{\text{c}}_{n}(t)|^{2}\langle|[\hat{W}_{n},\hat{M}_{i}]|^{2}\rangle=\sum_{n}|\varphi^{\text{c}}_{n}(t)|^{2}d_{n}.$
(16)

Here we use $\varphi^{\text{c}}_{n}(t)$ to denote the expansion coefficients
of $\hat{O}(t)$ of the closed system. On the other hand, the infinite
temperature OTOC $\langle|[\hat{O}(t),\hat{M}_{i}]|^{2}\rangle$ should be
bound by $\mathcal{C}\mathcal{K}(t)$ in the same closed system, where
$\mathcal{C}$ is certain constant. Hence, we have

$\displaystyle\sum_{n}d_{n}|\varphi^{\text{c}}_{n}(t)|^{2}\leq\mathcal{C}\mathcal{K}(t)=\mathcal{C}\sum_{n}n|\varphi^{\text{c}}_{n}(t)|^{2}.$
(17)

If this bound is valid, it requires $\delta<1$ and if the bound is tight, it
further requires $\delta=1$. Hence, this gives rise to a linear growth of
$d_{n}$.

Figure 4: Spectrum for the non-hermitian tight-binding model. (a-c) We choose
$\beta_{d}=2.80$, $\alpha_{d}=0.35$, $\beta_{b}=0.66$, $\alpha_{b}=0.34$ and
$n_{\text{s}}=1000$. (a) Eigen-energy
$\epsilon=\epsilon^{\prime}-i\epsilon^{\prime\prime}$.
$(\epsilon^{\prime},\epsilon^{\prime\prime})$ for
$\gamma=0.007<\gamma_{\text{c}}$ (b)
$(\epsilon^{\prime},\epsilon^{\prime\prime})$ for
$\gamma=0.04>\gamma_{\text{c}}$. Here we shift the imaginary part
$\epsilon^{\prime\prime}$ by a constant such that the smallest
$\epsilon^{\prime\prime}$ is zero. (c) The wave functions for two modes below
the gap in (b). (d) $\gamma_{\text{c}}$ versus $n_{\text{s}}$.

Similar argument can be applied to the situation that both $\hat{O}$ and
$\hat{M}$ are fermionic. In this case, the infinite temperature OTOC is also
defined in terms of anti-commutator
$\langle|\\{\hat{O}(t),\hat{M}_{i}\\}|^{2}\rangle$. Similarly, we have

$\displaystyle\sum_{i}\langle|\\{\hat{O}(t),\hat{M}_{i}\\}|^{2}\rangle=\sum_{i}\langle|\\{\sum_{n}\varphi^{\text{c}}_{n}(t)\hat{W}_{n},\hat{M}_{i}\\}|^{2}\rangle$
$\displaystyle\approx\sum_{i}\sum_{n}|\varphi^{\text{c}}_{n}(t)|^{2}\langle|\\{\hat{W}_{n},\hat{M}_{i}\\}|^{2}\rangle=\sum_{n}|\varphi^{\text{c}}_{n}(t)|^{2}d_{n}.$
(18)

Also using Eq. 17, we reach the same conclusion that $d_{n}$ increases
linearly in $n$.

Spectrum of Non-hermitian Hopping Model. Below we simplify the non-hermitian
tight-binding model Eq. 8 by only considering the diagonal components of
$d_{nm}$. We assume that $b_{n}=\beta_{b}+\alpha_{b}n$ and
$d_{n}=\beta_{d}+\alpha_{d}n$ up to $n=n_{\text{s}}$ and both remain constants
for $n>n_{\text{s}}$, where $\beta_{b}$, $\alpha_{b}$, $\beta_{d}$ and
$\alpha_{d}$ are always positive. We obtain reasonable values of these
parameters by fitting Fig. 3(b).

The spectrum of the non-hermitian tight-binding model determines the dynamical
behavior of $\mathcal{K}(t)$. We write the eigen-energies as
$\epsilon=\epsilon^{\prime}-i\epsilon^{\prime\prime}$. The imaginary part
$\epsilon^{\prime\prime}$ is always positive. We solve all the eigen-state
$\phi_{l}$ of the non-hermitian tight-binding model and plot their eigen-
energies $\\{(\epsilon_{l}^{\prime},\epsilon_{l}^{\prime\prime})\\}$ in Fig.
4. We find that for $\gamma>\gamma_{\text{c}}$, the imaginary part of the
spectrum acquires a gap $\sim\Delta$ as shown in Fig. 4(b). Note that the time
evolution in the half-infinite chain follows

$\varphi(t)=\sum\limits_{l}c_{l}e^{-i(\epsilon^{\prime}_{l}-i\epsilon^{\prime\prime}_{l})t}\phi_{l}.$
(19)

Hence, the eigen-modes with larger $\epsilon^{\prime\prime}$ decay faster.
When $t\gg 1/\Delta$, all modes above the gap damps out and the mode below the
gap dominates $\varphi(t)$. We show in Fig. 4(c), the wave-functions of these
two modes below the gap are localized around the edge of the half-infinite
chain. We note that the existence of such localized edge states is a universal
behavior of such non-hermitian tight-binding model. Therefore,
$\mathcal{K}(t)$ saturates to a much smaller value at long-time, and this
value is determined by the center position of the wave function shown in Fig.
4(c). Such a behavior of $\mathcal{K}(t)$ is shown by the solid line in Fig.
1, compared with the closed system with the same Hamiltonian. This can also be
understood from the fact that $d_{n}$ is larger for larger $n$, which means
that more complicated operators are subjected to stronger decay, and
therefore, simpler operators survive at long-time. Hence, we conclude that the
growth of Krylov complexity is suppressed by dissipation.

Here we also find that $\gamma_{\text{c}}$ depends on $n_{\text{s}}$. Fig.
4(c) shows that $\gamma_{\text{c}}\sim 1/n^{0.85}_{\text{s}}$. $n_{\text{s}}$
increases as the total system size increases. Hence, for an infinite system,
$n_{\text{s}}\rightarrow\infty$, and therefore, $\gamma_{\text{c}}\rightarrow
0$.

Summary and Outlook. In summary, we generalize Krylov complexity to an open
system governed by the Lindblad equation. We show that Krylov complexity
defined for the open system can be mapped to a non-hermitian tight-binding
model. This model also exhibits universal behavior, and its localized edge
modes determine the long-time behavior of Krylov complexity in an open system.
This work opens a new route to extend the discussion of operator complexity
and chaos to open quantum systems. For those systems such as the SYK model
with gravity interpretation, such discussion can also shed light on gravity
physics through holographic duality.

Acknowledgment. We thank Pengfei Zhang for pointing out the minus sign in the
operator Lindblad equation. We thank Zhenbin Yang for bringing our attention
to Krylov complexity. We thank Wei Yi, Zhong Wang, Xiaoliang Qi, Yingfei Gu
and Peng Zhang for helpful discussions. The project is supported by Beijing
Outstanding Young Scholar Program, NSFC Grant No. 11734010 and the XPLORER
Prize.

Note Added. When finishing this work, a new preprint appeared where Krylov
complexity in an open system has also been proposed H. Sahu . However, the
method and the focus are different from ours.

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##  I Supplementary Material

In this supplementary material, we derive the Lindblad equation for an
operator $\hat{O}(t)$. Especially, we should highlight that an extra minus
sign is required in front of the quantum jump term when both $\hat{O}$ and the
dissipation operator $\hat{M}$ are both fermionic ones. First of all, we
consider the duality between the Schrödinger picture and the Heisenberg
picture

$\text{Tr}[\hat{O}\hat{\rho}(t)]=\text{Tr}[\hat{O}(t)\hat{\rho}].$ (20)

Since it is known that the density matrix $\hat{\rho}(t)$ of an open system
obeys the following Lindblad equation

$\frac{d\hat{\rho}(t)}{dt}=-i[\hat{H},\hat{\rho}]+\gamma\sum\limits_{i}(2\hat{M}_{i}\hat{\rho}\hat{M}^{\dagger}_{i}-\\{\hat{M}^{\dagger}_{i}\hat{M}_{i},\hat{\rho}\\}),$
(21)

it is straightforward to show that $\hat{O}(t)$ obeys the adjoint equation

$\frac{d\hat{O}(t)}{dt}=i[\hat{H},\hat{O}]+\gamma\sum\limits_{i}(2\hat{M}^{\dagger}_{i}\hat{O}\hat{M}_{i}-\\{\hat{M}^{\dagger}_{i}\hat{M}_{i},\hat{O}\\}).$
(22)

However, Eq. 22 is not always correct. When the operator $\hat{O}$ is
fermionic, the expectation value of $\hat{O}(t)$ is always zero. As a result,
Eq. 20 cannot result in a unique equation for $\hat{O}(t)$ given the Lindblad
equation for $\hat{\rho}(t)$. Therefore, in order to include the situations
with both bosonic and fermionic operators $\hat{O}$, we shall derive the
operator Lindblad equation by first explicitly including the operators in bath
and then integrating out the bath operators.

We start with the general form

$\hat{H}=\hat{H}^{\text{S}}+\hat{H}^{\text{B}}+\hat{H}^{\text{int}},$ (23)

where $\hat{H}^{\text{S}}$ and $\hat{H}^{\text{B}}$ are respectively the
Hamiltonians for system and for bath. $\hat{H}^{\text{int}}$ represents the
interaction between system and bath, which is assumed to be

$\hat{H}^{\text{int}}=\lambda(\hat{M}^{\dagger}\hat{\xi}+\hat{\xi}^{\dagger}\hat{M}).$
(24)

The operator evolution in the entire system obeys the Heisenberg equation as

$i\partial_{t}\hat{O}=[\hat{O},\hat{H}].$ (25)

The effective evolution of the operator acting on system only can be derived
by tracing out the bath degree of freedom
$\hat{O}^{\text{S}}(t)=\text{Tr}_{\text{B}}(\hat{\rho}_{\text{B}}\hat{O}(t))$.
Below we turn into the interaction picture by introducing the unitary
transformation
$\hat{U}_{0}(t)=e^{-i(\hat{H}^{\text{S}}+\hat{H}^{\text{B}})t}$. We write down
the operator in the interaction picture as
$\hat{O}_{\text{I}}(t)=\hat{U}_{0}(t)\hat{O}(t)\hat{U}_{0}^{\dagger}(t)$.
Then, the evolution equation becomes

$i\partial_{t}\hat{O}_{\text{I}}=[\hat{O}_{\text{I}},\hat{H}^{\text{int}}_{\text{I}}].$
(26)

We assume a Markovian bath and apply the white noise approximation to the
Green’s function of the bath, that is,

$\langle\hat{\xi}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime}),$
(27)

and we also have

$\langle\hat{\xi}_{I}(t)\rangle=\langle\hat{\xi}_{I}^{\dagger}(t)\rangle=\langle\hat{\xi}_{I}^{\dagger}(t)\hat{\xi}_{I}(t^{\prime})\rangle=\langle\hat{\xi}_{I}(t)\hat{\xi}_{I}(t^{\prime})\rangle=\langle\hat{\xi}_{I}^{\dagger}(t)\hat{\xi}_{I}^{\dagger}(t^{\prime})\rangle=0.$
(28)

Formally integrating out Eq. 26 we obtain

$\hat{O}_{\text{I}}(t)=\hat{O}_{\text{I}}(0)-i\int_{0}^{t}dt^{\prime}[O_{\text{I}}(t^{\prime}),H^{\text{int}}_{\text{I}}(t^{\prime})].$
(29)

Substitute this equation back to Eq. 26 we obtain

$i\partial_{t}\hat{O}_{\text{I}}(t)=[\hat{O}_{\text{I}}(0),\hat{H}_{\text{I}}^{\text{int}}(t)]-i\int_{0}^{t}dt^{\prime}[[\hat{O}_{\text{I}}(t^{\prime}),\hat{H}_{\text{I}}^{\text{int}}(t^{\prime})],\hat{H}_{\text{I}}^{\text{int}}(t)].$
(30)

Now we trace out the bath degree of freedoms, and it yields

$\displaystyle i\partial_{t}\hat{O}^{S}_{\text{I}}(t)$
$\displaystyle=\text{Tr}_{\text{B}}\left[\hat{\rho}_{\text{B}}\left([\hat{O}^{\text{S}}_{\text{I}}(0),\hat{H}_{\text{I}}^{\text{int}}(t)]-i\int_{0}^{t}dt^{\prime}[[\hat{O}^{\text{S}}_{\text{I}}(t),\hat{H}_{I}^{\text{int}}(t^{\prime})],\hat{H}_{\text{I}}^{\text{int}}(t)]\right)\right]$
(31)
$\displaystyle=-i\int_{0}^{t}dt^{\prime}\text{Tr}_{\text{B}}\left[\hat{\rho}_{\text{B}}(\hat{O}^{\text{S}}_{\text{I}}(t)\hat{H}^{\text{int}}_{\text{I}}(t^{\prime})\hat{H}^{\text{int}}_{\text{I}}(t)+\hat{H}^{\text{int}}_{\text{I}}(t)\hat{H}^{\text{int}}_{\text{I}}(t^{\prime})\hat{O}^{\text{S}}_{\text{I}}(t)-\hat{H}^{\text{int}}_{I}(t^{\prime})\hat{O}^{\text{S}}_{\text{I}}(t)\hat{H}^{\text{int}}_{\text{I}}(t)-\hat{H}^{\text{int}}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{H}^{\text{int}}_{\text{I}}(t^{\prime}))\right].$
(32)

The first term in Eq.31 only contains single bath operator, therefore, thanks
to Eq. 28, it vanishes after tracing out the bath. Since the kernal in the
integration of Eq. 32 decays fast enough, we can extend the the upper limit of
the integration to infinity. Then, using the bath correlation function Eq. 27
and Eq. 28, we have

$\displaystyle i\partial_{t}\hat{O}^{\text{S}}_{\text{I}}(t)$
$\displaystyle=-i\lambda^{2}\int_{0}^{\infty}dt^{\prime}\text{Tr}_{B}[\hat{\rho}_{B}(\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}^{\dagger}(t)\hat{M}_{\text{I}}(t)+\hat{M}^{\dagger}_{\text{I}}(t)\hat{\xi}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{M}_{\text{I}}(t^{\prime})\hat{O}^{\text{S}}_{\text{I}}(t)$
(33)
$\displaystyle\quad-\hat{M}^{\dagger}_{\text{I}}(t)\hat{\xi}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{M}_{\text{I}}(t^{\prime})-\hat{M}^{\dagger}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}(t^{\prime})\hat{O}^{\text{S}}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t)\hat{M}_{\text{I}}(t))]$
$\displaystyle=-i\lambda^{2}\int_{0}^{\infty}dt^{\prime}\langle\hat{\xi}_{\text{I}}(t)\hat{\xi}^{\dagger}_{\text{I}}(t^{\prime})\rangle(\\{\hat{M}_{\text{I}}^{\dagger}(t)\hat{M}_{\text{I}}(t),\hat{O}^{\text{S}}_{\text{I}}(t)\\}-2(-1)^{\eta}\hat{M}_{\text{I}}^{\dagger}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t))$
$\displaystyle=-i\lambda^{2}(\\{\hat{M}_{\text{I}}^{\dagger}(t)\hat{M}_{\text{I}}(t),\hat{O}^{\text{S}}_{\text{I}}(t)\\}-2(-1)^{\eta}\hat{M}_{\text{I}}^{\dagger}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t)),$

where the index $\eta$ comes from exchanging operator $\hat{\xi}$ or
$\hat{\xi}^{\dagger}$ with operator $\hat{O}^{\text{S}}_{\text{I}}$ in system.
When $\hat{M}$ is fermionic, $\hat{\xi}$ should also be fermionic. Then, if
$\hat{O}$ is also fermionic, there will be an extra minus sign when exchanging
$\hat{\xi}$ or $\hat{\xi}^{\dagger}$ with $\hat{O}$, and therefore, $\eta=1$.
Otherwise, if either $\hat{\xi}$ or $\hat{O}$ is bosonic, or both of them are
bosonic, there will be no extra sign when exchanging $\hat{\xi}$ or
$\hat{\xi}^{\dagger}$ with $\hat{O}$, and therefore, $\eta=0$. Then, we return
to the Heisenberg picture from interaction picture by the unitary
transformation
$\hat{O}^{S}(t)=\hat{U}^{\dagger}_{0}(t)\hat{O}^{S}_{\text{I}}(t)\hat{U}_{0}(t)$.
Hence, we finally reach the operator Lindblad equation

$\frac{d\hat{O}^{S}(t)}{dt}=i[\hat{H},\hat{O}^{S}]-\gamma\left(\\{\hat{M}^{\dagger}\hat{M},\hat{O}^{S}\\}-(-1)^{\eta}2\hat{M}^{\dagger}\hat{O}^{S}\hat{M}\right).$
(34)

where $\gamma=\lambda^{2}$. When $\eta=0$, it is consistent with Eq. 22.

There is a tricky point that worths mentioning in Eq.33. Considering the term
$\hat{M}^{\dagger}_{\text{I}}(t)\hat{\xi}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{M}_{\text{I}}(t^{\prime})$,
we can also write it as

$\text{Tr}_{B}[\hat{\rho}_{B}\hat{M}^{\dagger}_{\text{I}}(t)\hat{\xi}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{M}_{\text{I}}(t^{\prime})]=\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]$
(35)

In order to trace out the bath operators, we need to move
$\hat{\rho}_{\text{B}}$, $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ together. There
are seemingly two different methods to do so. The first method is to utilize
the partial-trace’s permutation as follows:

$\displaystyle\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]$
$\displaystyle=\text{Tr}_{B}[\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})]$
$\displaystyle=\text{Tr}_{B}[\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)]\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})$
$\displaystyle=\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime}).$
(36)

The second method is to directly pass $\hat{\xi}^{\dagger}$ through
$\hat{M}^{\dagger}\hat{O}^{S}\hat{M}$

$\displaystyle\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]$
$\displaystyle=(-1)^{\eta}\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})]$
$\displaystyle=(-1)^{\eta}\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime}).$
(37)

This method generates a minus sign in the case when $\hat{M}^{\dagger}$,
$\hat{O}^{\text{S}}$, $\hat{M}$ and $\hat{\xi}^{\dagger}$ are all fermionic.
Obviously, these two methods contradict with each other once this minus sign
is present. Here we should argue that the first method is not correct when
$\hat{\xi},\hat{\xi}^{\dagger}$ are fermionic.

Let us consider the total Hilbert space $\mathcal{H}_{\text{total}}$ as a
tensor product of system and bath
$\mathcal{H}_{\text{S}}\otimes\mathcal{H}_{\text{B}}$. And let us consider
$\hat{M}^{\dagger}$, $\hat{M}$ and $\hat{O}^{\text{S}}$ are supported solely
on $\mathcal{H}_{\text{S}}$. When $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are
bosonic operators, their support are solely on $\mathcal{H}_{\text{B}}$. Then,
$\hat{\xi}^{\dagger}$ and $\hat{\xi}$ commute with
$\hat{M}^{\dagger}\hat{O}^{S}\hat{M}$. Both methods are consistent with each
other. However, when $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are fermionic,
although they are bath operators, they are supported on the total Hilbert
space $\mathcal{H}_{\text{total}}$ in order to fulfil the anti-commutation
relation with fermion operators in $\mathcal{H}_{\text{S}}$. In other word,
the matrix representation of $\hat{\xi}$ or $\hat{\xi}^{\dagger}$ requires a
sign that depends on the physical state of the system. This is reminiscent of
the Jordan-Wigner transformation where fermions carry non-local strings in its
matrix representation. Thus, the partial-trace’s permutation, i.e. the first
equality in Eq. 36, is not correct. This difference between bosonic and
fermionic operators is explicitly illustrated in Fig. 5. Moreover, in
fermionic case, one can alternatively assume that $\hat{\xi}$ and
$\hat{\xi}^{\dagger}$ are solely supported on $\mathcal{H}_{\text{B}}$. Then,
$\hat{M}^{\dagger}$, $\hat{M}$ and $\hat{O}^{\text{S}}$ should supported on
the entire $\mathcal{H}_{\text{total}}$ in order to fulfill the anti-
commutation relation. Then, the second equality in Eq. 36 is not correct. In
any case, the first method fails. The same discussion applies to another term
$\hat{M}^{\dagger}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}(t^{\prime})\hat{O}^{\text{S}}_{\text{I}}(t)\hat{\xi}_{\text{I}}^{\dagger}(t)\hat{M}_{\text{I}}(t)$
in Eq. 33.

Figure 5: A quantum circuit illustration of evaluation of
$\text{Tr}_{B}[\hat{\rho}_{B}\hat{\xi}_{\text{I}}(t)\hat{M}^{\dagger}_{\text{I}}(t)\hat{O}^{\text{S}}_{\text{I}}(t)\hat{M}_{\text{I}}(t^{\prime})\hat{\xi}_{\text{I}}^{\dagger}(t^{\prime})]$.
(a) When $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are fermionic, they are
supported on the total Hilbert space and the partial-trace’s permutation is
incorrect. (b) When $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are bosonic, they
are only supported on the Hilbert space of bath and the partial-trace’s
permutation is correct.