# Account Abstraction, Analysed

Qin Wang1, Shiping Chen1
1CSIRO Data61, Australia

###### Abstract

Ethereum recently unveiled its upcoming roadmap’s Splurge phase, highlighting
the integration of EIP-4337 as a foundational standard for account abstraction
(AA). AA aims to enhance user accessibility and facilitate the expansion of
functionalities. Anticipatedly, the deployment of AA is poised to attract a
broad spectrum of new users and ignite further innovation in DApps. In this
paper, we elucidate the underlying operating mechanisms of this new concept,
as well as provide a review of concurrent advancements in accounts, wallets,
and standards related to its development. We step further by conducting a
preliminary security evaluation to qualitatively assess the extent of security
enhancements achieved through AA updates.

###### Index Terms:

Ethereum, Account Abstraction, EOA

##  I Introduction

Accounts within the Ethereum ecosystem [1] serve as the bedrock for asset
querying, storage, and transactions, constituting a pivotal element of its
infrastructure. However, the present account design poses challenges for Web2
users due to its complicated design. Ethereum classifies accounts into two
types [2], namely, externally owned accounts (EOA) and contract accounts
(CA)111Across several documents, the term smart contract account has been used
interchangeably to denote the same concept within the context of this paper..
EOA is controlled by private keys held by its creators or users. Contract
accounts are controlled by codes without the involvement of private keys
(Fig.1). The reliance on private keys makes these two accounts act in
different roles: EOA can prove the validity of a transaction and trigger the
state transition in CAs but with limitations:

  * •

Expense: The functioning of the contract wallet necessitates initiation by an
EOA, essentially entailing a contract invocation. Every transaction within
this process incurs an extra 21,000 Gas cost, which includes charges for ECDSA
signature verification, Nonce value verification, and ensuring adequate
account balance.

  * •

Elevated barrier: EOAs must possess a substantial ether balance to cover Gas
expenses (involving management of two separate accounts), or alternatively
depend on a Relayer to manage Gas payments, potentially introducing
centralization concerns.

  * •

Flunction: In addition to the fee in ether (ETH), users are consequently
required to hold ether, which exposes them to the potential volatility of its
price.

  * •

User perception: From a user’s perspective, grasping the nuances of gas price,
gas limit, and transaction congestion is far from straightforward.

The effort to integrate two distinct types of accounts while preserving their
core functionalities has been a longstanding subject of deliberation within
Ethereum communities. Two primary technical pathways have come to the
forefront.

  * •

EOA delegates control to smart contracts, in which the contract logic can
inplement the core functionalities from EOA transactions such as gas payment
(e.g., EIP222Ethereum Request for Comment (ERC) is one type of Ethereum
Improvement Proposals (EIP) that requires on-chain agreements. Throughout this
paper, we use EIP to refer to all standards for the sake of simplicity.-3074).

  * •

EOA accounts are designed to be armed with several key functionalities from
smart contracts (EIP-4337).

The concept of account abstraction (AA), indexed by EIP-4337 [3], aligns with
the second approach, aiming to bestow EOA with the programmable functionality
akin to CAs. The incorporation of EIP-4337 into the present roadmap [4]
signifies Ethereum’s definitive stance in these dual directions. By adopting
AA updates, Ethereum sidesteps the potential hurdle of transitioning existing
users. This is attributed to its implementation on the application layers
rather than the consensus layer, which inherently offers robust backward
compatibility capabilities. Ethereum transactions are required to emanate from
EOAs, which are typically accessed through various wallets like MetaMask,
Phantom, and Rainbow. The account abstraction approach ensures the retention
of current users while empowering them with an effortless means of engaging
with smart contracts. Beyond that, the functionalities elucidated in EIP-4337
can be further applied across Ethereum-compatible blockchains, including
platforms such as BNB Chain (formerly BSC), Polygon, Avalanche, Optimism,
Arbitrum, and Base (also refer to Sec.III-D).

Figure 1: Ethereum accounts [5]

As of August 2023, a sequence of indicators333Dune Analytics
https://dune.com/niftytable highlights the escalating attention garnered by
this impactful concept. The cumulative count of active accounts has reached
739,295, and the aggregate count of successfully executed
$\mathsf{UserOperation}$s (detailed explanations refer to Sec.III) has reached
1,307,197. In addition, the total number of bundled transactions has amounted
to 1,197,871. Besides, in August, a remarkable upswing is observed across
various market indicators compared to previous months (starting from March
2023, which marks the launch of EIP-4337): The count of monthly active
EIP-4337 smart accounts has surged to an impressive 420k (inclusive of all
platforms). The earnings from monthly revenues and UserOperation fees have
surpassed 24k and 220k respectively. Notably, the monthly tally of
successfully executed UserOperations stands at 710k, while the gas expenses
covered by paymasters have exceeded $360k USD.

Our attempts. Given its nascent stage, only a limited number of studies have
delved into this emerging concept. We are aligned with the growing momentum
surrounding account abstraction and further expand the boundaries of its
research landscape. In particular,

  * •

Concept refinement (Sec.II&III). We conduct a thorough exploration of diverse
resources, including academic literature, blogs, forum discussions, and Git
repositories. Drawing from such analyses, we offer a coherent and succinct
elucidation of the fundamental concepts and operational mechanisms of account
abstraction.

  * •

Security framework (Sec.IV). We further delve into the security risks that may
exist in AA. Delving into historical contract and account-related
vulnerabilities, we construct a framework (Tab.III) to engage in a proper
discussion about the security implications of this account paradigm.

  * •

Further discussion (Sec.V). Additionally, we provide more discussion about the
principle of account abstraction designs and potential opportunities inspired.

Security results. Our security discussion (majorly Sec.IV) led to the
following insights. It is evident that integrating AA into current account
systems can alleviate a multitude of vulnerabilities in both contract usage
and block creation domains. However, AA’s effectiveness remains limited when
addressing intrinsic aspects of solidity language and design intricacies, such
as structural elements, configuration constraints, and verification
mechanisms. Upon further analysis, we have identified that AA’s strength in
fortifying security arises from its decoupling of key account abstraction
components. By reallocating functions previously encompassed within accounts,
AA can protect against numerous vulnerabilities in Ethereum’s application
layer: critical functions like gas payment mechanisms are seamlessly
transferred to the paymaster, avoiding vulnerabilities related to gas fees;
the intricate process of transaction packaging now benefits from the
management of bundler; and complex trading and swapping operations, previously
reliant on CEX or DEX, are elegantly executed within the same transaction. The
decoupling process has reduced the threat that was heavily dependent on
contract functions. thereby increasing the overall security quotient.

Available resources. The Ethereum official documentation [4] has introduced
formal documentation detailing its account abstraction concept and roadmap
that is accompanied by a suite of relevant standards, encompassing EIP-2771,
4337, 2938, and 3047. Notably, EIP-4337 encapsulates the central concept
driving this transition. In a study by Singh et al. [6], an initial
exploration into the Ethereum account abstraction is provided, outlining its
distinctive features and functional methodologies. Binance Research [7] has
released an insightful analysis that delves into recent market trends and
noteworthy advancements within this area. Based on these contributions, a
range of media sources [8, 9, 10, 11] have embarked on elucidating this
intricate concept, offering surrounding insights and explanations. Besides,
several studies have initiated the development of on-top solutions aimed at
addressing various challenges within the Web3 ecosystem [12], such as address
incompatibility [13].

##  II Ethereum Accounts

###  II-A Recall Ethereum Account

EOA. EOAs serve the purpose of storing and transferring ether, and ERC-20
tokens. EOAs are generated from a public key, which is a 20-byte hexadecimal
identifier (e.g., 0xF57D1D6b84db4053cE452B35B7DB77878dCbdc65). These accounts
are managed by a private key, which includes the EOA’s password or seed
phrase, held exclusively by the account’s owner. Transactions involving EOAs
do not rely on code or smart contract logic for their validity. As long as the
private key remains known, the account’s owner possesses the capability to
execute transactions. The transaction’s verification is contingent upon the
user’s signature and nonce.

Contract account. CA is a type of account that executes operations based on
its pre-programmed logic, thereby enabling the creation of decentralized
applications (DApps) and facilitating various functionalities within the
Ethereum network. CA autonomously executes code in response to transactions,
potentially modifying the contract’s state, and exchanging ether or tokens. CA
accounts are assigned unique addresses akin to individual identification
numbers, which permit interaction with other accounts. Once a CA is deployed,
its code and state become immutable, contributing to transparency and ensuring
the integrity of transactions. They also offer storage capabilities for
manipulating data on the blockchain (Fig.1). These accounts can interact with
EOAs and generate events that facilitate communication with other accounts.

Two types of transaction. These different accounts give rise to two distinct
practical types of transactions for communication: contract creation and
message call. Contract creation involves the generation of a new smart
contract, with the transaction carrying an initialization code segment to
define the new contract’s properties. This process results in the assignment
of a unique address to the newly created contract, which includes both its
code and storage within the corresponding account state. Conversely, a message
call signifies the modification of a smart contract’s state. In this case, the
transaction includes input data to update the contract’s internal data. A
message call does not create a new contract in the world state; instead, it
alters the existing contract’s state.

###  II-B Challenges in Account Design

Coming with private keys, EOA has many fundamental functions including
claiming the ownership of the account and signing the permissions of
transactions. However, this may present a number of significant concerns:

  * •

Risk of private key loss. Users who lose their private keys (due to loss or
hacking) would face the irreversible loss of all their assets.

  * •

Restricted signature options. The native protocol exclusively supports ECDSA
signature and verification algorithms for transaction validation.

  * •

Single signer authority. The absence of inherent multi-signature capability
(which can only be achieved through smart contracts) means that a single
signature is all that’s required to execute any operation.

###  II-C Ethereum Roadmap

Account abstraction is a crucial functional improvement outlined in the sixth
phase of Ethereum’s roadmap (Tab.I). This upgrade involves a series of smaller
refinements and adjustments aimed at ensuring seamless network operations
subsequent to the implementation of other upgrades.

TABLE I: Ethereum’s roadmap

Merge | Move to proof-of-stake consensus
---|---
Surge | Rollup-centric scaling to 100,000+ transactions per second
Scourge | Avoid centralization and other protocol risks from MEV
Verge | Verifying blocks to become super easy
Purge | Simplify the protocol and reduce the costs of running nodes
Splurge | Fix everything else (account abstraction included)

##  III Account Abstraction

###  III-A Previous Attempts

A series of continuous efforts have been made to make existing accounts
extensible with more functions. We list a series related token standards in
Tab.II.

TABLE II: AA-related Ethereum token standards

EIP- | Time | Status |  Key
---|---|---|---
101 | 2015 | Stagnant | Serenity currency and crypto abstraction
86 | 2017 | Stagnant | Abstraction of transaction origin and signature
859 | 2018 | Stagnant | Account abstraction for main chain
2718 | 2020 | n/a | Typed transaction envelope
2938 | 2020 | Stagnant | Account abstraction
3607 | 2021 | n/a | Reject transactions from senders with deployed code
5003 | 2022 | Stagnant | Insert Code into EOAs with AUTHUSURP
5189 | 2022 | Stagnant | Account abstraction via endorsed operations
2771 | 2020 | Final | Secure Protocol for native meta transactions
3074 | 2020 | Review | Allow EOA to delegate control to a contract
6900 | 2021 | Draft | Modular Smart Contract Accounts and Plugins
4337 | 2021 | Draft | Account abstraction using alt mempool
6551 | 2023 | Review | Non-fungible token bound accounts

In Ethereum’s early phases, attempts were made to distinguish between EOA and
CA by introducing various new transaction types. EIP-86/208 leverage the
differentiation between two accounts to design customizable and collision-
resistant contract addresses. This, in turn, led to the implementation of
EIP-1014 and EIP-2470. EIP-859 introduced transaction-initiated contract
deployment. It allowed for on-the-spot deployment of contract addresses if
none existed, forming the basis for the fundamental functionality of EIP-4337.
EIP-2718 ushered in compatibility for future Ethereum iterations with any
newly suggested transaction types. EIP-293 systematically cataloged various
advantages of contract accounts, encompassing recovery, key rotation,
customized identity verification algorithms, and meta-transactions, which
collectively entrenched the role of contract accounts.

However, these efforts resulted in unwieldy complexity. Altering transaction
types requires concurrent modifications to the underlying signature
verification algorithms. This encompasses considerations like miner acceptance
of new types, ensuring incentives are not lower than regular transactions to
encourage verification, and addressing concerns about account address
management and conflicts. Unfortunately, this approach lacked both backward
and forward compatibility. Ultimately, these attempts have coalesced around
two primary directions as stated in our Sec.I.

###  III-B Account Abstraction

Account abstraction introduces a suite of functions designed to personalize
fundamental elements that enable smart contract functionalities. These
customizations encompass various aspects, including user operations, fee
payment methods, and transaction packing mechanisms.

Overview. The workflow is summarized as follows: ① Users initiate interactions
with the frontend abstraction (also referred to as application) layer, where
their actions are translated into underlying transactions (via
$\mathsf{UserOperation}$). ② Bundlers aggregate multiple
$\mathsf{UserOperation}$ instances, consolidating them into a single
transaction that is then transmitted to the $\mathsf{EntryPoint}$ contract. ③
Within the $\mathsf{EntryPioint}$ contract, user signatures are verified and
transactions initiated by the abstraction layer are processed. ④ The entries
logged in $\mathsf{UserOperation}$ prompt the activation of relevant smart
contracts, initiating state transitions tailored to their specific
prerequisites. ⑤ Optionally, the user’s operations can be aggregated and
authenticated using BLS signatures. Meanwhile, transaction fees for user
actions are managed by the $\mathsf{Paymaster}$ contract. Consequently, on-
chain applications interact with user actions in a manner similar to their
interaction with standard external accounts.

Figure 2: Account abstraction

✑ User operation. $\mathsf{UserOperation}$ serves as a transaction-like
entity, representing a user’s request achieved by transaction events. Within
$\mathsf{UserOperation}$, multiple requests and supplementary data can be
encapsulated, facilitating the execution of transactions that a CA can
perform. While sharing similarities with conventional transactions,
$\mathsf{UserOperations}$ exhibit several distinctive characteristics.

  * •

Additional fields. $\mathsf{UserOperation}$ includes new fields in the
transaction structure, e.g., $\mathsf{EntryPoint}$, $\mathsf{Bundler}$,
$\mathsf{Paymaster}$ and $\mathsf{Aggregator}$.

  * •

Alternate mempool. $\mathsf{UserOperation}$ is sent to a separate mempool,
where bundlers can package them into transactions which get included in a
block.

  * •

Intent based. Today, transaction inputs are specific, e.g., swap 2k USDC for
1.2ETH. In contrast, $\mathsf{UserOperation}$ can be decorated with additional
metadata to be more outcome-focused, e.g., I want to trade 2k USDC for the
most amount of ETH possible.

✑ Bundler. Bundlers constitute a pivotal component of the infrastructure
necessary for the realization of EIP-4337. These entities are a specific type
of Ethereum node dedicated to facilitating the processing of
$\mathsf{UserOperation}$s. These $\mathsf{UserOperation}$s are directed to a
network of bundlers, which diligently monitor the mempool. Bundlers
efficiently consolidate multiple $\mathsf{UserOperation}$s into a singular
transaction, expertly packaging and submitting them to the blockchain on
behalf of users. In exchange for this service, bundlers receive compensation
for their efforts in executing these tasks.

✑ EntryPoint. It is a singleton smart contract that undertakes the
verification and execution of $\mathsf{UserOperation}$s.

  * •

Verification loop. The process checks the account balance. It involves
assessing whether the wallet possesses sufficient funds to cover the maximum
potential gas expenditure (derived from the gas field in the UserOp).
Transactions lacking adequate funds will be declined.

  * •

Execution loop. Upon verification, the transaction will be executed.
Correspondingly, an amount from CA is deducted to reimburse the bundler. This
reimbursement is required to cover the gas expenses.

✑ Aeggregator (optional). In scenarios where multiple messages are signed
using distinct keys, an aggregator plays a role in producing an authenticated
signature. The signature inherently validates the authenticity of all
individual signatures within the collection. For accounts associated with
particular signature types that enable aggregation (such as BLS), the
verification process of the account’s signature is deferred to an external
contract. This external contract, in turn, is responsible for verifying a
solitary signature across an entire bundle, thus streamlining the validation
process.

Through the consolidation of multiple signatures into a singular entity,
aggregators contribute to the efficient management of data availability. This
allows for the validation of numerous bundled $\mathsf{UserOperation}$s in a
single consolidated process.

✑ Paymaster (optional). The Paymaster takes charge of implementing gas payment
policies, offering developers the capability to offer their end users gas-free
interactions through sponsored or ERC-20 gas policies. These policies
introduce flexibility in how gas is paid, such as the choice of currency
(e.g., native ETH or ERC-20 tokens), and by whom it is paid. This effectively
eliminates the need for users to possess native blockchain tokens to engage
with the blockchain.

###  III-C AA Benefits

Based on its design, AA brings a series of direct benefits that previous CA
and EOA do not have.

  * •

Transaction batching. Capitalizing on the capabilities of the
$\mathsf{EntryPoint}$ contract, the consolidation of multiple transactions can
guard against frontrunning attacks and potential MEV threats.

  * •

Multisig. Departing from reliance solely on a single signer (EOA, private key
holder), AA introduces the capability of multi-signatures sourced from a
spectrum of entities.

  * •

Gas abstraction. ERC-20 tokens are accepted as valid payment for these fees,
unshackling the restrictions previously confined to ether (ETH).

  * •

Social recovery. AA is engineered to encompass a mechanism for recovery
through a social network of trusted associates. This mechanism offers an
alternative route to regain access to the wallet.

  * •

Custom module. Users are empowered with the ability to craft bespoke modules
catering to specific functions.

###  III-D AA Surroundings

Supportive standards. EIP-2771 is designed to enable gasless transactions for
users by introducing the concept of meta-transactions. It permits third
parties to cover a user’s gas expenses without necessitating modifications to
Ethereum. The standards include the following key steps: The transaction
signer signs and dispatches transactions to a gas relay. The Gas Relay,
operating off-chain, receives these signed requests and subsidizes the gas
costs, transforming them into valid transactions that are then routed through
a trusted forwarder. This trusted forwarder is essentially a contract
entrusted by recipients to accurately verify signatures and nonces.

EIP-3047 is designed to enable users to delegate control of their EOA to a
contract, a parallel approach discussed in Sec.I. This feature allows any EOA
to function as a smart contract wallet without requiring the deployment of a
separate contract. As part of this standard, two new opcodes are introduced to
the Ethereum Virtual Machine (EVM): AUTH and AUTHCALL. These opcodes
facilitate proper invocation from smart contracts. Although this standard are
incorporated into future updates of Ethereum’s account infrastructure, it
still provides valuable insights.

EIP-6900 introduces a standard aimed at synchronizing the implementation
efforts of CA developers and plugin developers. It outlines the creation of a
modular CA with the capability to accommodate all plugins adhering to standard
protocols. In this standard, CAs are rendered as fully programmable accounts
capable of housing assets within on-chain smart contracts. In parallel,
account plugins serve as interfaces for smart contracts that facilitate the
integration of compositional logic within CAs. This approach enhances data
portability for users and alleviates the need for plugin developers to target
specific wallet implementations.

EIP-6551 introduces a token standard primarily focused on NFTs [14]. This
standard enables the binding of one or multiple Ethereum accounts to an NFT.
By doing so, NFTs gain the capability to possess assets and engage with
applications, all without necessitating modifications to current smart
contracts or infrastructure. These accounts, which are associated with tokens,
seamlessly integrate with existing on-chain asset standards and have the
potential to encompass forthcoming asset standards. Ethereum’s abstract
account model offers an enhanced approach to implementing this standard,
exemplified by the Web3 game project Sapienz [15].

Supportive platforms. In light of the forthcoming updates to Ethereum, a
diverse range of competitive blockchain platforms are also stepping up their
game. Notable contenders such as BNB Chain, Avalanche, Optimism, Polygon,
StarkNet, and zkSync have all unveiled their respective initiatives for
implementing crucial updates.

Supportive wallet. Complementing the infrastructural advancements, an equally
pivotal component lies within the realm of wallets. Serving as the hub where
users manage, utilize, and safeguard their accounts, wallets play a pivotal
role. Multiple wallet service providers, including MetaMash, Argent, Beam,
Safe, Trust Wallet, Braavos, and OKX Wallet, have expressed their commitment
to seamlessly incorporate these anticipated updates into their products.

###  III-E Real Case Adoption

Visa. As demonstrated in [16], Visa is actively exploring a solution that
leverages the Paymaster smart contract. This solution aims to abstract the
core interactions of blockchain technology and enhance the user payment
experience by introducing a self-custodial smart contract wallet. The primary
goal is to streamline the process for users conducting transactions within
their wallets. One of the intriguing aspects is that users can now use any
token to pay for gas fees, and these gas costs will be covered by Paymaster.
In more detail, users have the capability to make payments to the Paymaster
contract using USDC. Subsequently, the Paymaster contract converts the USDC
into ETH and facilitates the transaction on the blockchain network. After a
period of on-chain processing, the recipient of the transaction can receive an
equivalent amount converted back to USDC. As of now, this particular use case
is still in the conceptual verification stage.

Web3 wallet. By seamlessly integrating AA into existing EOA wallets, the
accounts are elevated to the status of smart contract wallets, enriched with
programmable logic and functionalities. For instance, Gnosis Safe [17] brings
forth a smulti-signature approach, necessitating multiple authorized entities
to provide signatures for the same account, rather than relying solely on
individual private keys. Argent [18] introduces the concept of social
recovery, enabling users to recover lost or forgotten private keys. It allows
users to utilize email addresses and phone numbers for offline recovery,
introducing a familiar two-factor authentication mechanism. Users in Braavos
[19] can access their wallets using the biometric features of their
smartphones, such as facial or fingerprint recognition, adding an extra layer
of security.

##  IV Security Analysis

We explore the potential security issues that may exist in new account formats
after its update. We recall a series of account threats and related smart
contract risks and develop a framework to evaluate potential risks in account
abstraction.

Focused scope. Conventional analyses of Ethereum often partition it into four
distinct layers (e.g., [20]): the application layer (including accounts, smart
contracts, and EVM), the data layer (including transactions, blocks, and
events), the consensus layer (involving PoX mechanisms and incentives), and
the network layer (comprising node discovery, message propagation, and
verification). Given the central theme of this endeavor around accounts, we
deliberately narrow our purview to the application layer, exploring the
potential vulnerabilities targeting accounts, smart contracts, and related
solidity language. Additionally, our coverage extends to certain elements
within the transaction and block layers.

Framework design (Tab.III). Inspired by a series of elegant investigations
[21, 22, 23, 24, 25, 20, 26, 27], we distill a succinct overview of
vulnerabilities present within the current Ethereum application layer. These
vulnerabilities can be broadly attributed to three factors: design pitfalls,
vulnerabilities intrinsic to the Solidity programming language, and errors
stemming from contract programming. Within each of these categories, we
identify and elaborate upon sub-vulnerabilities (Vulnerability) that manifest
across diverse functions and processes (CauseFmWhere). Our discussion then
delves into the potential of account abstraction to alleviate these
vulnerabilities. We examine the feasibility of employing account abstraction
to mitigate these issues and explore strategies for implementation that could
effectively address these concerns (byWhichFunc).

TABLE III: Vulnerabilities in Ethereum application layer

| Vulnerability | CauseFmWhere | byWhichFunc
---|---|---|---
Design | Transaction disorder | BlockCreate | $\mathsf{Bundler}$
Timestamp manipulation | BlockCreate | $\mathsf{Bundler}$
Short address | InputCheck |  $\mathsf{Bdr}$/$\mathsf{EntryPoint}$
Randomness reliance | BlockCreate | ✗
Empty account | StateTrie | ✗
Under-price opcodes | GasCost | ✗
Stack limit | Execution | ✗
Solidity | Unbounded operations | GasUsage | $\mathsf{Paymaster}$
Inconsistent call returns | Exception | ✗
Pointer unreset | BlankField | ✗
Outdated compiler | Compiler | ✗
Unclear constructor name | Syntax | ✗
Type casts | Design | ✗
Smart Contract | Reentrancy | Dependence | $\mathsf{Paymaster}$
Frozen ether | Dependence | $\mathsf{Bundler}$
contract Upgradeability | Dependence | ✗
Delegatecall injection | Dependence | ✗
Unexpected revert | Dependence | ✗
Manipulated balance | Validation |  $\mathsf{PyMs}$/$\mathsf{UsrOpn}$
Integer overflow | Validation | ✗
Insufficient signature | Authentication |  $\mathsf{Bdr}$/$\mathsf{MultiSig}$
Self-destruct contract | Authentication | $\mathsf{EntryPoint}$
$\mathsf{tx.origin}$ | Authentication |  $\mathsf{UsrOpn}$/$\mathsf{Bdr}$
Wrong payment | Authentication |  $\mathsf{UsrOpn}$/$\mathsf{Bdr}$
Lack of confidentiality | Authentication | ✗
Erroneous visibility | Authentication | ✗
| AA updates

  * Abbr. PyMs - Paymaster; UsrOpn - UserOperation; Bdr - Bundler.

Analyses. Our discussions are categorized into two facets, contingent upon the
effectiveness of AA updates.

✓ Transaction disorder. This vulnerability centers around a concurrency issue
where the blockchain’s future state is contingent upon the sequence of
transaction execution, a process influenced by miners who group transactions
into blocks based on incentives. The design of AA effectively mitigates this
problem by employing bundlers to manage and submit transactions. Unlike miners
or validators, bundlers have a limited scope of impacting transaction orders,
as they handle a relatively small number of transactions.

✓ Timestamp manipulation. This vulnerability happens when a contract utilizes
$\mathsf{block.timestamp}$ in crucial operations or as a means of generating
randomness. It can be exploited by a malicious miner who has the capability to
manipulate the value of $\mathsf{block.timestamp}$ (evidenced by [28]). The
introduction of account abstraction helps address this concern, as AA reduces
reliance on $\mathsf{block.timestamp}$ and shifts focus to bundlers, which are
better suited for managing transactions and ensuring more controlled and
secure interactions.

✓ Short address. The vulnerability stems from the EVM’s absence of address
validity checking. In the process of encoding during contract invocation, if
the encoded arguments’ length is insufficient, the EVM compensates by adding
extra zeros to ensure 32 bytes. The AA design, through the involvement of the
$\mathsf{bundler}$ and $\mathsf{EntryPoint}$ contract, can mitigate this
issue. The design can effectively check the length of Ethereum addresses via
$\mathsf{msg.data}$.

✗ Randomness reliance. This vulnerability revolves around the manipulation of
seed (a form of randomness) by malicious miners in gambling and lottery
contracts that use pseudorandom number generation. AA might not directly
enhance the security of random seed creation. This is because the process of
generating random seeds typically involves functions within the contract logic
rather than being directly embedded in accounts, which is the primary focus of
AA.

✗ Empty account. Malicious actors leverage empty accounts (devoid of nonce,
balance, code, and storage) to initiate a DoS attack. These empty accounts
still necessitate tracking within the Ethereum state trie and result in
extended transaction processing times. AA is not equipped to mitigate this
vulnerability, as it currently lacks verification mechanisms for empty
accounts. However, the issue has been addressed through the implementation of
EIP-161.

✗ Under-price opcodes. Ethereum uses the gas mechanism to deter abuse of
computing resources, but this vulnerability emerges when the gas cost for
resource consumption is improperly set. The exploitation of under-priced
opcodes will consume a disproportionate amount of computing resources. This
issue is not specifically addressed by the design of account abstraction, as
it pertains to the gas pricing and resource allocation mechanisms within
Ethereum.

✗ Stack limit. Solidity would trigger an exception and terminate the call due
to its rigid limit of 1,024 frames in the EVM call stack. Malicious actors
could recursively call a contract, causing it to reach the maximum stack depth
and fail subsequent external calls. While account abstraction cannot
inherently address this specific vulnerability, changes in the Ethereum
protocol (e.g., EIP-150) can indirectly impact AA as well as other aspects of
the Ethereum ecosystem.

✓ Unbounded operations. The vulnerability arises when a contract’s execution
demands a greater amount of gas than the permitted block gas limit. This
occurs due to the inclusion of unbounded operations, such as loops, within the
contract. AA design offers effective solutions to counter such issues in two
primary ways: Firstly, the paymaster contract defines a fee policy that
establishes clear limitations on gas limits. This mechanism ensures that
transactions adhere to predefined gas constraints, preventing situations where
execution demands exceed the allowable limit. Secondly, after AA’s update,
every loop in smart contracts is invoked by an internal account rather than an
external account. This setup grants the contract owner the ability to
terminate loops if necessary, enhancing control over potential gas consumption
issues.

✗ Inconsistent call returns. This vulnerability stems from the inconsistency
between the two return values from different calling routes: (i) directly
referencing the callee contract instance and (ii) using low-level methods like
send and call. Route-(i) will return the call back to the caller while
Route-(ii) returns false to the caller. The inconsistency in behavior persists
in the current state of the Solidity compiler, lacking rectification. AA
design does not inherently address this issue, but updates or changes in the
Solidity compiler or EVM could potentially influence AA as well.

✗ Pointer unreset. The vulnerability arises from Solidity’s behavior regarding
uninitialized compound local variables, which, when not explicitly
initialized, leads to the reference defaulting to slot 0 and overwriting
content from that point onwards. AA design does not address this particular
language-level issue, but it’s fortunate that the vulnerability was rectified
by updating the version of Solidity.

✗ Outdated compiler. This vulnerability surfaces when a contract is compiled
using an outdated compiler that harbors bugs. While AA design may not directly
address this concern, it aligns with the solution: opting for an up-to-date
compiler can substantially reduce the associated risks. Keeping the compiler
current is a key practice to ensure the solidity and security of the compiled
contracts.

✓ Unclear constructor name. The vulnerability stems from the misnaming of the
constructor function, inadvertently allowing unauthorized parties to take
ownership of the contract. While AA design cannot directly solve this issue,
it is fortunate that the problem was mitigated through the introduction of the
constructor keyword

✗ Type casts. A contract can invoke another contract’s function by directly
referencing the callee contract’s instance. However, the verification process
carried out by the Solidity compiler only assesses whether the invoked
function has been declared, and it doesn’t extend to verifying child functions
that the invoked function inherits or parent functions that trigger it. This
loophole can lead to unintended function execution. AA design does not offer a
solution for this vulnerability, as it stems from Solidity’s insufficient type
system. In this context, AA’s mechanisms don’t directly address this issue.

✓ Reentrancy. The vulnerability arises when an external contract initiates a
function within the caller contract before the latter’s ongoing execution
concludes—essentially manifesting as a cyclic call. This exploitable
invocation persists until the caller contract depletes its gas resources. The
foundation of this vulnerability can be attributed to two pivotal factors: (i)
a contract’s decision-making hinges on specific state variables, which
necessitate updating prior to invoking another contract; and (ii) the absence
of a predefined gas limit for the transition. The introduction of AA can offer
a partial solution to this issue. AA’s advantages, including reduced gas
consumption and the ability to control transactions precisely through the
paymaster contract, contribute to its potential mitigation. Additionally, user
operations are treated as distinct inputs through independent transactions,
fostering an environment free from dependencies on other contracts. This
segregation of operations enhances security by reducing the risk of
vulnerability to such cyclic call-related attacks.

✓ Frozen ether. The vulnerability arises from a scenario in which users can
deposit funds into their contract accounts but are unable to later access or
spend those deposited funds. Two primary factors contribute to this
vulnerability: (i) contracts not furnishing a function for the expenditure of
funds, and (ii) the inadvertent or deliberate termination of the callee
contract. Account abstraction can offer partial mitigation of this
vulnerability due to its transaction batching made by bundlers. With each
user-requested operation, such as depositing funds into a smart contract, the
transaction bundler gains an enhanced capability to inspect the contract
before sending tokens. This heightened level of examination surpasses that of
a standard user. The transaction bundler possesses the potential to ensure
that operations transpire within the confines of the originating contract’s
security.

✗ Upgraded contract. Contract upgrading can be approached through two methods:
(i) Dividing a contract into a proxy contract (non-updatable) and a logic
contract (updatable). (ii) Employing a registry contract to maintain a record
of the updated contracts. While the design of account abstraction is separate
from these complexities, its contract functionalities align with its
principles of upgradability.

✗ Delegatecall injection. EVM offers an opcode called $\mathsf{delegatecall}$,
allowing the bytecode of a callee contract to be inserted into the bytecode of
the caller contract. The vulnerability arises due to the possibility that a
callee contract can modify state variables within the caller contract. Account
abstraction does not inherently provide a distinct method to mitigate this
particular vulnerability. The effectiveness of AA’s security in mitigating the
issue hinges on its inherent design and solution. Fortunately, addressing this
can be achieved via a straightforward means by enhancing its declaration.

✗ Unexpected revert. This vulnerability stems from the situation where a
callee contract causes the execution of a caller contract to be reverted. The
design of account abstraction does not offer a significantly improved
mitigation for this issue, as it fundamentally stems from contract programming
practices. Consequently, this vulnerability might still be pertinent even
within AA’s contract functionalities.

✓ Manipulated balance. This vulnerability surfaces when a contract’s decision-
making relies on the values of $\mathsf{this.balance}$ or
$\mathsf{address(this).balance}$. These values can be manipulated by an
attacker to gain unauthorized access to funds. The design update introduced by
AA can potentially influence this vulnerability. In AA, all account addresses
are explicitly bound with balances, unlike the implicit nature of the former
design. This alteration could impact the dynamics of this vulnerability.

✗ Integer overflow. This vulnerability arises when the result of an arithmetic
operation surpasses the range of a Solidity data type. Regrettably, neither
the Solidity compiler nor the EVM incorporates mechanisms to identify integer
overflow or underflow. Account abstraction design limitations prevent it from
overcoming these challenges. However, a solution to mitigate this
vulnerability is within reach through the utilization of the SafeMath library,
which has already proven effective in handling such arithmetic issues.

✓ Insufficient signature. The vulnerability surfaces when a sender channels
funds to multiple recipients via a proxy contract, bypassing individual
transactions. In this mechanism, the proxy contract evaluates the authenticity
of digital signatures from senders. If these signatures lack essential data
(such as nonce or proxy contract address), a malicious recipient can exploit
the situation to replay the message repeatedly, facilitating the withdrawal of
surplus payments. In AA’s design, the role of the bundler mirrors that of a
proxy agent. The bundler manages the accumulation of multiple transactions,
functioning as a trusted entity. This approach not only aligns with the
principles of a proxy contract but also serves as a safeguard against replay
attacks. Further, the adoption of multi-sig techniques in AA can also
contribute to the mitigation of issues.

✓ Self-destruct contract. When the owner of a contract or a third party
utilizes the self-destruct method to terminate a contract, the contract’s
bytecode and storage will be erased. The vulnerability emerges from inadequate
authentication mechanisms embedded within the contract. AA can indeed mitigate
this issue by introducing multi-factor authentication, a more robust approach.
This mandates the approval of multiple parties before a suicide operation is
executed. By implementing this safeguard, AA enhances security by fortifying
the authentication process and reducing the likelihood of unauthorized
termination.

✓ $\mathsf{tx.origin}$. The problem originates from the utilization of
$\mathsf{tx.origin}$, a global variable within Solidity that identifies the
original EOA responsible for triggering the transaction. The vulnerability
arises when a contract employs $\mathsf{tx.origin}$ for authorization, thereby
exposing it to phishing attacks. AA addresses this concern through its
integration of EOA with contract invocation. The integration is pivotal as AA
replaces $\mathsf{tx.origin}$ with $\mathsf{msg.sender}$ in default
authentication. This shift is now a necessary step to bolster security against
vulnerabilities stemming from $\mathsf{tx.origin}$ usage.

✓ Wrong payment. This security issue arises due to the absence of identity
verification when a caller invokes a function to transfer Ether to a random
address. As seen in the mitigation strategies employed for previous balance-
related vulnerabilities, the authentication mechanisms and the role of the
bundler within account abstraction can aid in alleviating this problem. This
ensures that only authorized entities are permitted to initiate such actions,
thwarting unauthorized withdrawals.

✗ Lack of confidentiality. Transaction details within a blockchain are
inherently public due to the nature of the technology. While designating a
state variable as private can limit other contracts’ access, the value of such
a variable can still be inferred from transaction data. It’s worth noting that
AA design does not specifically address the intricacies of transaction
confidentiality; however, it aligns with the overall progression of smart
contract evolution. Mitigation strategies encompass the implementation of
cryptographic techniques like commitment schemes [29] and zero-knowledge
proofs [30]. Additionally, hardware-based solutions [31] could also be
explored to enhance the security of sensitive data within an untrusted
environment.

✗ Erroneous visibility. This vulnerability stems from inaccurately defining
the visibility of a function, where the default public visibility can be
exploited by attackers to improperly access functions. AA design doesn’t
directly address this issue but aligns with its solution. This vulnerability
is mitigated by requiring explicit specification of function visibility, a
measure AA’s design also adheres to.

##  V Further Discussions

###  V-A More about AA Design

AA represents an application-level update with its primary impact centered
around the application layer. Aligned with the idea outlined in EIP-4337, AA
introduces perfect backward compatibility, minimizing its effects on consensus
layers and underlying structures. The fact is evident in security
enhancements, which effectively tackle many challenges originating from the
application layer rather than fundamental design flaws (refer to Tab.III).

Rather than directly merging EOA and CA, AA incorporates key functionalities
and delegates subsequent tasks to related contracts in tandem. Notably,
separating the functions required to be executed within the smart contract and
allocating a portion of them to the user’s account enhances overall
efficiency. The corresponding batch processing isn’t limited to user
operations but also extends to fee payments.

The user experience is markedly improved when employing an AA account as
opposed to an EOA. While a series of interconnected contracts work
harmoniously to support the functionality of an AA account, users perceive it
as effortlessly accessible as popular Web2 Apps (e.g., shopping, and banking).
Complex technical jargon is relegated to the backend, and this will spare
regular users from unnecessary cognitive load. The user interface offers a
friendly entry point.

###  V-B AA Opportunities

Ease accessibility to Web3. Account abstraction not only streamlines the
interaction with DApps but also significantly enhances the user experience
within Web3. By embracing AA, users gain the ability to tailor their accounts
to operate exclusively under specific conditions, which significantly differs
from previous solutions that rely on complicated invocation across functions
or contracts [32]. Users are empowered with a greater degree of control over
their customized programming functions. For instance, unlike the traditional
multi-sig setup, where the conditions for executing transactions are
relatively standardized, account abstraction allows for a more personalized
set of conditions. This adaptability ensures that the operations are only
executed when predetermined criteria are met, such as a predefined number of
signatories.

Promoting Layer-2 (L2). The fundamental L2 projects revolve around alleviating
the substantial computational load on L1 chains. As a result, a significant
focus of various L2 solutions lies in streamlining the intricate processing
logic inherent in DApps. This optimization not only reduces the overall gas
cost but also enriches the user experience through the introduction of
extended functionalities. To realize this goal, numerous L2 projects have
embraced the notion of incorporating account abstraction, marking a pivotal
step toward achieving their mission. Notable examples [7] include zkSync,
which seamlessly integrates the IAccount interface, and StarkNet, with the
well-regarded platform Argent. Similarly, Optimistic rollups also witness the
integration of customized APIs.

##  VI Conclusion

In this paper, we explore the notion of account abstraction (featured by
EIP-4337), which is formally involved the sixth stage of Ethereum’s roadmap.
We study its operating mechanism, key features, and surrounding developments.
We also examine its security by assessing a set of related criteria. Our
results reveal the scope and extent of the security improvements introduced by
the adoption of account abstraction. To our knowledge, this work provides the
first formal AA study.

## References

  * [1] G. Wood _et al._ , “Ethereum: A secure decentralised generalised transaction ledger,” _Ethereum project yellow paper_ , vol. 151, no. 2014, pp. 1–32, 2014.
  * [2] @jmcook1186, “Ethereum accounts,” _Retrivedhttps://ethereum.org/en/developers/docs/accounts/_, 2023.
  * [3] “Ethereum: ERC-4337,” _https://www.erc4337.io/_ , 2023.
  * [4] Ethereum, “Ethereum roadmap: Account abstraction,” _https://ethereum.org/en/roadmap/account-abstraction/_ , 2023.
  * [5] T. T., “Ethereum EVM illustrated,” _https://takenobu-hs.github.io/downloads/ethereum_evm_illustrated.pdf_ , 2023\.
  * [6] A. K. Singh, I. U. Hassan, G. Kaur, S. Kumar _et al._ , “Account abstraction via singleton entrypoint contract and verifying paymaster,” _International Conference on Edge Computing and Applications (ICECAA)_ , pp. 1598–1605, 2023.
  * [7] C. Colin, “Biance research: A primer on account abstraction,” _https://research.binance.com/static/pdf/a-primer-on-account-abstraction.pdf_ , 2023\.
  * [8] A. Emmanuel, “Account abstraction: Past, present, future,” _https://metamask.io/news/latest/account-abstraction-past-present-future/_ , 2023\.
  * [9] Alchemy, “How ERC-4337 supports account abstraction,” _https://www.alchemy.com/learn/account-abstraction_ , 2023.
  * [10] Moderate, “Coinmarketcap: Account abstraction,” _https://coinmarketcap.com/alexandria/glossary/account-abstraction_ , 2023\.
  * [11] aleksandrmalyshev, “What is account abstraction and why is everyone talking about it?” _https://hackernoon.com/what-is-account-abstraction-and-why-is-everyone-talking-about-it_ , 2023\.
  * [12] Q. Wang, R. Li, Q. Wang, S. Chen, M. Ryan, and T. Hardjono, “Exploring Web3 from the view of blockchain,” _arXiv preprint arXiv:2206.08821_ , 2022.
  * [13] S. Park, J. H. Lee, S. Lee, J. H. Chun, H. Cho, M. Kim, H. K. Cho, and S.-M. Moon, “Beyond the blockchain address: Zero-knowledge address abstraction,” _Cryptology ePrint Archive_ , 2023.
  * [14] Q. Wang, R. Li, Q. Wang, and S. Chen, “Non-fungible token (NFT): Overview, evaluation, opportunities and challenges,” _arXiv preprint arXiv:2105.07447_ , 2021.
  * [15] “Sapienz wallet,” _https://www.sapienz.xyz/_ , 2023.
  * [16] B. Andrew, G. Catherine, R. Srini, M. Mohsen, and K. Ranjit, “Visa: Auto payments for self-custodial wallets,” _https://usa.visa.com/solutions/crypto/auto-payments-for-self-custodial-wallets.html_ , 2023\.
  * [17] “Safe wallet,” _https://safe.global/wallet_ , 2023.
  * [18] “Argent wallet,” _https://www.argent.xyz/_ , 2023.
  * [19] “Bravvos wallet,” _https://braavos.app/_ , 2023.
  * [20] H. Chen, M. Pendleton, L. Njilla, and S. Xu, “A survey on Ethereum systems security: Vulnerabilities, attacks, and defenses,” _ACM Computing Surveys (CSUR)_ , vol. 53, no. 3, pp. 1–43, 2020.
  * [21] P. Tsankov, A. Dan, D. Drachsler-Cohen, A. Gervais, F. Buenzli, and M. Vechev, “Securify: Practical security analysis of smart contracts,” in _Proceedings of the ACM SIGSAC Conference on Computer and Communications Security (CCS)_ , 2018, pp. 67–82.
  * [22] L. Su, X. Shen, X. Du, X. Liao, X. Wang, L. Xing, and B. Liu, “Evil under the sun: Understanding and discovering attacks on ethereum decentralized applications,” in _USENIX Security Symposium (USENIX Sec)_ , 2021, pp. 1307–1324.
  * [23] P. Tolmach, Y. Li, S.-W. Lin, Y. Liu, and Z. Li, “A survey of smart contract formal specification and verification,” _ACM Computing Surveys (CSUR)_ , vol. 54, no. 7, pp. 1–38, 2021.
  * [24] W. Zou, D. Lo, P. S. Kochhar, X.-B. D. Le, X. Xia, Y. Feng, Z. Chen, and B. Xu, “Smart contract development: Challenges and opportunities,” _IEEE Transactions on Software Engineering (TSE)_ , vol. 47, no. 10, pp. 2084–2106, 2019\.
  * [25] J. Krupp and C. Rossow, “teEther: Gnawing at Ethereum to automatically exploit smart contracts,” in _USENIX Security Symposium (USENIX Sec)_ , 2018, pp. 1317–1333.
  * [26] L. Brent, N. Grech, S. Lagouvardos, B. Scholz, and Y. Smaragdakis, “Ethainter: a smart contract security analyzer for composite vulnerabilities,” in _Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (SIGPLAN)_ , 2020, pp. 454–469.
  * [27] Z. Wan, X. Xia, D. Lo, J. Chen, X. Luo, and X. Yang, “Smart contract security: A practitioners’ perspective,” in _IEEE/ACM International Conference on Software Engineering (ICSE)_. IEEE, 2021, pp. 1410–1422.
  * [28] X. Zhang _et al._ , “Time-manipulation attack: Breaking fairness against proof of authority Aura,” in _Proceedings of the ACM Web Conference (WWW)_ , 2023, pp. 2076–2086.
  * [29] Q. Wang, B. Qin, J. Hu, and F. Xiao, “Preserving transaction privacy in Bitcoin,” _Future Generation Computer Systems (FGCS)_ , vol. 107, pp. 793–804, 2020.
  * [30] A. Kosba, A. Miller, E. Shi, Z. Wen, and C. Papamanthou, “Hawk: The blockchain model of cryptography and privacy-preserving smart contracts,” in _IEEE Symposium on Security and Privacy (SP)_. IEEE, 2016, pp. 839–858.
  * [31] R. Li _et al._ , “SoK: TEE-assisted confidential smart contract,” _Proceedings on Privacy Enhancing Technologies_ , vol. 3, pp. 711–731, 2022\.
  * [32] G. Yu, X. Wang _et al._ , “Towards web3 applications: Easing the access and transition,” _arXiv preprint arXiv:2210.05903_ , 2022.


# Thermalization condition for non-Hermitian quantum systems

Yiting Mao  Beijing Computational Science Research Center, Beijing 100084,
China  Peigeng Zhong  Beijing Computational Science Research Center, Beijing
100084, China  Haiqing Lin<EMAIL_ADDRESS>Beijing Computational Science
Research Center, Beijing 100084, China  School of Physics, Zhejiang
University, Hangzhou, 310058, China  Xiaoqun Wang<EMAIL_ADDRESS>School of Physics, Zhejiang University, Hangzhou, 310058, China  Shijie Hu
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100084, China  Department of Physics, Beijing Normal University, Beijing,
100875, China

###### Abstract

The application of the eigenstate thermalization hypothesis to non-Hermitian
quantum systems has become one of the most important topics in dissipative
quantum chaos, recently giving rise to intense debates. The process of
thermalization is intricate, involving many time-evolution trajectories in the
reduced Hilbert space of the system. By considering two different expansion
forms of the density matrices adopted in the biorthogonal and right-state time
evolutions, we have derived two versions of the Gorini-Kossakowski-Sudarshan-
Lindblad master equations describing the non-Hermitian systems coupled to a
bosonic heat bath in thermal equilibrium. By solving the equations, we have
identified a sufficient condition for thermalization under both time
evolutions, resulting in Boltzmann biorthogonal and right-eigenstate
statistics, respectively. This finding implies that the recently proposed
biorthogonal random matrix theory needs an appropriate revision. Moreover, we
have exemplified the precise dynamics of thermalization and thermodynamic
properties with test models.

Chaotic systems are known for their high sensitivity to initial conditions and
their unpredictable long-term dynamics [1]. The famous Berry-Tabor [2] and
Bohigas-Giannoni-Schmit [3] conjectures assert that classically integrable
systems follow Poisson statistics of uncorrelated random variables, while
chaotic quantum systems in the semiclassical limit exhibit statistics based on
random matrix theory (RMT), which depends solely on symmetries rather than
details of the models. For non-Hermitian chaotic systems, such as dissipative
quantum systems described by Lindbladian, or non-unitary quantum dynamics
governed by non-Hermitian Hamiltonians, or others [4], energy levels are
predicted to follow Ginibre level statistics [5, 6, 7] and to show notable
cubic level repulsion [8]. Despite these established theories, conflicts have
also arisen in recent years [9, 10, 11, 12, 13, 14].

Figure 1:  (Color online) When the system s is coupled to environments, it
eventually reaches a stationary state described by either the Boltzmann
biorthogonal statistics or the Boltzmann right-eigenstate statistics,
depending on the time evolution assumption, BTE (1) or RTE (2). These
environments, denoted as $\mathcal{E}$1, $\cdots$, $\mathcal{E}$n, introduce
non-Hermitian effects due to the post-selection. The bosonic heat bath remains
in thermal equilibrium at temperature $T=1/\beta$ and interacts with the
system through the coupling term $A\otimes B$. Rightmost plots: In the
statistics, the transitions between two levels with an energy difference of
$\omega$ hold the detailed balance.

The construction of the eigenstate thermalization hypothesis (ETH) [15, 16,
17] to describe thermalization in general non-Hermitian systems has been a
topic of debate in the past year [18, 19]. Metaphysically, the time evolution
of the unit, consisting of a system and a heat bath, is governed by a non-
Hermitian Hamiltonian $H$, with corresponding wave functions for the left
eigenstate $\ket{\Phi(t)}$ and the right eigenstate $\ket{\Psi(t)}$. Since the
unit is not inherently “closed”, it is challenging to determine which bases
are uniformly suitable for expanding the density matrix $\rho(t)$ of a state
toward the long-term limit $t=+\infty$. Among different choices evolving from
the same initial density matrix $\rho(t=0)$: the biorthogonal time evolution
(BTE) gives a mixed state with

$\displaystyle\rho(t)=\mathinner{|{\Psi(t)}\rangle\langle{\Phi(t)}|}=e^{-iHt}\rho(0)e^{iHt}\,,$
(1)

and the right-state time evolution (RTE) yields a pure state with

$\displaystyle\rho(t)=\mathinner{|{\Psi(t)}\rangle\langle{\Psi(t)}|}=e^{-iHt}\rho(0)e^{iH^{\dagger}t}\,,$
(2)

as shown in Fig. 1. The reference [18] extended ETH to non-Hermitian systems
using RTE (2) and proposed modifications in the off-diagonal measurement due
to non-zero overlaps between different right eigenstates [18]. The resulting
Boltzmann right-eigenstate statistics (BRS) is in agreement with the previous
works [20, 21, 22] on the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master
equation [23, 24]. In contrast, another study [19] argued that RMT violates
ETH when evaluating the biorthogonal measurement of the operator under the
assumption of BTE (1). This argument is supported by significant fluctuations
in both the diagonal and off-diagonal measurements as well as strong
correlations between neighboring energy levels, which are absent in chaotic
Hermitian systems [19]. Although intriguing, to date, the biorthogonal chaotic
dynamics in RMT have not yet been validated through model simulations.

In this work, we mainly study the thermalization process in the biorthogonal
GKSL master equation using BTE (1). In contrast to RMT [19], our results show
that non-Hermitian systems consistently experience decoherence when the
thermalization condition is satisfied, eventually leading to thermalization.
In the long term, the density matrix is shown to obey the Boltzmann
biorthogonal statistics (BBS), which is further illustrated by two models.

Biorthogonal GKSL master equation. We start with a system (s) that is coupled
to multiple environments (b, $\mathcal{E}$1, …, $\mathcal{E}$n), one of which
serves as a bosonic heat bath (b) in thermal equilibrium at temperature
$T=1/\beta$, as shown in Fig. 1. The post-selection condition [25, 26, 27, 28]
remains valid for other environments, such as those encountered in setups with
superconducting qubits [29, 30] and optical lattices [25, 31, 27]. The reduced
Hilbert space of the unit (s plus b) $\mathcal{H}$ is a product of those for
the system $\mathcal{H}_{\text{s}}$ (the dimension $d\geq 2$) and the heat
bath $\mathcal{H}_{\text{b}}$, i.e.,
$\mathcal{H}=\mathcal{H}_{\text{s}}\otimes\mathcal{H}_{\text{b}}$. The total
Hamiltonian can be expressed as
$H=H_{\text{s}}\otimes\mathbbm{1}_{\text{b}}+\mathbbm{1}_{\text{s}}\otimes
H_{\text{b}}+H_{\text{sb}}$, where $H_{\text{s}}$, $H_{\text{b}}$ and
$H_{\text{sb}}$ represent the Hamiltonians for the system, heat bath and their
interaction, respectively. In a general scenario without trivial symmetries,
the non-Hermitian Hamiltonian $H^{\phantom{{\dagger}}}_{\text{s}}\neq
H_{\text{s}}^{\dagger}$ has the left eigenstate $\ket{m_{\text{L}}}$ and the
right eigenstate $\ket{m_{\text{R}}}$ for the $m$th non-degenerate state with
energy $e_{m}$, satisfying both the biorthonormalization condition
$\braket{m_{\text{L}}}{n_{\text{R}}}=\delta_{mn}$ and the normalization
condition $\braket{m_{\text{R}}}{m_{\text{R}}}=1$. The coupling term can be
simply written as $H_{\text{sb}}=A\otimes B$, where the operator $A$ defined
in the system and the operator $B$ defined in the heat bath are Hermitian. For
the Hermitian system, the Hermiticity of the operators $A$ and $B$ is the key
to preserving the energy conservation of the unit [32, 33].

Following BTE (1), we get the Heisenberg equation in the interaction picture,
i.e., $\text{d}\rho^{\text{I}}(t)/\text{d}t=-i[V(t),\,\rho^{\text{I}}(t)]$,
where the coupling term $V(t)=e^{iH_{0}t}H_{\text{sb}}e^{-iH_{0}t}$ and
$\rho^{\text{I}}(t)=e^{iH_{0}t}\rho(t)e^{-iH_{0}t}$ are rotated by the
Hamiltonian
$H_{0}=H_{\text{s}}\otimes\mathbbm{1}_{\text{b}}+\mathbbm{1}_{\text{s}}\otimes
H_{\text{b}}$. Assuming that the correlation time of the bath is much shorter
than the relaxation time of the system when the Born-Markov approximation
(BMA) is valid [1], we get
$\rho^{\text{I}}(t)\approx\rho^{\text{I}}_{\text{s}}(t)\otimes\rho^{\text{I}}_{\text{b}}(t)$
and $[V(t),\,\rho^{\text{I}}(0)]\approx 0$, where the density matrices in the
interaction picture
$\rho^{\text{I}}_{\text{s}}(t)=e^{iH_{\text{s}}t}\rho^{\phantom{\text{I}}}_{\text{s}}(t)e^{-iH_{\text{s}}t}$
and
$\rho^{\text{I}}_{\text{b}}(t)=e^{iH_{\text{b}}t}\rho^{\phantom{\text{I}}}_{\text{b}}(t)e^{-iH_{\text{b}}t}$
are linked to the ordinary ones $\rho^{\phantom{\text{I}}}_{\text{s}}(t)$ and
$\rho^{\phantom{\text{I}}}_{\text{b}}(t)$, respectively. In addition, the heat
bath always maintains its thermal equilibrium, i.e.,
$\rho^{\text{I}}_{\text{b}}(t)\approx\rho^{\text{I}}_{\text{b}}(0)=\bar{\rho}^{\phantom{{\dagger}}}_{\text{b}}=e^{-\beta
H_{\text{b}}}/Z_{\text{b}}$ with $Z_{\text{b}}=\text{tr}_{\text{b}}e^{-\beta
H_{\text{b}}}$. Consequently, one can obtain an equation:

$\displaystyle\frac{\text{d}\rho_{\text{s}}^{\text{I}}(t)}{\text{d}t}=-\int_{0}^{\infty}\text{d}\tau\
\text{tr}^{\phantom{{\dagger}}}_{\text{b}}[V(t),\,[V(\tau),\,\rho^{\text{I}}_{\text{s}}(t)\otimes\bar{\rho}^{\phantom{{\dagger}}}_{\text{b}}]]\,.$
(3)

Using the biorthogonal bases $\ket{m_{\text{L}}}$ and $\ket{m_{\text{R}}}$, we
get the expansion of $V(t)=\sum_{\omega\neq 0}e^{-i\omega t}A_{\omega}\otimes
B(t)$ as convention [1], where $A_{\omega}=\sum_{\omega_{mn}=\omega}A_{mn}$
sums over all selected pairs of $m$ and $n$ satisfying the level spacing
$\omega_{mn}\equiv e_{n}-e_{m}=\omega$, and the operator for a pair of $m$ and
$n$ is given by
$A_{mn}=\mathbbm{A}_{mn}\mathinner{|{m_{\text{R}}}\rangle\langle{n_{\text{L}}}|}$
with the expansion coefficient
$\mathbbm{A}_{mn}=\braket{m_{\text{L}}}{A}{n_{\text{R}}}$. We assume that the
system has no trivial symmetries so that $\mathbbm{A}_{mn}\neq 0$ for any $m$
and $n$. In addition, $B(t)=e^{iH_{\text{b}}t}Be^{-iH_{\text{b}}t}$. In this
work, we consider only the case of the real number $\omega_{mn}$, specifically
in the $\mathcal{PT}$-unbroken region of the models.

We continue to use the definition of the correlation function for the bosonic
heat bath [1], i.e.,
$\Gamma(\omega)=\lim_{t\to+\infty}\int^{t}_{0}\text{d}\tau\,e^{i\omega\tau}\text{tr}_{\text{b}}[B(t)B(t-\tau)\bar{\rho}^{\phantom{{\dagger}}}_{\text{b}}]=\gamma(\omega)/2+i\eta(\omega)$,
due to the Kubo-Martin-Schwinger (KMS) condition
$\gamma(-\omega)/\gamma(\omega)=e^{-\beta\omega}$. Within the rotating-wave
approximation (RWA), only the zero-frequency microscopic processes provide the
effective contribution [34]. We also ignore the imaginary part $\eta$, which
only gives rise to the Lamb-shift term, affecting the evolution process
instead of the steady state [1]. At last, we get the biorthogonal GKSL master
equation:

$\displaystyle\frac{\text{d}\rho_{\text{s}}}{\text{d}t}=\mathcal{L}_{\text{s}}[\rho_{\text{s}}]+\mathcal{L}_{\text{b}}[\rho_{\text{s}}]=\mathcal{L}_{\text{s}}[\rho_{\text{s}}]-\frac{1}{2}\mathcal{L}_{\text{d}}[\rho_{\text{s}}]+\frac{1}{2}\mathcal{L}_{\text{j}}[\rho_{\text{s}}]\,,\
$ (4)

where

$\displaystyle\mathcal{L}_{\text{s}}[\rho_{\text{s}}]$ $\displaystyle=$
$\displaystyle-i[H_{\text{s}},\,\rho_{\text{s}}]\,,\
\mathcal{L}_{\text{d}}[\rho_{\text{s}}]=\sum_{\omega\neq
0}\gamma(\omega)\\{A_{-\omega}A_{\omega},\,\rho_{\text{s}}\\}\,,$
$\displaystyle\mathcal{L}_{\text{j}}[\rho_{\text{s}}]$ $\displaystyle=$
$\displaystyle\sum_{\omega\neq
0}\gamma(\omega)(A_{\omega}\rho_{\text{s}}A_{-\omega}+A_{\omega}\rho_{\text{s}}A_{-\omega})$
(5)

represent the Liouvillians for the system, dissipation, and quantum jumps,
respectively. And
$[o^{\phantom{{\dagger}}}_{1},\,o^{\phantom{{\dagger}}}_{2}]$
($\\{o^{\phantom{{\dagger}}}_{1},\,o^{\phantom{{\dagger}}}_{2}\\}$) denotes
the regular (anti)commutator.

Thermalization condition. As we can see that the transition from the projector
$\mathinner{|{m_{\text{R}}}\rangle\langle{n_{\text{L}}}|}$ to the other one
$\mathinner{|{(m^{\prime})_{\text{R}}}\rangle\langle{(n^{\prime})_{\text{L}}}|}$
in Eq. (4) conserves the energy, that is
$\omega_{m^{\prime}m}=\omega_{n^{\prime}n}$, so that the density matrix
$\rho_{\text{s}}(t)$ can be written as a direct sum of partial density
matrices $\rho^{(\Delta)}_{\text{s}}(t)$ for the sectors labeled by the bias
$\Delta=e_{n}-e_{m}$.

To assume that $M$ pairs of $m$ and $n$ construct

$\displaystyle\rho^{(\Delta)}_{\text{s}}(t)=\sum^{M}_{p=1}c_{p}(t)\mathinner{|{(m_{p})_{\text{R}}}\rangle\langle{(n_{p})_{\text{L}}}|}\,,$
(6)

we get a Pauli master equation
$\text{d}{\bm{c}}(t)/\text{d}t=\mathbbm{L}{\bm{c}}(t)$ [35], where a vector of
${\bm{c}}(t)=(c^{\phantom{{\dagger}}}_{1}(t),\,\cdots,\,c^{\phantom{{\dagger}}}_{M}(t))^{\intercal}$
and two sets of
${\bm{v}}^{\phantom{{\dagger}}}_{m}=\\{m^{\phantom{{\dagger}}}_{1},\,\cdots,\,m^{\phantom{{\dagger}}}_{M}\\}$
and
${\bm{v}}^{\phantom{{\dagger}}}_{n}=\\{n^{\phantom{{\dagger}}}_{1},\,\cdots,\,n^{\phantom{{\dagger}}}_{M}\\}$
are introduced. The detailed derivation is provided in the Supplementary
Material. In the time-independent matrix $\mathbbm{L}$, the diagonal elements
$\mathbbm{L}_{pp}=-[\sum_{m^{\prime}\neq
m_{p}}\gamma(\omega_{m^{\prime}m_{p}})\kappa_{m_{p}m^{\prime}}+\sum_{n^{\prime}\neq
n_{p}}\gamma(\omega_{n^{\prime}n_{p}})\kappa_{n_{p}n^{\prime}}]/2$ are
generated by the dissipation $\mathcal{L}_{\text{d}}$ in Eq. (Thermalization
condition for non-Hermitian quantum systems), where symmetric
$\kappa_{mn}=\mathbbm{A}_{mn}\mathbbm{A}_{nm}$ represents the transition rate,
and both $m^{\prime}$ and $n^{\prime}$ run over all $d$ levels of the system.
We divide $\mathbbm{L}_{pp}$ into the part $-\sum_{q\neq
p}\gamma(\omega_{m_{q}m^{\phantom{{\dagger}}}_{p}})(\kappa_{m_{p}m_{q}}+\kappa_{n_{p}n_{q}})/2$
from the sector $\Delta$, and the rest from
$m^{\prime}\notin{\bm{v}}^{\phantom{{\dagger}}}_{m}$ and
$n^{\prime}\notin{\bm{v}}^{\phantom{{\dagger}}}_{n}$. In contrast, the off-
diagonal matrix element
$\mathbbm{L}_{qp}=\gamma(\omega_{m_{q}m_{p}})\mathbbm{A}_{m_{q}m_{p}}\mathbbm{A}_{n_{p}n_{q}}$
for $q\neq p$ is only determined by the quantum jumps $\mathcal{L}_{\text{j}}$
within the sector $\Delta$ as specified in Eq. (Thermalization condition for
non-Hermitian quantum systems). For a steady state ${\bm{c}}(t)=e^{\lambda
t}\bar{\bm{c}}$, we get an equation
$\mathbbm{L}\bar{\bm{c}}=\lambda\bar{\bm{c}}$ for the right eigenstate
$\bar{\bm{c}}$ of $\mathbbm{L}$ with $\lambda$ representing the eigenvalue, as
well as the one
$\bar{\bm{c}}^{\intercal}_{\text{L}}\mathbbm{L}=\lambda\bar{\bm{c}}^{\intercal}_{\text{L}}$
for the corresponding left eigenstate $\bar{\bm{c}}_{\text{L}}$.

We have two scenarios to consider: (i) For the partial density matrix
$\rho^{(0)}_{\text{s}}(t)$ in diagonal [36] with $n_{p}\equiv m_{p}$ in Eq.
(6), the equality $\sum^{M}_{q=1}\mathbbm{L}_{qp}=0$ holds for any $p$,
ensuring the existence of at least a stable left eigenstate
$\bar{{\bm{c}}}_{\text{L}}=(1,\,\cdots,\,1)^{\intercal}$ with the zero
eigenvalue $\lambda=0$. (ii) For the off-diagonal term $\rho^{(\Delta\neq
0)}_{\text{s}}(t)$, the possible presence of nonzero $\mathbbm{L}_{qp}$
indicates the existence of the level spacing degeneracy ($M\geq 2$),
facilitating coherent transitions from
$\mathinner{|{(m_{p})_{\text{R}}}\rangle\langle{(n_{p})_{\text{L}}}|}$ to
$\mathinner{|{(m_{q})_{\text{R}}}\rangle\langle{(n_{q})_{\text{L}}}|}$.
Remarkably, if

$\displaystyle\mathbbm{A}_{mn}=(\mathbbm{A}_{nm})^{*}\quad\text{or}\quad\kappa_{mn}=(\kappa_{mn})^{*}=|\mathbbm{A}_{mn}|^{2}$
(7)

is satisfied for arbitrary $m\neq n$, it is easy to prove that the matrix
$\mathbbm{L}$ is strictly diagonally dominant because
$|\mathbbm{L}_{pp}|-\sum_{q\neq p}|\mathbbm{L}_{qp}|>\sum_{q\neq
p}\gamma(\omega_{m_{q}m_{p}})(|\mathbbm{A}_{m_{p}m_{q}}|-|\mathbbm{A}_{n_{p}n_{q}}|)^{2}/2\geq
0$ [37]. In this case, the real parts of eigenvalues for sectors $\Delta\neq
0$ are always negative because of $\mathbbm{L}_{pp}<0$, resulting in
decoherence accordingly. Overall, the condition (7) is sufficient but
unnecessary for thermalization in the long term, where only
$\rho^{(0)}_{\text{s}}(t)$ survives. For the convenience of the following
discussion, we get a simple form of
$\rho^{(0)}_{\text{s}}(t)=\sum^{d}_{m=1}c_{m}(t)\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{L}}}|}$,
because $M=d$ is given for $\Delta=0$.

Boltzmann biorthogonal statistics. When the thermalization condition (7) is
valid, one only considers the diagonal partial density matrix in the long
term, with $\mathcal{L}_{\text{s}}[\rho_{\text{s}}^{(0)}]=0$ and

$\displaystyle\mathcal{L}[\rho^{(0)}_{\text{s}}]=\sum_{m,n}\left(\gamma(\omega_{nm})W_{mn}+\gamma(\omega_{mn})W_{nm}\right)\,,$
(8)

where the transition superoperator is given by
$W_{mn}=\kappa_{mn}(\mathinner{|{n_{\text{R}}}\rangle\langle{n_{\text{L}}}|}-\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{L}}}|})c_{m}(t)$
for indicating the probability transition from the $m$th level to the $n$th
level. The KMS condition establishes the detailed balance between those
levels: $\gamma(\omega_{nm})/\gamma(\omega_{mn})=e^{-\beta\omega_{mn}}$, as
shown in the upper rightmost plot of Fig. 1. It is easy to get the right
eigenstate
$\bar{\bm{c}}=(\chi^{\phantom{{\dagger}}}_{1},\,\cdots,\,\chi^{\phantom{{\dagger}}}_{M})^{\intercal}$
for the stable steady state, where the Boltzmann weight for the $m$th level
$\chi^{\phantom{{\dagger}}}_{m}=e^{-\beta e_{m}}/Z_{\text{s}}$ and the
partition function $Z_{\text{s}}=\sum_{m}e^{-\beta e_{m}}$ are given. As a
result, the system evolves to a thermal state described by the density matrix
for Boltzmann biorthogonal statistics, i.e.,

$\displaystyle\bar{\rho}_{\text{s}}^{\text{lr}}=\sum^{d}_{m=1}\chi_{m}\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{L}}}|}\,.$
(9)

In contrast, in the $\mathcal{PT}$-broken region, the perturbative quantum
jumps and the dissipative terms in Eq. (Thermalization condition for non-
Hermitian quantum systems) can be simultaneously ignored compared to the
dissipation caused by the non-Hermiticity of the system. As a result, the
system evolves to the eigenstate corresponding to the eigenvalue with the
maximal imaginary part, which has the highest growth rate of the amplitude and
becomes dominant in the long term.

Boltzmann right-eigenstate statistics. Under RTE (2), we get the solution to
Eq. (4) similar to Ref. [20, 21, 22], with Liouvillian defined as:

$\displaystyle\mathcal{L}_{\text{s}}[\rho_{\text{s}}]$ $\displaystyle=$
$\displaystyle-i[H_{\text{s}},\,\rho_{\text{s}}]_{\dagger}\,,\
\mathcal{L}_{\text{d}}[\rho_{\text{s}}]=\sum_{\omega\neq
0}\gamma(\omega)\\{A_{-\omega}A_{\omega},\,\rho_{\text{s}}\\}_{\dagger}\,,$
$\displaystyle\mathcal{L}_{\text{j}}[\rho_{\text{s}}]$ $\displaystyle=$
$\displaystyle\sum_{\omega\neq
0}\gamma(\omega)(A_{\omega}\rho_{\text{s}}(A_{\omega})^{\dagger}+A_{\omega}\rho_{\text{s}}(A_{\omega})^{\dagger})\,.$
(10)

where
$[o^{\phantom{{\dagger}}}_{1},\,o^{\phantom{{\dagger}}}_{2}]_{\dagger}=o^{\phantom{{\dagger}}}_{1}o^{\phantom{{\dagger}}}_{2}-o^{\phantom{{\dagger}}}_{2}o^{\dagger}_{1}$
and
$\\{o^{\phantom{{\dagger}}}_{1},\,o^{\phantom{{\dagger}}}_{2}\\}_{\dagger}=o^{\phantom{{\dagger}}}_{1}o^{\phantom{{\dagger}}}_{2}+o^{\phantom{{\dagger}}}_{2}o^{\dagger}_{1}$
denote the modified commutator and anticommutator, respectively.

Analogous to BTE (1), starting from an initial density matrix that can be
written as the diagonal form of non-orthogonal right eigenstates
$\ket{m_{\text{R}}}$, the thermalization condition (7) yields a long-term
stable thermal state with the density matrix for Boltzmann right-eigenstate
statistics, i.e.,

$\displaystyle\bar{\rho}_{\text{s}}^{\text{rr}}=\sum^{d}_{m=1}\chi_{m}\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{R}}}|}\,.$
(11)

As shown in the lower rightmost plot of Fig. 1, the detailed balances can also
be activated by the Liouvillian (8) with the redefined transition
superoperator
$W_{mn}=\kappa_{mn}(\mathinner{|{n_{\text{R}}}\rangle\langle{n_{\text{R}}}|}-\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{R}}}|})c_{m}(t)$.
Nevertheless, the condition (7) can not guarantee that the off-diagonal terms
in the density matrix disappear in the long term in general.

Figure 2:  (Color online) The thermalization of the qubit (12) is studied by
analyzing the time evolution from an initial state
$\mathinner{|{\uparrow}\rangle\langle{\uparrow}|}$, using either (a) the
biorthogonal GKSL master equation (Thermalization condition for non-Hermitian
quantum systems) or (b) the right-state one (Thermalization condition for non-
Hermitian quantum systems) at a temperature of $T/h_{x}=1$. We focus on the
expansion coefficients $c_{\uparrow\uparrow}$ (red line),
$c_{\downarrow\downarrow}$ (blue line) and $c_{\uparrow\downarrow}$ (black
line) in the density matrix
$\rho_{\text{s}}(t)=\sum_{\sigma,\sigma^{\prime}=\uparrow/\downarrow}c_{\sigma\sigma^{\prime}}(t)\mathinner{|{\sigma}\rangle\langle{\sigma^{\prime}}|}$
as a function of the time $h_{x}t$. The dashed lines represent the anticipated
coefficients in ETH. In the long term, under (c) BTE and (d) RTE, the vector
$\braket{\bm{\sigma}}$ (points or vectors) is plotted in 3-dimensional
coordinates as the temperature varies from $T=0.1$ (blue) to $5$ (red). Here,
we use $h_{y}/h_{x}=0.5$ in the $\mathcal{PT}$-unbroken region.

A qubit. In the simplest example of the $\mathcal{PT}$-symmetric qubit [29],
the system Hamiltonian

$H_{\text{s}}=h_{x}\sigma^{x}-ih_{y}\sigma^{y}$ (12)

has the biorthogonal eigenstates $\ket{\text{e}_{\text{L}}}$,
$\ket{\text{e}_{\text{R}}}$, $\ket{\text{g}_{\text{L}}}$ and
$\ket{\text{g}_{\text{R}}}$ for the excited state (e) and the ground state (g)
with the energy $e_{\text{e}}$, $e_{\text{g}}=\pm\sqrt{h^{2}_{x}-h^{2}_{y}}$,
respectively. $\sigma^{x,y,z}$ denote the Pauli matrices in three axes and
form a vector operator
${\bm{\sigma}}=(\sigma^{x},\,\sigma^{y},\,\sigma^{z})^{\intercal}$. We take
$h_{x}=1$ as the energy unit and investigate the long-term behavior in the
$\mathcal{PT}$-unbroken region $|h_{y}|<1$. We choose $A=\sigma^{z}$,
satisfying the thermalization condition (7).

After the initial preparation of a fully polarized pure state at temperature
$T/h_{x}=1$, Figure 2 shows the trajectory of the time-evolving matrix
elements of $\rho_{\text{s}}$, according to Eq. (4). When $h_{x}t\gg 1$, it
can be easily verified that the stable steady state satisfies BBS (9) in Fig.
2(a) and BRS (11) in Fig. 2(b), neglecting the normalization coefficients.

Figure 3:  (Color online) In the $\mathcal{PT}$-unbroken region of the non-
Hermitian Ising model (13), we plot the variance of the long-term ($Jt\gg 1$)
solution of either (a) the biorthogonal GKSL master equation (Thermalization
condition for non-Hermitian quantum systems) and (b) the right-state one
(Thermalization condition for non-Hermitian quantum systems) compared to the
anticipated Boltzmann distributions (9) and (11), respectively, in the
($h_{y}/J$, $h_{z}/J$)-plane. The variance near the exceptional point
$|h_{y}|=|h_{z}|$ grows fast due to strong numerical instability. (c) When
$h_{y}/J=0.2$ and $h_{z}/J=0.75$, we analyze the behavior of the averaged
polarization in the $z$-axis
$\bar{\sigma}^{z}=\sum^{L}_{\ell=1}\braket{\sigma^{z}_{\ell}}/L$ over time
under BTE (1), starting from either a mixed state at infinity temperature
$\sum_{m}\mathinner{|{m_{\text{R}}}\rangle\langle{m_{\text{L}}}|}$ (solid
line) or a pure state at zero
temperature$\mathinner{|{1_{\text{R}}}\rangle\langle{1_{\text{L}}}|}$ (dashed
line). We consider two cases of $A_{\ell}=\sigma^{z}_{\ell}$ (red) and
$\sigma^{x}_{\ell}$ (black). The dashed line indicates the anticipated value
according to ETH. Here, we set $L=4$ and $T/J=1$ for these calculations.

In Figs. 2(c) and (d), we also plot the vector
$\braket{\bm{\sigma}}=\text{tr}[\rho_{\text{s}}(t){\bm{\sigma}}]_{h_{x}t\gg
1}=(\braket{\sigma^{x}},\,\braket{\sigma^{y}},\,\braket{\sigma^{z}})^{\intercal}$
in three dimensions. When we lower the temperature in Figs. 2(c),
$\braket{\bm{\sigma}}$ under BTE (1) persists along $x$-axis in the equatorial
plane, since $\text{Re}\braket{\sigma^{y,z}}=0$. In contrast, under RTE (2),
the orientation of $\braket{\bm{\sigma}}$ continuously rotates in the
equatorial plane. It is worthy noticing that under BTE (1) the vector
$\braket{\bm{\sigma}}$ can move out of the Bloch sphere at low temperatures
because $|\braket{\text{e}_{\text{L}}}{\sigma^{x}}{\text{e}_{\text{R}}}|>1$.

Non-Hermitian Ising model. In the model, we consider a real magnetic field
$h_{z}$ in the $z$-axis and an imaginary one $ih_{y}$ in the $y$-axis. The
latter can be implemented by the post-selection using Lindblad operators
$A_{\ell}=\sqrt{h_{z}}(\sigma_{\ell}^{x}-i\sigma_{\ell}^{y})$ for all sites
[38, 39]. Thus, the Hamiltonian of the system is defined as

$\displaystyle
H_{\text{s}}=\sum_{\ell=1}^{L-1}J\sigma_{\ell}^{x}\sigma_{\ell+1}^{x}+\sum_{\ell=1}^{L}(ih_{y}\sigma_{\ell}^{y}+h_{z}\sigma_{\ell}^{z}).$
(13)

where the index $\ell$ ranges from $1$ to the size of the chain $L$, and the
coupling strength $J=1$ sets the energy unit. The operator $A_{\ell}$ is
coupled to the individual heat bath at each site-$\ell$, i.e.,
$H_{\text{sb}}=\sum^{L}_{\ell=1}A_{\ell}\otimes B_{\ell}$, to avoid the (non-
Markov) memory effects caused by the long-range correlations [40, 41].
Although it is not a strict limitation, this assumption is physically
plausible. More general forms of coupling could also be argued similarly.

We study the thermalization of the system in the $\mathcal{PT}$-unbroken
region $|h_{z}|>|h_{y}|$ [39]. To compare the solution of the GKSL master
equation (4) with statistics (9) and (11) at a typical temperature $T/J=1$, we
investigate the variance of
$V^{\text{lr},\text{rr}}=\|\rho_{\text{s}}(t)-\bar{\rho}^{\text{lr},\text{rr}}_{\text{s}}]\|_{\infty}$
as $Jt\gg 1$, where all density matrices have been normalized and
$\|\cdots\|_{\infty}$ denotes the infinity-norm [42]. First,
$A_{\ell}=\sigma_{\ell}^{x}$ has been taken in Fig 3(a) and (b) to ensure the
system can be thermalized under BTE (1) and RTE (2) according to Eq. (7), even
if more level spacing degeneracies ($M\gg 1$) exist and rich coherent
transitions are allowed. In the long term, the system can be described by
density matrices (9) and (11) within small error bars. In Fig. 3(c), we also
consider the case of $A_{\ell}=\sigma_{\ell}^{z}$, which violates the
thermalization condition (7). Under BTE (1), this violation only leads to a
longer period of decoherence. In contrast, under RTE (2), the thermalization
cannot occur at all (not shown).

Figure 4:  (Color online) After the thermalization of the non-Hermitian Ising
model (13), the discrepancy of the thermal entropies
$\delta\mathcal{S}=\mathcal{S}_{\text{von}}-\mathcal{S}_{\text{gib}}$ is
plotted in the ($h_{y}/J$, $h_{z}/J$)-plane. We consider (a) BBS (9) and (b)
BRS (11). Inset: Along a red dashed line $h_{z}/J=0.8$ and
$h_{y}/J\in[0,\,0.8)$, we make a comparison of three thermal entropies:
$\mathcal{S}^{\text{lr}}_{\text{von}}$ (black),
$\mathcal{S}^{\text{rr}}_{\text{von}}$ (blue) and $\mathcal{S}_{\text{gib}}$
(red). Here, we set $L=4$ and $T/J=1$ for these calculations.

Thermal entropy. Two distinct statistics may bring us qualitative differences
in measuring thermodynamic quantities. Again taking the non-Hermitian Ising
model as an example, we analyzed the thermal von-Neumann entropy (vNE) [43] of
stable steady states described by density matrices (9) and (11) after
thermalization. In Fig. 4(a), the biorthogonal vNE
$\mathcal{S}^{\text{lr}}_{\text{von}}=-\text{tr}\bar{\rho}_{\text{s}}^{\text{lr}}\ln\bar{\rho}_{\text{s}}^{\text{lr}}$
is equal to the Gibbs entropy
$\mathcal{S}_{\text{gib}}=-\sum^{d}_{m=1}\chi_{m}\log\chi_{m}$ [44], trivially
because of the biorthonormalization condition between the left and right
eigenstates. While in Fig. 4(b), we found that the right-eigenstate vNE
$\mathcal{S}^{\text{rr}}_{\text{von}}=-\text{tr}\bar{\rho}_{\text{s}}^{\text{rr}}\ln\bar{\rho}_{\text{s}}^{\text{rr}}$
becomes lower than the Gibbs entropy $\mathcal{S}_{\text{gib}}$. This
phenomenon arises from the fact that the dissipation under the incomplete
measurement lowers the information by reweighing bases or even erasing certain
during the reorthogonalization of the right eigenstates in
$\bar{\rho}_{\text{s}}^{\text{rr}}$, exemplified with the skin effects [45].

Conclusion and discussions. We model the dynamics of non-Hermitian systems
interacting with a bosonic heat bath in thermal equilibrium and establish the
GKSL master equations using both the biorthogonal and the right-state time
evolutions, where the latter has been obtained in previous studies [20, 21,
22]. Our analysis reveals a sufficient condition (7) for thermalization,
ensuring that the long-term solutions of the equations exhibit BBS (9) and BRS
(11), respectively. For a single qubit and a non-Hermitian Ising model in the
$\mathcal{PT}$-unbroken region, we have shown that the density matrices agree
with the anticipated statistics in the long term.

The early biorthogonal RMT suggested that the non-Hermitian system could
generally not undergo decoherence due to the lack of entropic suppression in
the off-diagonal terms of the biorthogonal measurement. However, the
biorthogonal GKSL master equation in this work indicates that a thermal state
with BBS (9) can be achieved once the system satisfies the thermalization
condition (7). This conflict may be attributed to a failure to consider the
hidden relationship between the left and right eigenstates in the current
version of RMT. Further investigation of the potential entropic effects on
decoherence in the biorthogonal RMT is warranted. Additionally, our work also
extends the previously proposed BRS (11) by adding a thermalization condition.

There are certain cases that may occur unexpectedly. For instance, if the
thermalization condition (7) is not met, we cannot guarantee that the diagonal
density matrix survives in the long term. Additionally, the matrix
$\mathbbm{A}$ may not be dense due to extra symmetries. In such cases, a
thorough examination is necessary.

We are grateful to Su-Peng Kou for fruitful discussions. This work is
supported by grants: MOST 2022YFA1402700, NSFC 12174020, NSFC 12088101, NSFC
11974244 and NSAF U2230402. Computational resources from Tianhe-2JK at the
Beijing Computational Science Research Center and Quantum Many-body I cluster
at SPA, Shanghai Jiaotong University are also highly appreciated.

## References

  * Breuer and Petruccione [2007] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University PressOxford, 2007).
  * Berry and Tabor [1977] M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 356, 375 (1977).
  * Bohigas  _et al._ [1984] O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Physical Review Letters 52, 1 (1984).
  * Sá  _et al._ [2020] L. Sá, P. Ribeiro, and T. Prosen, Complex spacing ratios: A signature of dissipative quantum chaos, Physical Review X 10, 021019 (2020).
  * Grobe  _et al._ [1988] R. Grobe, F. Haake, and H.-J. Sommers, Quantum distinction of regular and chaotic dissipative motion, Physical Review Letters 61, 1899 (1988).
  * Haake [1991] F. Haake, Quantum signatures of chaos, in  _NATO ASI Series_ (Springer US, 1991) pp. 583–595.
  * Akemann  _et al._ [2019] G. Akemann, M. Kieburg, A. Mielke, and T. Prosen, Universal signature from integrability to chaos in dissipative open quantum systems, Physical Review Letters 123, 254101 (2019).
  * Grobe and Haake [1989] R. Grobe and F. Haake, Universality of cubic-level repulsion for dissipative quantum chaos, Physical Review Letters 62, 2893 (1989).
  * Altland and Zirnbauer [1997] A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Physical Review B 55, 1142 (1997).
  * Kawabata  _et al._ [2019] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Physical Review X 9, 041015 (2019).
  * Hamazaki  _et al._ [2020] R. Hamazaki, K. Kawabata, N. Kura, and M. Ueda, Universality classes of non-hermitian random matrices, Physical Review Research 2, 023286 (2020).
  * Li  _et al._ [2021] J. Li, T. Prosen, and A. Chan, Spectral statistics of non-hermitian matrices and dissipative quantum chaos, Physical Review Letters 127, 170602 (2021).
  * Chan  _et al._ [2022] A. Chan, S. Shivam, D. A. Huse, and A. De Luca, Many-body quantum chaos and space-time translational invariance, Nature Communications 13, 7484 (2022).
  * Shivam  _et al._ [2023] S. Shivam, A. De Luca, D. A. Huse, and A. Chan, Many-body quantum chaos and emergence of ginibre ensemble, Physical Review Letters 130, 140403 (2023).
  * Deutsch [1991] J. M. Deutsch, Quantum statistical mechanics in a closed system, Physical Review A 43, 2046 (1991).
  * Srednicki [1994] M. Srednicki, Chaos and quantum thermalization, Physical Review E 50, 888 (1994).
  * Srednicki [1999] M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, Journal of Physics A: Mathematical and General 32, 1163 (1999).
  * Roy  _et al._ [2023] S. S. Roy, S. Bandyopadhyay, R. C. de Almeida, and P. Hauke, Unveiling eigenstate thermalization for non-hermitian systems (2023), arXiv:2309.00049 [quant-ph] .
  * Cipolloni and Kudler-Flam [2024] G. Cipolloni and J. Kudler-Flam, Non-hermitian hamiltonians violate the eigenstate thermalization hypothesis, Physical Review B 109, l020201 (2024).
  * Du  _et al._ [2022] Q. Du, K. Cao, and S.-P. Kou, Physics of $\mathcal{PT}$-symmetric quantum systems at finite temperatures, Physical Review A 106, 032206 (2022).
  * Cao  _et al._ [2023] K. Cao, Q. Du, and S.-P. Kou, Many-body non-hermitian skin effect at finite temperatures, Physical Review B 108, 165420 (2023).
  * Cao and Kou [2023] K. Cao and S.-P. Kou, Statistical mechanics for non-hermitian quantum systems, Physical Review Research 5, 033196 (2023).
  * Albash  _et al._ [2012] T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, Quantum adiabatic markovian master equations, New Journal of Physics 14, 123016 (2012).
  * Mai and Yin [2013] P. Mai and S. Yin, Derivation of lindblad master equation for the quantum ising model interacting with a heat bath (2013), arXiv:1303.3366 [cond-mat.stat-mech] .
  * Fukuhara  _et al._ [2013] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauß, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi, C. Gross, I. Bloch, and S. Kuhr, Quantum dynamics of a mobile spin impurity, Nature Physics 9, 235 (2013).
  * Magaña-Loaiza  _et al._ [2014] O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, Amplification of angular rotations using weak measurements, Physical Review Letters 112, 200401 (2014).
  * Ashida  _et al._ [2017] Y. Ashida, S. Furukawa, and M. Ueda, Parity-time-symmetric quantum critical phenomena, Nature Communications 8, 15791 (2017).
  * Minganti  _et al._ [2020] F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-liouvillian formalism connecting exceptional points of non-hermitian hamiltonians and liouvillians via postselection of quantum trajectories, Physical Review A 101, 062112 (2020).
  * Naghiloo  _et al._ [2019] M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nature Physics 15, 1232 (2019).
  * Chen  _et al._ [2021] W. Chen, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum jumps in the non-hermitian dynamics of a superconducting qubit, Physical Review Letters 127, 140504 (2021).
  * Fukuhara  _et al._ [2015] T. Fukuhara, S. Hild, J. Zeiher, P. Schauß, I. Bloch, M. Endres, and C. Gross, Spatially resolved detection of a spin-entanglement wave in a bose-hubbard chain, Physical Review Letters 115, 035302 (2015).
  * Gaspard and Nagaoka [1999] P. Gaspard and M. Nagaoka, Non-markovian stochastic schrödinger equation, The Journal of Chemical Physics 111, 5676 (1999).
  * de Vega and Alonso [2017] I. de Vega and D. Alonso, Dynamics of non-markovian open quantum systems, Reviews of Modern Physics 89, 015001 (2017).
  * Fleming  _et al._ [2010] C. Fleming, N. I. Cummings, C. Anastopoulos, and B. L. Hu, The rotating-wave approximation: consistency and applicability from an open quantum system analysis, Journal of Physics A: Mathematical and Theoretical 43, 405304 (2010).
  * Pauli [1928] W. Pauli,  _Festschrift zum 60. Geburtstage A. Sommerfeld_ (Hirzel, Leipzig, 1928) pauli master equation; p. 30.
  * Bulnes Cuetara  _et al._ [2016] G. Bulnes Cuetara, M. Esposito, and G. Schaller, Quantum thermodynamics with degenerate eigenstate coherences, Entropy 18, 447 (2016).
  * Timm [2009] C. Timm, Random transition-rate matrices for the master equation, Physical Review E 80, 021140 (2009).
  * Deguchi and Ghosh [2009] T. Deguchi and P. K. Ghosh, The exactly solvable quasi-hermitian transverse ising model, Journal of Physics A: Mathematical and Theoretical 42, 475208 (2009).
  * Yang  _et al._ [2022] F. Yang, H. Wang, M.-L. Yang, C.-X. Guo, X.-R. Wang, G.-Y. Sun, and S.-P. Kou, Hidden continuous quantum phase transition without gap closing in non-hermitian transverse ising model, New Journal of Physics 24, 043046 (2022).
  * Laine  _et al._ [2012] E.-M. Laine, H.-P. Breuer, J. Piilo, C.-F. Li, and G.-C. Guo, Nonlocal memory effects in the dynamics of open quantum systems, Physical Review Letters 108, 210402 (2012).
  * Breuer  _et al._ [2016] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-markovian dynamics in open quantum systems, Reviews of Modern Physics 88, 021002 (2016).
  * Rajput  _et al._ [2022] A. Rajput, A. Roggero, and N. Wiebe, Hybridized methods for quantum simulation in the interaction picture, Quantum 6, 780 (2022).
  * von Neumann [2018] J. von Neumann,  _Mathematical Foundations of Quantum Mechanics: New Edition_, edited by N. A. Wheeler (Princeton University Press, 2018).
  * Kardar [2007] M. Kardar,  _Statistical Physics of Particles_ (Cambridge University Press, 2007).
  * Kawabata  _et al._ [2023] K. Kawabata, T. Numasawa, and S. Ryu, Entanglement phase transition induced by the non-hermitian skin effect, Physical Review X 13, 021007 (2023).