Submitted:
07 December 2024
Posted:
09 December 2024
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Abstract
Quantum cryptography is a promising way to secure communication in the era of quantum information technologies (Nx-ICT). However, ensuring the confidentiality and authenticity of quantum communication is challenging in the presence of noisy quantum channels and a lack of trust between communicating parties. Unconditional mistrustful quantum cryptography protocols address these challenges by providing security guarantees even when parties do not trust each other. This paper explores the fundamental concepts and methodologies behind unconditional mistrustful quantum cryptography protocols in Nx-ICT scenarios. We examine the theoretical underpinnings of these protocols, which are based on cryptographic techniques from quantum mechanics, such as quantum key distribution (QKD) and entanglement-based protocols. We also review the latest experimental developments and achievements in the implementation of unconditional mistrustful quantum cryptography protocols. We discuss the progress that has been made in mitigating noise-induced vulnerabilities, as noisy quantum channels are common in practical quantum communication systems. Finally, we discuss the future prospects and open research questions in this field. We emphasize the importance of continuous research to improve the efficiency, scalability, and security of unconditional mistrustful quantum cryptography protocols in the face of evolving Nx-ICT applications and advancements.
Keywords:
Unconditional Mistrustful Quantum Cryptography
; Quantum Cryptography Protocols
; Nx-ICT Scenarios
; Noisy Quantum Channels
Quantum cryptography is a promising technology for secure communication. It uses the principles of quantum mechanics to create unbreakable encryption, even against attacks from quantum computers(Padmavathi A, 2021). Traditional quantum key distribution (QKD) methods rely on trust in the involved devices and channels(Mehic et al., 2021). However, in some cases, trust cannot be guaranteed. For instance, if the devices or channels are not under the control of the communication parties, or if there is a risk of eavesdropping(Al-Ghamdi et al., 2020). In these cases, unconditional mistrustful quantum cryptography protocols are required.
These protocols provide security even when trust cannot be guaranteed(Grasselli, 2021). They achieve this by using techniques such as entanglement-based approaches(Yin et al., 2020), quantum error-correcting codes(Yang et al., 2022), and post-quantum cryptographic primitives(Malina et al., 2021). The Nx-ICT scenario is characterized by a complex and interconnected information and communication technology infrastructure(Knud Erik Skouby, 2022). This makes the need for secure communication even more critical. Noisy quantum channels can pose a significant challenge in this context, as they can lead to data loss, key rate decreases, or even unauthorized access.
This paper investigates existing unconditional mistrustful quantum cryptography protocols in the Nx-ICT scenario, considering the unique constraints given by noisy quantum channels. It examines the strengths and weaknesses of the protocols, their theoretical basis, and their robustness in the presence of noise. It also looks at mitigation measures used to improve communication reliability in the face of noise, such as error correction codes and quantum error correction. The discussions of this study can help to guide the development of new unconditional mistrustful quantum cryptography protocols that are suitable for the Nx-ICT scenario. They can also help to improve the security of existing protocols in the presence of noisy quantum channels.
II. Background
Trust assumptions in quantum cryptography are the assumptions made about the behavior of the devices and entities involved in the communication. Traditional quantum key distribution (QKD) protocols assume that the devices used are trusted and free from any vulnerabilities or tampering. However, in practical scenarios, it is challenging to ensure the trustworthiness of all devices and parties involved. This limitation has led to the development of unconditional mistrustful quantum cryptography protocols(Bozzio et al., 2022; Nadeem, 2014a).
Unconditional mistrustful quantum cryptography techniques seek to provide secure communication without relying on trust assumptions(Al-Ghamdi et al., 2020). These protocols strive towards unconditional security, which means that the security of the communication is assured independent of an adversary’s computational capacity. They accomplish this by establishing additional levels of protection and deploying cryptographic approaches that can withstand attacks from formidable adversaries(Sharma et al., 2022).
The existence of noise in quantum channels is one of the most difficult difficulties in establishing secure quantum communication systems. Any unwanted contact or disruption that affects the transmitted quantum states is referred to as noise. Since quantum systems are particularly sensitive to noise, even minor disturbances can introduce errors or change the information being conveyed(Gebhart et al., 2023). Noise can be caused by a variety of factors, including heat fluctuations, faulty equipment, or eavesdropping attempts.
In the Nx-ICT landscape, which includes advanced technology interwoven into various aspects of our life, noisy quantum channels represent a substantial threat to the reliability and security of quantum communication(Knud Erik Skouby, 2022). The integration of quantum technologies with information and communication technology introduces new complexities and obstacles. The inclusion of noise in quantum channels thus reduce the fidelity of quantum states, resulting in errors and jeopardizing the communication’s security.
To address the challenges posed by noisy quantum channels, there have been proposals for various strategies and protocols that aim to mitigate the effects of noise and enhance the security and reliability of quantum communication. They employ techniques such as error correction codes(Yang et al., 2022), quantum error correction, and adaptive protocols (Eastman et al., 2019)to overcome the limitations imposed by noise and maintain the integrity and confidentiality of quantum states.
Understanding the background of unconditional mistrustful quantum cryptography protocols in the Nx-ICT scenario, specifically in the presence of noisy quantum channels, is crucial for developing robust and secure communication systems. By exploring the theoretical foundations, existing protocols, and mitigation strategies, we can gain insights into the advancements made in this field and identify the open challenges that need to be addressed to ensure secure communication in real-world scenarios.
III. Nx-ICT
Nx-ICT stands for Next-Generation Information and Communication Technology. It refers to the advancements and integration of cutting-edge technologies in various domains, including communication, computing, networking, and information systems. Nx-ICT encompasses a wide range of emerging technologies and applications that aim to revolutionize the way we communicate, access information, and interact with digital systems.
In the context of Nx-ICT, advanced technologies are utilized to enhance the efficiency, reliability, and security of information and communication processes. Some key technologies that fall under the umbrella of Nx-ICT include: cloud computing, Internet of Things (IoT), big data analytics, artificial intelligence (AI) and machine learning (ML), cybersecurity, and autonomous systems(Knud Erik Skouby, 2022).
Figure.
NX-ICT (source: Researcher).
The integration of these advanced technologies in the Nx-ICT scenario has the potential to transform industries, improve efficiency, and enhance the overall quality of life(Kirankumar & Momaya, 2020). However, it also introduces new challenges, including the need for robust and secure communication protocols that can withstand potential vulnerabilities and threats associated with these technologies.
Figure.
NX-ICT Integration (source: Researcher).
Quantum cryptography protocols that do not rely on trust assumptions can be used to ensure the confidentiality, integrity, and authenticity of information exchanged in the Nx-ICT scenario, even in the presence of noise and potential adversaries. These protocols are essential for developing resilient and secure systems in the era of advanced information and communication technologies.
In Nx-ICT scenarios, the integration of quantum communication into existing ICT infrastructure introduces the challenge of noisy quantum channels. These channels can degrade quantum states, leading to decoherence and loss, which can reduce signal fidelity. Decoherence occurs when quantum states interact with the environment, leading to the loss of quantum coherence and the introduction of errors(Albash & Lidar, 2015). Loss occurs when quantum states are lost as they travel through the channel.
In addition to decoherence and loss, noisy quantum channels can also be exposed to environmental noise. This noise can come from a variety of sources, such as electromagnetic radiation, temperature fluctuations, and electromagnetic interference. Environmental noise can introduce errors and disturbances into the transmitted quantum states, which can impact the reliability and security of quantum communication(Pirandola et al., 2017).
In addition to decoherence, loss, and environmental noise, noisy quantum channels can also suffer from channel impurities. These impurities can arise from the characteristics of the physical components used in the channel, such as imperfect qubits, imperfect gates, or imperfect measurements(Li et al., 2023). Channel impurities can contribute to errors and introduce noise into the communication process, which can degrade the quality of the transmitted quantum states.
Additionally, noisy quantum channels can be subject to channel disturbances. These disturbances can be caused by external factors, such as fluctuations in channel parameters, transmission delays, phase shifts, or amplitude variations. They can also be caused by intentional attacks, such as quantum hacking or eavesdropping. Channel disturbances can lead to errors and misalignment of quantum states, which can further compromise the security of the communication in noisy channels.
The impact of noisy quantum channels in Nx-ICT scenarios can be mitigated by using a combination of theoretical advancements, experimental techniques, and practical implementations. These strategies and techniques include quantum error correction, noise estimation and characterization, measurement calibration, error extrapolation, noise-adaptive strategies, quantum repeaters and amplifiers, and adaptive protocols and channel monitoring. By addressing the challenges of noisy quantum channels, we can develop more reliable and secure quantum communication systems.
IV. Noisy Quantum Channels
The effects of noise on quantum communication can vary depending on the type of noise, the noise levels, and the specific quantum communication protocol being used. A noisy quantum channel can be described using the Kraus operator formalism(Udayakumar & Kumar-Eslami, 2019) or the Choi matrix representation(Datta et al., 2018).
a) Kraus Operator Formalism
A noisy quantum channel is represented by a set of Kraus operators {K_i}, where each operator K_i acts on the input state to produce the corresponding output state. The Kraus operators satisfy the condition
∑_i K_i^† K_i = I, ensuring that the probabilities are preserved.
The action of the noisy quantum channel on an input state ρ can be expressed as:
ρ’ = ∑_i K_i ρ K_i^†
Here, ρ’ represents the output state after the noisy channel acts on the input state ρ.
b) Choi Matrix Representation
The Choi matrix is a convenient mathematical representation of a quantum channel. It is a square matrix that fully characterizes the action of the channel. Let’s consider a noisy quantum channel that maps an input density matrix ρ to an output density matrix ρ’. The Choi matrix C associated with this channel is defined as:
where |Φ⟩ = ∑_ij |ii⟩/√d represents a maximally entangled state between the input and output Hilbert spaces, and d is the dimension of the Hilbert space.
C = (∑_i (K_i ⊗ I) |Φ ⟩⟨ Φ| (K_i ⊗ I) ^†),
The Choi matrix C provides a compact representation of the noisy quantum channel and can be used to analyze its properties, such as quantum process tomography, entanglement-breaking property, and channel capacity(Torlai et al., 2023).
Both the Kraus operator formalism and the Choi matrix representation provide a mathematical framework to describe the action of noisy quantum channels. These representations allow researchers to analyze the effects of noise, design error correction schemes, evaluate channel capacities, and develop strategies for secure quantum communication in the presence of noise.
There are many types of noise that can occur in quantum communication systems, including decoherence, photon loss, photon misidentification, depolarizing noise, scattering and attenuation, and eavesdropping. These noises can degrade the signal quality, introduce errors, and decrease the overall efficiency of quantum communication. The effects of noise can vary depending on the type of noise, the noise levels, and the specific quantum communication protocol being used.
| Type of noise | Effect |
| Decoherence | Causes quantum systems to lose coherence and information. |
| Photon loss | Leads to a reduction in the number of photons that reach the receiver. |
| Photon misidentification | Can lead to errors in quantum state measurements. |
| Depolarizing noise | Randomly changes the polarization of transmitted photons. |
| Scattering and attenuation | Introduce losses and alter the properties of the transmitted quantum states. |
| Eavesdropping | Can introduce additional noise into the quantum channel and compromise the security of quantum communication. |
Dealing with noisy quantum channels is a crucial aspect of designing secure and reliable quantum communication protocols. Researchers propose various strategies to mitigate the effects of noise, including error correction codes, quantum error correction techniques, and adaptive protocols that can adapt to changing channel conditions.
V. Unconditional Mistrustful Quantum Cryptography
Unconditional mistrustful quantum cryptography is a branch of quantum cryptography that deals with the problems that arise when two parties who do not trust each other need to communicate securely(Nadeem, 2014b). In classical cryptography, this is typically done using a trusted third party, such as a bank or a government agency. However, in quantum cryptography, it is possible to achieve unconditional security without the need for a trusted third party. This is because the laws of quantum mechanics prevent certain types of cheating(Giacomini et al., 2019), since, it is impossible to copy a quantum state perfectly, which means that a dishonest party cannot simply record a message and then replay it later.
Additionally, the laws of quantum mechanics also prevent certain types of quantum measurements from being made without the knowledge of the other party(Nadeem, 2014b). The Cryptographic tasks that can be achieved using unconditional mistrustful quantum cryptography include:
a) Quantum Key Distribution (QKD)
This a process of generating a shared secret key between two parties who do not trust each other. The main equation used in quantum key distribution (QKD) is the secret key rate equation(Papanastasiou et al., 2023). This equation gives the rate at which a secret key can be generated between two parties, given the error rate of the quantum channel and the amount of noise in the system. The equation is as follows:
where:
R = k * (1 - h(e))
R is the secret key rate, in bits per second.
k is a constant that depends on the specific QKD protocol being used.
e is the error rate of the quantum channel.
h(e) is the binary entropy function, which gives the amount of information that can be extracted from a binary signal with error rate e.
The secret key rate equation shows that the secret key rate decreases as the error rate of the quantum channel increases, and this is because the eavesdropper can use the errors to gain information about the secret key.
The secret key rate also decreases as the noise in the system increases, and this is because the noise can also be used by Eve to gain information about the secret key. The secret key rate equation is a fundamental equation in QKD. It is used to design QKD protocols and to estimate the performance of QKD systems(Lo et al., 2014).
b) Quantum Bit Commitment
This a process of one party, X, committing to a value, such as a bit, and then later revealing the value without being able to change it. There is no single quantum bit commitment (QBC) equation, as the security of QBC protocols depends on a number of factors, including the specific protocol being used(Song & Yang, 2020) and the amount of noise in the system(Bennett & DiVincenzo, 2000). However, there are a number of equations that are used to analyze the security of QBC protocols. One important equation is the binding equation. The binding equation gives the probability that a cheating sender can open a commitment to a value that they did not actually commit to. The binding equation is as follows:
where:
P(cheat) < 1 - 2^{-c}
P(cheat) is the probability that the sender can cheat.
c is a constant that depends on the specific QBC protocol being used.
The binding equation shows that the probability of cheating decreases as the constant c increases. This is because a larger value of c means that the sender needs to store more quantum information in order to cheat.
Another important equation is the concealing equation(Y. Gao et al., 2020). This equation gives the probability that a cheating receiver can learn the value of a commitment without the sender’s cooperation. The concealing equation is as follows:
where:
P(learn) < 1 - 2^{-s}
P(learn) is the probability that the receiver can learn the value of the commitment.
s is a constant that depends on the specific QBC protocol being used.
The concealing equation shows that the probability of learning decreases as the constant s increases. This is because a larger value of s means that the receiver needs to store more quantum information in order to learn the value of the commitment. The binding and concealing equations are used to analyze the security of QBC protocols.
c) Quantum Oblivious Transfer (OT)
This is the process of one party, Alice, sending a message to another party, Bob, such that Bob can choose which message he receives, but Alice cannot learn which message he chose. There is no single quantum oblivious transfer (OT) equation, as the security of OT protocols depends on a number of factors, including the specific protocol being used and the amount of noise in the system(Erven et al., 2014).
However, there are a number of equations that are used to analyze the security of OT protocols. One important equation is the security equation. This equation gives the probability that an eavesdropper (Eve) can learn the value of a message that is being transferred in an OT protocol. The security equation is as follows:
where:
P(Eve learns message) < 1 - 2^{-k}
P(Eve learns message) is the probability that Eve can learn the value of the message.
k is a constant that depends on the specific OT protocol being used.
The security equation shows that the probability of Eve learning the message decreases as the constant k increases. This is because a larger value of k means that Eve needs to store more quantum information in order to learn the value of the message. The security equation is used to analyze the security of OT protocols. The goal of an OT protocol is to have a security equation with high probability(Bose, 2013).
This means that Eve cannot learn the value of the message that is being transferred. Additionally, other equations that are used to analyze the security of OT protocols, calculate the amount of noise that can be tolerated in the system, to optimize the performance of OT protocols, and to design new OT protocols.
Other common equations used in quantum oblivious transfer include: -
- -
- The commitment equation(He, 2015) is used to ensure that the sender cannot cheat by sending a different message than the one they committed to.
- -
- The opening equation(Tsarev et al., 2023) is used to ensure that the receiver cannot learn the value of the message without the sender’s cooperation.
- -
- The privacy equation(Watkins et al., 2023) is used to ensure that the sender cannot learn which message the receiver chose.
These equations are used to prove the security of quantum oblivious transfer protocols. They show that it is impossible for either party to cheat or learn the value of the message without the cooperation of the other party. Quantum oblivious transfer is a powerful cryptographic primitive that has a number of applications. It can be used to build secure voting systems, secure auctions, and secure electronic contracts. It is a promising technology with the potential to revolutionize secure communication.
Unconditional mistrustful quantum cryptography is a relatively new field of research, and there are still many challenges that need to be overcome. However, the potential benefits of this technology are significant, and it is likely to play an important role in future secure communication systems. Some of the challenges that need to be overcome in order to realize the full potential of unconditional mistrustful quantum cryptography.
VI. Existing Quantum Cryptography Protocols
Some of the most well-known protocols include:
- ▪
- BB84 (Bennett-Brassard 1984): This is the first quantum key distribution protocol to be proposed, and it is still one of the most widely used protocols today. It is based on the idea of sending qubits in a random basis, and it is relatively simple to implement. However, it is also relatively susceptible to noise.
- ▪
- E91 (Ekert 1991): This protocol is based on the use of entangled pairs of photons, and it is more secure than BB84 against noise. However, it is also more complex to implement.
- ▪
- BBM92 (Bennett, Brassard, Mermin, and Popescu 1992): This protocol is a hybrid of BB84 and E91, and it offers a good compromise between security and complexity.
- ▪
- SARG04 (Scarani, Acín, Ribordy, Gisin, and Kurtsiefer 2004): This protocol is based on the use of four different bases, and it is more secure than BB84 against noise. However, it is also more complex to implement.
- ▪
- MDI-QKD (Measurement-Device-Independent QKD): This protocol does not require the trusted setup of the devices, and it is therefore more secure against attacks on the devices. However, it is also more complex to implement.
- ▪
- As the technology continues to develop, new protocols are being developed all the time. The goal is to develop protocols that are both secure and efficient, so that they can be used for practical applications.
VII. Security Challenges in Nx-ICT Scenarios
Security challenges in Nx-ICT scenarios arise due to the integration of advanced technologies and the increasing complexity of interconnected systems. These challenges can have significant implications for the confidentiality, integrity, and availability of information, as well as the overall trustworthiness of the systems involved. Here are some key security challenges in Nx-ICT scenarios:
(i) Cybersecurity Threats
As Nx-ICT systems become more interconnected and reliant on digital infrastructure, they become attractive targets for cybercriminals. Threats such as hacking, data breaches, malware, ransomware, and distributed denial-of-service (DDoS) attacks pose serious risks to the security and stability of Nx-ICT systems. The ever-evolving nature of cyber threats requires continuous vigilance and robust security measures.
(ii) Privacy Concerns
Nx-ICT systems often deal with large amounts of personal and sensitive data. The collection, storage, and processing of such data raise privacy concerns, especially in cases where user consent, data anonymization, and secure data handling practices are not properly implemented. Unauthorized access, data leaks, or improper data usage can lead to privacy breaches and loss of user trust.
(iii) Trustworthiness of Devices and Infrastructure
Nx-ICT scenarios involve a wide range of interconnected devices, including sensors, actuators, gateways, and cloud-based infrastructure. Ensuring the trustworthiness of these devices is a significant challenge, as compromised or malicious devices can introduce vulnerabilities, manipulate data, or disrupt the entire system. Supply chain attacks and the verification of device authenticity are critical concerns in Nx-ICT security.
(iv) Data Integrity and Authenticity
Maintaining the integrity and authenticity of data in Nx-ICT scenarios is crucial. Malicious actors may attempt to manipulate or forge data, leading to incorrect decision-making, financial losses, or safety risks. Robust data integrity and authentication mechanisms, such as digital signatures, secure timestamps, and secure data storage, are necessary to mitigate these risks.
(v) Complexity and Interoperability
Nx-ICT environments involve the integration of diverse technologies, platforms, and protocols, leading to complex ecosystems. The interoperability of different components and systems can introduce security challenges, as vulnerabilities in one component can potentially impact the security of the entire system. Ensuring secure communication and seamless integration between heterogeneous systems is a key challenge in Nx-ICT security.
(vi) Insider Threats
Insiders with privileged access, such as employees, contractors, or administrators, pose a significant security risk in Nx-ICT scenarios. Insider threats can range from unintentional errors or negligence to deliberate malicious actions. Implementing strong access controls, monitoring systems, and comprehensive security policies are essential to mitigate the risks associated with insider threats.
(vii) Emerging Technologies
The adoption of emerging technologies in Nx-ICT, such as artificial intelligence (AI), machine learning (ML), blockchain, and quantum computing, introduces both opportunities and security challenges. These technologies often have unique security requirements and potential vulnerabilities that need to be addressed. Adequate security measures and practices specific to each technology must be implemented to ensure their secure integration and usage.
Addressing the security challenges in Nx-ICT scenarios requires a multi-layered approach that includes robust security protocols, regular security assessments, employee awareness and training, secure software development practices, and collaboration among stakeholders. Continuous monitoring, threat intelligence, and rapid incident response capabilities are also essential to detect and respond to security incidents in a timely manner.
VIII. Evaluation of Noisy Quantum Channels
The evaluation of noisy quantum channels is a complex and challenging problem. The evaluation involves assessing their performance, quantifying the effects of noise, and understanding their impact on various quantum communication protocols.
Important factors that need to be considered when evaluating noisy quantum channels include; the type of noise that is present in the channel, such as decoherence, loss, and errors. The severity of the noise which can vary from channel to channel. Some channels are very noisy, while others are relatively noise-free. The length of the channel, which also affect the amount of noise that is introduced.
Longer channels are more likely to be affected by noise than shorter channels, and the type of quantum information that is being transmitted can also affect the amount of noise that is introduced. Some types of quantum information are more resilient to noise than others. Some common methods used to evaluate noisy quantum channels:
a) Quantum Process Tomography
Quantum process tomography is a technique used to experimentally determine the action of a quantum channel. By preparing a set of input states, applying the channel, and measuring the resulting output states, one can reconstruct the quantum process and assess the effects of noise(Torlai et al., 2023b). Quantum process tomography provides a complete characterization of the channel, allowing for the evaluation of its fidelity, coherence, and other properties.
It involves preparing a set of input states, applying the quantum process to these states, and measuring the resulting output states. The measured data is then used to reconstruct the complete quantum process. The formula for quantum process tomography depends on the specific representation chosen for the quantum process.
One common representation is the operator-sum representation, which uses Kraus operators to describe the action of the channel. In this case, the formula for quantum process tomography can be expressed as follows:
Given a set of d-dimensional input states {|ψ_i⟩}, and their corresponding d-dimensional output states {|φ_i⟩} after the quantum process, the quantum process tomography formula is:
where E(ρ) represents the reconstructed quantum process acting on the input state ρ, and M_i are the estimated Kraus operators obtained from the measurement data. The operators M_i should satisfy the condition ∑_i M_i† M_i = I, ensuring that the reconstructed process is completely positive and trace-preserving.
E(ρ) = ∑_i (M_i ρ M_i†),
The goal of quantum process tomography is to find the set of Kraus operators {M_i} that best fits the measured data, such that the reconstructed quantum process accurately represents the true action of the channel. Various optimization techniques, such as maximum likelihood estimation or least squares fitting, are employed to find the best estimate of the Kraus operators based on the measurement outcomes. The formula provided here represents a general framework for quantum process tomography using the operator-sum representation.
b) Fidelity and Trace Distance
The fidelity and trace distance are measures used to quantify the similarity between two quantum states. In the context of noisy quantum channels, these measures can be employed to evaluate the fidelity of transmission through the channel. A high fidelity indicates that the channel preserves the transmitted quantum states well, while a low fidelity implies significant degradation due to noise(Choi et al., 2023). The trace distance provides an upper bound on the error probability of discriminating between input states after they have passed through the channel.
The formulas for fidelity and trace distance are as follows: The fidelity between two quantum states ρ and σ is defined as the overlap between their square root density matrices:
In terms of the density matrices ρ and σ, the fidelity can be expressed as:
The fidelity ranges from 0 to 1, with a value of 1 indicating perfect overlap or identical states(Waters et al., 2014). The trace distance between two quantum states ρ and σ is defined as half of the trace norm of their difference:
In terms of the density matrices ρ and σ, the trace distance can be expressed as:
The trace distance ranges from 0 to 1, with a value of 0 indicating that the states are indistinguishable, while a value of 1 means the states are perfectly distinguishable.
F (ρ, σ) = Tr(√(√ρσ√ρ)) ²
F(ρ, σ) = Tr(ρσ) + 2 √[√(ρ) √(σ) √(ρ)]
D (ρ, σ) = 1/2 ||ρ - σ||₁
D (ρ, σ) = 1/2 Tr (|ρ - σ|)
Both fidelity and trace distance provide measures of the similarity or distinguishability between quantum states. In the context of evaluating noisy quantum channels, a high fidelity indicates that the channel preserves the transmitted quantum states well, while a low trace distance implies significant degradation due to noise and makes the states more distinguishable(Hahn & Tan, 2022). These measures are valuable for assessing the performance of noisy quantum channels and evaluating the fidelity of transmission through them.
c) Quantum Capacity
The quantum capacity of a noisy quantum channel represents the maximum rate at which reliable quantum information can be transmitted through the channel(Larocca et al., 2023). Evaluating the quantum capacity provides insights into the channel’s performance and its ability to transmit quantum states with high fidelity.
Various techniques, including entanglement-assisted capacities, coherent information, and entropic quantities, can be employed to estimate or bound the quantum capacity (Smith et al., 2011). The quantum capacity of a noisy quantum channel quantifies the maximum rate at which reliable quantum information can be transmitted through the channel.
The exact formula for the quantum capacity depends on the specific channel model and noise characteristics(Velu & Putra, 2023). However, there are some general formulas and bounds that are commonly used in quantum information theory. Coherent Information, which provides an upper bound on the quantum capacity and is defined as the mutual information between the input and output states of the channel after the optimal measurement is performed.
The coherent information of a quantum channel is given by:
where ρ is the input state, (ρ) is the corresponding maximally entangled state between the input and an ancillary system, (ρ) represents the output state after the channel acts on ρ, and S (⋅) denotes the von Neumann entropy.
Q() = max_ρ [S((ρ)) - S(((ρ)))],
Entanglement-assisted Quantum Capacity: For certain classes of channels (Velu & Putra, 2023), the entanglement-assisted quantum capacity provides a higher achievable rate than the coherent information. It considers the assistance of pre-shared entanglement between the sender and receiver. The entanglement-assisted quantum capacity can be calculated using various entanglement measures, such as the distillable entanglement or the relative entropy of entanglement (Piveteau et al., 2022).
Quantum Channel Capacities: Different quantum channel capacities are defined based on specific criteria or restrictions. For example, the zero-error capacity represents the maximum rate at which information can be transmitted with zero probability of error(Liu et al., 2023). The private capacity quantifies the maximum rate at which classical information can be transmitted privately, without revealing any information to a potential eavesdropper. These capacities may involve complex optimization problems or are generally hard to compute.
d) Entanglement Measures
Noisy quantum channels can degrade the amount of entanglement between quantum systems (Lami & Regula, 2023). Evaluating the entanglement measures, such as entanglement entropy, concurrence, or entanglement of formation, before and after the channel’s action helps assess the impact of noise on the entanglement properties.
A decrease in the entanglement measures indicates the degradation caused by the noisy channel. Entanglement measures used to quantify the entanglement between quantum systems include.
Entanglement Entropy: is a measure of the amount of entanglement in a quantum state (Luo et al., 2017), calculated using the von Neumann entropy formula, which is based on the density matrix of the quantum state. The entanglement entropy:
where ρ_A is the reduced density matrix obtained by tracing out the degrees of freedom of subsystem B from the joint density matrix ρ_AB of a bipartite system A and B.
S_ent is: S_ent = -Tr (ρ_A log₂ ρ_A),
Concurrence(Schneeloch et al., 2023), is a measure of entanglement specifically designed for two-qubit (bipartite) quantum systems. It quantifies the entanglement between the two qubits, and is expressed as
where λ_i are the square roots of the eigenvalues of the matrix ρ_AB * (σ_y ⊗ σ_y) * ρ_AB^* * (σ_y ⊗ σ_y), and σ_y is the Pauli Y matrix.
C = max (0, λ_1 - λ_2 - λ_3 - λ_4),
Entanglement of Formation: This is a measure of entanglement that quantifies the minimum amount of entanglement needed to create a given quantum state(Hrmo et al., 2023). It is defined for bipartite quantum systems. The formula for the entanglement of formation EoF is not straightforward and often requires optimization procedures. However, for pure states, the formula simplifies to:
where S_ent is the entanglement entropy of the state.
EoF = S_ent,
These formulas provide a way to compute entanglement measures for specific quantum states or density matrices, enabling the quantification of entanglement between quantum systems. Different entanglement measures have different properties and applications, and the choice of measure depends on the specific context and requirements of the quantum system under consideration.
e) Error Correction Performance
Noisy quantum channels pose challenges to quantum error correction schemes. Evaluating the performance of error correction codes and protocols in the presence of noise provides insights into the channel’s characteristics. Metrics such as the logical error rate, the threshold for fault-tolerant operations, or the decoding failure rate are used to assess the effectiveness of error correction under noisy channel conditions(Wang et al., 2022).
The evaluation of error correction performance in the context of noisy quantum channels involves analyzing the effectiveness of error correction codes and protocols in mitigating the effects of noise.
Logical Error Rate (p_logical): The logical error rate is a measure of the error probability per encoded qubit or logical qubit(Katabarwa & Geller, 2015). It represents the probability that an error occurs in the encoded information that cannot be corrected by the error correction code. The logical error rate can be expressed as:
where p_error is the error probability per physical qubit, and n is the number of physical qubits used to encode each logical qubit.
p_logical = (p_error) ^n,
The threshold error rate represents the maximum error rate per physical qubit that a quantum error correction code can tolerate while maintaining a low logical error rate. It serves as a critical performance metric for error correction codes. The threshold error rate can vary depending on the specific error correction code and the channel model considered. It is typically determined through numerical simulations or analytical calculations.
The decoding failure rate quantifies the probability that the error correction decoder fails to recover the original encoded information correctly. It represents the likelihood that the decoding process introduces additional errors or fails to correct errors that occurred during transmission through the noisy channel. The decoding failure rate can be estimated by considering the error correction code’s decoding algorithm and the noise model of the channel.
The fidelity of the error correction process measures the similarity between the original encoded state and the state after error correction. It quantifies how well the error correction code preserves the encoded information despite the presence of noise. Fidelity can be calculated using the formula:
where |ψ⟩ is the original encoded state, and ρ_corrected is the state after error correction.
F = |⟨ψ|ρ_corrected|ψ⟩|,
It is important to note that the formulas mentioned above provide general frameworks for evaluating error correction performance. The specific formulas and calculations may vary depending on the type of error correction code, the encoding scheme, the decoding algorithm, and the noise model considered in the analysis.
f) Security Analysis
Noisy quantum channels can introduce vulnerabilities and compromise the security of quantum communication protocols. Evaluating the security of quantum cryptographic schemes, such as quantum key distribution (QKD) or quantum secure direct communication (QSDC), in the presence of noise involves analyzing the error rates, information leakage, or vulnerability to eavesdropping attacks.
Security proofs and analysis techniques specific to each cryptographic protocol are employed to evaluate the impact of noise on their security guarantees. Security analysis is a highly specialized and rigorous field that requires expertise in cryptography, mathematics, and information theory. The specific techniques and formulas employed in security analysis vary depending on the cryptographic scheme under study and the specific security properties being evaluated.
By utilizing these evaluation methods, researchers can gain a deeper understanding of the performance, limitations, and characteristics of noisy quantum channels. This knowledge is crucial for designing robust quantum communication protocols, optimizing error correction schemes, and developing strategies to mitigate the effects of noise on quantum information processing.
IX. Mitigation Strategies and Countermeasures
Mitigating the security risks and challenges associated with unconditional mistrustful quantum cryptography protocols in Nx-ICT scenarios, specifically in the presence of noisy quantum channels, requires the implementation of appropriate countermeasures and mitigation strategies. Some key strategies to consider:
I. Error Correction Codes
Implementing quantum error correction codes (QECCs) is crucial to mitigate the impact of noise and errors introduced by noisy quantum channels. QECCs enable the detection and correction of errors, thereby enhancing the reliability and security of quantum communication(Majumdar & Sur-Kolay, 2020). By encoding quantum information redundantly, errors can be identified and rectified, ensuring the integrity and confidentiality of transmitted data.
II. Noise Estimation and Characterization
Understanding the characteristics and behavior of noisy quantum channels is essential for developing effective countermeasures. Implementing techniques for noise estimation and characterization allows for better modeling and analysis of the noise sources, enabling the design of tailored strategies to mitigate their effects. Quantum process tomography and other estimation techniques can be employed to gather information about the noise present in the channel.
III. Quantum Error Mitigation Techniques
In addition, various error mitigation techniques can be employed to reduce the impact of noise in quantum systems. The techniques involve estimating and compensating for the errors induced by the noisy quantum channel. Examples include error mitigation based on measurement calibration, error extrapolation, and noise-adaptive strategies. Such techniques can improve the reliability and fidelity of quantum communication protocols(Ravi et al., 2022).
IV. Channel Monitoring and Adaptive Protocols
Continuous monitoring of the noisy quantum channels allows for real-time assessment of their performance. By monitoring the noise levels, error rates, and other relevant parameters, adaptive protocols can be implemented to dynamically adjust the communication parameters or switch to alternative channels with lower noise. Adaptive protocols(Z. Gao et al., 2015) can optimize the communication process and mitigate the effects of noisy channels.
V. Quantum Key Distribution (QKD) Protocols
QKD protocols are specifically designed to address security concerns in quantum communication(Mehic et al., 2022). Implementing QKD protocols with proper key reconciliation and privacy amplification techniques can enhance the security of quantum communication in the presence of noise and mistrustful scenarios. QKD protocols allow for the secure distribution of encryption keys, ensuring confidentiality even in the presence of potential eavesdroppers.
VI. Quantum Authentication and Verification
Implementing robust authentication and verification mechanisms is crucial to ensure the integrity and authenticity of quantum communication in noisy channels. Techniques such as quantum signatures, quantum message authentication codes (MACs)(Rao & Jayaraman, 2023), or quantum digital certificates can be employed to verify the integrity and origin of quantum information. These mechanisms help detect tampering or unauthorized modifications during transmission.
VII. Quantum Repeaters and Amplifiers
In long-distance quantum communication scenarios, the use of quantum repeaters and amplifiers can mitigate the effects of noise in noisy quantum channels. Quantum repeaters(Ghalaii & Pirandola, 2020) allow for the distribution of entanglement over long distances, while amplifiers help boost the weak quantum signals and improve their reliability. These technologies can extend the reach of quantum communication and mitigate the detrimental effects of noise.
VIII. Quantum Cryptographic Protocols with Post-Processing
Post-processing techniques, such as error correction, privacy amplification, and key distillation, play a crucial role in mitigating the effects of noise and mistrustful scenarios in quantum cryptography protocols. These techniques help extract secure cryptographic keys, eliminate noise-induced errors, and enhance the security of the communication process.
It is important to note that the implementation of these mitigation strategies should be tailored to the specific requirements and constraints of the Nx-ICT scenario. The choice of countermeasures will depend on the level of noise, the characteristics of the quantum channels, the specific cryptographic protocols being used, and the desired security objectives. Continuous research and development in the field of quantum cryptography are crucial for advancing these mitigation strategies and ensuring the security of quantum communication in noisy channels.
IX. Future Directions and Open Challenges
Unconditional mistrustful quantum cryptography protocols in Nx-ICT scenarios, especially in the presence of noisy quantum channels, present several open challenges and opportunities for future research and development.
Noise-Resilient Quantum Protocols: Developing quantum cryptographic protocols that are specifically designed to operate effectively in the presence of noise is an important research direction. Designing protocols with built-in noise resilience, error correction capabilities, and adaptive strategies can enhance the reliability and security of quantum communication in noisy quantum channels.
Quantum Error Correction in Noisy Channels: Advancements in quantum error correction codes and techniques are essential to combat the detrimental effects of noise in quantum channels. Developing more efficient, fault-tolerant error correction codes that can handle various types of noise is crucial for improving the performance and reliability of quantum communication protocols.
Quantifying and Characterizing Noisy Quantum Channels: Further research is needed to understand and quantify the characteristics of noisy quantum channels in Nx-ICT scenarios. Developing comprehensive models for characterizing noise sources, studying the impact of different noise types and levels, and devising methods for accurately estimating and mitigating noise in real-time are important challenges.
Scalability and Long-Distance Quantum Communication: Scaling up quantum communication networks and achieving long-distance quantum communication are significant challenges. Overcoming limitations related to the degradation of quantum signals over long distances, developing efficient repeater technologies, and addressing scalability issues are important directions for future research.
Hybrid Classical-Quantum Security Approaches: Investigating hybrid approaches that combine classical and quantum security measures can provide additional layers of protection in Nx-ICT scenarios. Developing protocols that leverage both classical and quantum cryptographic techniques to mitigate the effects of noise, address trust issues, and enhance overall security is an emerging research area.
Addressing these future directions and open challenges will require interdisciplinary collaboration among researchers, engineers, and policymakers, and will involve advancements in quantum technology, cryptography, communication protocols, and system design to create secure and reliable unconditional mistrustful quantum cryptography protocols in the presence of noisy quantum channels.
References
- Albash, T.; Lidar, D. A. Decoherence in adiabatic quantum computation. Physical Review A-Atomic, Molecular, and Optical Physics 2015, 91(6). [Google Scholar] [CrossRef]
- Al-Ghamdi, A.; Al-Sulami, A.; Aljahdali, A. O. On the security and confidentiality of quantum key distribution. Security and Privacy 2020, 3(5). [Google Scholar] [CrossRef]
- Bennett, C. H.; DiVincenzo, D. P. Quantum information and computation. Nature 2000, 404(6775), 247–255. [Google Scholar] [CrossRef] [PubMed]
- Bose, S. Quantum togetherness. Nature 2013, 502(7469), 40–41. [Google Scholar] [CrossRef]
- Bozzio, M.; Vyvlecka, M.; Cosacchi, M.; Nawrath, C.; Seidelmann, T.; Loredo, J. C.; Portalupi, S. L.; Axt, V. M.; Michler, P.; Walther, P. Enhancing quantum cryptography with quantum dot single-photon sources. Npj Quantum Information 2022, 8(1), 104. [Google Scholar] [CrossRef]
- Choi, J.; Shaw, A. L.; Madjarov, I. S.; Xie, X.; Finkelstein, R.; Covey, J. P.; Cotler, J. S.; Mark, D. K.; Huang, H.-Y.; Kale, A.; Pichler, H.; Brandão, F. G. S. L.; Choi, S.; Endres, M. Preparing random states and benchmarking with many-body quantum chaos. Nature 2023, 613(7944), 468–473. [Google Scholar] [CrossRef] [PubMed]
- Datta, C.; Sazim, Sk.; Pati, A. K.; Agrawal, P. Coherence of quantum channels. Annals of Physics 2018, 397, 243–258. [Google Scholar] [CrossRef]
- Eastman, J. K.; Szigeti, S. S.; Hope, J. J.; Carvalho, A. R. R. Controlling chaos in the quantum regime using adaptive measurements. Physical Review A 2019, 99(1), 012111. [Google Scholar] [CrossRef]
- Erven, C.; Ng, N.; Gigov, N.; Laflamme, R.; Wehner, S.; Weihs, G. An experimental implementation of oblivious transfer in the noisy storage model. Nature Communications 2014, 5(1), 3418. [Google Scholar] [CrossRef]
- Gao, Y.; Al-Sarawi, S. F.; Abbott, D. Physical unclonable functions. Nature Electronics 2020, 3(2), 81–91. [Google Scholar] [CrossRef]
- Gao, Z.; Dai, L.; Wang, Z.; Chen, S. Spatially Common Sparsity Based Adaptive Channel Estimation and Feedback for FDD Massive MIMO. IEEE Transactions on Signal Processing 2015, 63(23), 6169–6183. [Google Scholar] [CrossRef]
- Gebhart, V.; Santagati, R.; Gentile, A. A.; Gauger, E. M.; Craig, D.; Ares, N.; Banchi, L.; Marquardt, F.; Pezzè, L.; Bonato, C. Learning quantum systems. Nature Reviews Physics 2023, 5(3), 141–156. [Google Scholar] [CrossRef]
- Ghalaii, M.; Pirandola, S. Capacity-approaching quantum repeaters for quantum communications. Physical Review A 2020, 102(6), 062412. [Google Scholar] [CrossRef]
- Giacomini, F.; Castro-Ruiz, E.; Brukner, Č. Quantum mechanics and the covariance of physical laws in quantum reference frames. Nature Communications 2019, 10(1), 494. [Google Scholar] [CrossRef]
- Grasselli, F. Quantum Cryptography; Springer International Publishing, 2021. [Google Scholar] [CrossRef]
- Hahn, T. A.; Tan, E. Y.-Z. Fidelity bounds for device-independent advantage distillation. Npj Quantum Information 2022, 8(1), 145. [Google Scholar] [CrossRef]
- He, G. P. Security bound of cheat sensitive quantum bit commitment. Scientific Reports 2015, 5(1), 9398. [Google Scholar] [CrossRef]
- Hrmo, P.; Wilhelm, B.; Gerster, L.; van Mourik, M. W.; Huber, M.; Blatt, R.; Schindler, P.; Monz, T.; Ringbauer, M. Native qudit entanglement in a trapped ion quantum processor. Nature Communications 2023, 14(1), 2242. [Google Scholar] [CrossRef]
- Katabarwa, A.; Geller, M. R. Logical error rate in the Pauli twirling approximation. Scientific Reports 2015, 5(1), 14670. [Google Scholar] [CrossRef]
- Kirankumar, D.; Momaya, S. Technology Management, IB and Competitiveness. In Shailesh J. Mehta School of Management (SJMSOM); 2020. [Google Scholar]
- Knud Erik Skouby, I. W. A. G. Handbook on ICT in Developing Countries: Next Generation ICT Technologies; CRC Press, 2022. [Google Scholar]
- Lami, L.; Regula, B. No second law of entanglement manipulation after all. Nature Physics 2023. [Google Scholar] [CrossRef]
- Larocca, M.; Ju, N.; García-Martín, D.; Coles, P. J.; Cerezo, M. Theory of overparametrization in quantum neural networks. Nature Computational Science 2023, 3(6), 542–551. [Google Scholar] [CrossRef] [PubMed]
- Li, C.-L.; Fu, Y.; Liu, W.-B.; Xie, Y.-M.; Li, B.-H.; Zhou, M.-G.; Yin, H.-L.; Chen, Z.-B. Breaking universal limitations on quantum conference key agreement without quantum memory. Communications Physics 2023, 6(1), 122. [Google Scholar] [CrossRef]
- Liu, Y.; Ramanathan, R.; Horodecki, K.; Rosicka, M.; Horodecki, P. Optimal measurement structures for contextuality applications. Npj Quantum Information 2023, 9(1), 63. [Google Scholar] [CrossRef]
- Lo, H.-K.; Curty, M.; Tamaki, K. Secure quantum key distribution. Nature Photonics 2014, 8(8), 595–604. [Google Scholar] [CrossRef]
- Luo, Y.; Zhang, F.-G.; Li, Y. Entanglement distribution in multi-particle systems in terms of unified entropy. Scientific Reports 2017, 7(1), 1122. [Google Scholar] [CrossRef]
- Majumdar, R.; Sur-Kolay, S. Approximate Ternary Quantum Error Correcting Code with Low Circuit Cost. 2020 IEEE 50th International Symposium on Multiple-Valued Logic (ISMVL) 2020, 34–39. [Google Scholar] [CrossRef]
- Malina, L.; Dzurenda, P.; Ricci, S.; Hajny, J.; Srivastava, G.; Matulevicius, R.; Affia, A.-A. O.; Laurent, M.; Sultan, N. H.; Tang, Q. Post-Quantum Era Privacy Protection for Intelligent Infrastructures. IEEE Access 2021, 9, 36038–36077. [Google Scholar] [CrossRef]
- Mehic, M.; Niemiec, M.; Rass, S.; Ma, J.; Peev, M.; Aguado, A.; Martin, V.; Schauer, S.; Poppe, A.; Pacher, C.; Voznak, M. Quantum Key Distribution. ACM Computing Surveys 2021, 53(5), 1–41. [Google Scholar] [CrossRef]
- Mehic, M.; Rass, S.; Fazio, P.; Voznak, M. Quantum Key Distribution Networks; Springer International Publishing, 2022. [Google Scholar] [CrossRef]
- Nadeem, M. Unconditionally secure commitment in position-based quantum cryptography. Scientific Reports 2014a, 4(1), 6774. [Google Scholar] [CrossRef]
- Nadeem, M. Unconditionally secure commitment in position-based quantum cryptography. Scientific Reports 2014b, 4(1), 6774. [Google Scholar] [CrossRef] [PubMed]
- Padmavathi A. Secure Communication Through Identity Based Cryptography Using Quantum Mechanics. In Turkish Journal of Computer and Mathematics Education; 2021; Vol. 12, Issue 13. [Google Scholar]
- Papanastasiou, P.; Mountogiannakis, A. G.; Pirandola, S. Composable security of CV-MDI-QKD with secret key rate and data processing. Scientific Reports 2023, 13(1), 11636. [Google Scholar] [CrossRef] [PubMed]
- Pirandola, S.; Laurenza, R.; Ottaviani, C.; Banchi, L. Fundamental limits of repeaterless quantum communications. Nature Communications 2017, 8(1), 15043. [Google Scholar] [CrossRef]
- Piveteau, A.; Pauwels, J.; Håkansson, E.; Muhammad, S.; Bourennane, M.; Tavakoli, A. Entanglement-assisted quantum communication with simple measurements. Nature Communications 2022, 13(1), 7878. [Google Scholar] [CrossRef]
- Rao, B. D.; Jayaraman, R. A novel quantum identity authentication protocol without entanglement and preserving pre-shared key information. Quantum Information Processing 2023, 22(2), 92. [Google Scholar] [CrossRef]
- Ravi, G. S.; Smith, K. N.; Gokhale, P.; Mari, A.; Earnest, N.; Javadi-Abhari, A.; Chong, F. T. VAQEM: A Variational Approach to Quantum Error Mitigation. 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA) 2022, 288–303. [Google Scholar] [CrossRef]
- Schneeloch, J.; Tison, C. C.; Jacinto, H. S.; Alsing, P. M. Negativity vs. purity and entropy in witnessing entanglement. Scientific Reports 2023, 13(1), 4601. [Google Scholar] [CrossRef]
- Sharma, A.; Paradkar, A.; Rao, V. N. Quantum Technologies II: Cryptography, Blockchains, and Sensing; Springer, 2022; pp. 55–102. [Google Scholar] [CrossRef]
- Smith, G.; Smolin, J. A.; Yard, J. Quantum communication with Gaussian channels of zero quantum capacity. Nature Photonics 2011, 5(10), 624–627. [Google Scholar] [CrossRef]
- Song, Y.; Yang, L. Semi-Counterfactual Quantum Bit Commitment Protocol. Scientific Reports 2020, 10(1), 6531. [Google Scholar] [CrossRef]
- Torlai, G.; Wood, C. J.; Acharya, A.; Carleo, G.; Carrasquilla, J.; Aolita, L. Quantum process tomography with unsupervised learning and tensor networks. Nature Communications 2023a, 14(1), 2858. [Google Scholar] [CrossRef] [PubMed]
- Tsarev, M.; Thurner, J. W.; Baum, P. Nonlinear-optical quantum control of free-electron matter waves. Nature Physics 2023. [Google Scholar] [CrossRef]
- Udayakumar, P.; Kumar-Eslami, P. Kraus operator formalism for quantum multiplexer operations for arbitrary two-qubit mixed states. Quantum Information Processing 2019, 18(12), 361. [Google Scholar] [CrossRef]
- Velu, C.; Putra, F. H. R. How to introduce quantum computers without slowing economic growth. Nature 2023, 619(7970), 461–464. [Google Scholar] [CrossRef]
- Wang, W.; Chen, Z.-J.; Liu, X.; Cai, W.; Ma, Y.; Mu, X.; Pan, X.; Hua, Z.; Hu, L.; Xu, Y.; Wang, H.; Song, Y. P.; Zou, X.-B.; Zou, C.-L.; Sun, L. Quantum-enhanced radiometry via approximate quantum error correction. Nature Communications 2022, 13(1), 3214. [Google Scholar] [CrossRef] [PubMed]
- Waters, C. A.; Strande, N. T.; Pryor, J. M.; Strom, C. N.; Mieczkowski, P.; Burkhalter, M. D.; Oh, S.; Qaqish, B. F.; Moore, D. T.; Hendrickson, E. A.; Ramsden, D. A. The fidelity of the ligation step determines how ends are resolved during nonhomologous end joining. Nature Communications 2014, 5(1), 4286. [Google Scholar] [CrossRef]
- Watkins, W. M.; Chen, S. Y.-C.; Yoo, S. Quantum machine learning with differential privacy. Scientific Reports 2023, 13(1), 2453. [Google Scholar] [CrossRef]
- Yang, Y.; Mo, Y.; Renes, J. M.; Chiribella, G.; Woods, M. P. Optimal universal quantum error correction via bounded reference frames. Physical Review Research 2022, 4(2), 023107. [Google Scholar] [CrossRef]
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