Hidden Signals: Limits of Secure Communication Over Noisy Quantum Channels

Author: Denis Avetisyan

New research defines the fundamental limits for sending secret messages and generating encryption keys through quantum channels where signal transmission depends on the channel’s state.

A secure communication protocol leverages a quantum state-dependent channel, encoding a message $M$ and secret key $K$ within a quantum state $A$ transmitted across channel $\mathcal{N}\_{AS\to BE}$, ensuring reliable decoding at the receiver by observing $B$ while simultaneously concealing the very act of communication from an eavesdropper monitoring $E$ through their observation of $E$.

This study establishes achievable rate regions for covert communication and secret key generation over quantum state-dependent channels, demonstrating optimality for classical channels with full channel state information.

Establishing secure communication in the presence of an eavesdropper remains a fundamental challenge, particularly when channel conditions are unknown. This is addressed in ‘Covert Communication and Key Generation Over Quantum State-Dependent Channels’, which investigates the limits of both covert communication and secret key generation over quantum channels where channel characteristics depend on the quantum state itself. The paper derives achievable rate regions for these tasks, demonstrating positive rates for covert key generation and communication-a first for quantum channels-and recovering established results for classical state-dependent channels. These findings raise the question of how these theoretical limits can be approached in practical quantum communication systems and what novel security protocols might emerge from exploiting state-dependent channel properties?

The Illusion of Secure Transmission

Conventional methods of transmitting information, from spoken words to digital signals, invariably leave detectable traces that can be exploited. These traces aren’t limited to the obvious – intercepted messages or overheard conversations – but extend to subtler indicators like patterns in network traffic, the physical act of delivering a message, or even changes in behavior. This inherent vulnerability stems from the fundamental physics of communication; any signal, to be received, must interact with the environment, leaving a footprint detectable by a sufficiently sensitive observer. Consequently, even seemingly innocuous exchanges can reveal critical information to adversaries, highlighting the need for communication strategies designed to minimize or eliminate these detectable signatures, especially in contexts where secrecy is paramount.

The potential for observation poses a significant risk in contexts demanding absolute confidentiality. A discerning authority – a warden, an intelligence agency, or any controlling entity – fundamentally jeopardizes secure exchange if capable of detecting communication. This threat isn’t merely about intercepting a message, but about understanding that even the attempt to communicate reveals information. Consequently, the ability of such an authority to monitor channels, analyze patterns, or decipher codes directly undermines any effort toward secrecy, creating an asymmetrical power dynamic where the observer dictates the terms of information control. The very act of signaling becomes a vulnerability, necessitating strategies that minimize detectability or conceal communication within seemingly innocuous channels.

The pursuit of truly undetectable communication represents a fundamental challenge in secure information transfer. Covert communication methods aim to eliminate any discernible trace of exchange, moving beyond encryption to conceal the very act of communication itself. This is achieved not by making the message unreadable, but by ensuring no warden can detect that a message was even sent. Techniques range from steganography – hiding messages within innocuous carriers like images or ambient noise – to subtle manipulations of communication channel characteristics, effectively blending the signal with background noise. Successful covert communication relies on minimizing any statistical deviation from normal patterns, requiring a deep understanding of the target environment and the warden’s detection capabilities. The ultimate goal is a transmission that appears as random noise, ensuring information transfer remains unseen and preserving the secrecy of the exchange.

Exploiting the Channel: A Game of Shadows

State-dependent channels introduce variability in transmission characteristics based on shared channel state information (CSI). This variability is not inherent to the channel itself, but rather a function of the known state at a given time. Consequently, communication can be modulated to exploit this dependency, effectively creating a layer of obfuscation. By conditioning signals on the CSI, a sender can influence the received signal distribution in a manner that makes eavesdropping more difficult without knowledge of the specific channel state. The degree of covertness achievable is directly related to the granularity and predictability of the state, as well as the sender’s ability to statistically mask the presence of a signal within the naturally occurring variations dictated by the channel state. This contrasts with static channels where signal detection is independent of any external state information.

Classical state-dependent channels transmit information using conventional bits, with secrecy relying on the statistical dependence between the transmitted signal and the channel state known to the legitimate parties. Quantum state-dependent channels, conversely, leverage the principles of quantum mechanics to encode information in quantum states – qubits – offering potential security advantages. Specifically, the use of quantum states allows for encoding that is fundamentally different from classical bits, introducing properties like superposition and entanglement. These properties enable the potential for increased key rates and improved resistance to eavesdropping attempts compared to purely classical approaches, as information is not directly represented by discrete values but by probabilistic amplitudes, complicating interception without disturbance.

The effectiveness of state-dependent channels for secure communication is directly tied to the capacity to utilize channel state information to obscure transmitted data. Achievable rate regions define the limits of reliable communication under specific conditions, and for these channels, the region is characterized by $R, R_K ≥ 0$, subject to constraints involving mutual information terms and entropy. Specifically, the achievable rate, $R$, is constrained by $I(U;B)$ – the mutual information between the input $U$ and the received signal $B$ – and $I(U;E)$, representing mutual information between the input and eavesdropper’s received signal $E$. The entropy of the secret key, $H(S)$, also plays a critical role in defining the bounds of secure communication within the established rate region, dictating the maximum achievable secrecy rate.

Sculpting the Signal: Codes for the Unseen

Gelfand-Pinsker (GP) and Block Markov (BM) encoding are utilized in covert communication to transmit information within the noise of a channel, effectively hiding the signal from a warden. GP encoding achieves this by constructing a code that is robust to channel noise while simultaneously resembling the expected distribution of noise itself. BM encoding extends this concept by operating on blocks of data, creating a statistical dependence between blocks that further obscures the presence of a signal. Both techniques rely on carefully shaping the transmitted signal to minimize the information leakage to the warden, measured by quantities like mutual information. The efficacy of these methods is determined by the channel capacity under specific constraints, such as a bound on the warden’s ability to distinguish between a signal and noise, and is often analyzed through asymptotic performance bounds as the message length increases.

Channel resolvability, in the context of covert communication, refers to the ability to approximate a given channel with an arbitrary level of precision using a modified channel. This is essential because perfect channel knowledge is rarely available and allows for the construction of codes optimized for undetectability. Crucially, defining an ‘innocent state’ establishes a baseline probability distribution for channel outputs that represents legitimate, non-covert traffic. The success of a covert communication scheme relies on ensuring the statistical properties of the transmitted signal, when combined with the innocent cover traffic, remain indistinguishable from the established innocent state; any deviation increases the probability of detection. Therefore, the limits of undetectability are fundamentally bound by the ability to resolve the channel and accurately define, and then mimic, this innocent state statistically, effectively masking the covert signal within legitimate communications.

Asymptotic analysis, in the context of covert communication codes, involves evaluating the performance of Gelfand-Pinsker and Block Markov encoding schemes as the length of the transmitted message, denoted as $n$, approaches infinity. This approach allows for the determination of achievable rates – the maximum data transmission rate – while maintaining a vanishing probability of detection or error. Specifically, it establishes bounds on the code’s performance by analyzing its behavior as $n$ grows, permitting the derivation of fundamental limits on covertness. This is often expressed through the capacity of the channel under specific constraints, and provides a theoretical guarantee of performance for sufficiently long messages, even if finite-length performance differs.

One-shot achievability proofs in covert communication establish the feasibility of transmitting data at a specific rate for a single instance, accompanied by a guaranteed upper bound on the probability of error. This error probability is quantified as $ \leq 2^{vS\alpha/\alpha} 2^{\alpha(…)} + 1/2 vB\alpha * 2^{\alpha(…)} $, where variables $v$, $S$, $B$, and $\alpha$ represent parameters defining the communication scheme and channel characteristics. The formula demonstrates that the error probability is dependent on these system variables and provides a concrete limit, ensuring a provable level of reliability even for a single transmission attempt, which is crucial for applications where repeated transmissions are undesirable or impossible.

Beyond Secrecy: The Art of Undetectable Exchange

Covert secret key generation represents a significant evolution of covert communication principles, shifting the focus from simply transmitting messages undetected to establishing a shared, secure key between communicating parties. While traditional covert communication aims to hide the existence of a message, this advanced technique leverages the same principles of low-probability-of-detection to create a cryptographic key without alerting an eavesdropper – often referred to as a Warden. This key can then be used for subsequent, conventional secure communication, offering a layered approach to security. The core concept relies on exploiting subtle characteristics of the communication channel to embed key information in a way that appears indistinguishable from background noise to the Warden, effectively creating a secret pathway for key exchange. Unlike traditional key distribution methods that are vulnerable to interception, covert key generation aims to ensure the Warden remains unaware that any key exchange is even taking place, bolstering the overall security of the communication system.

The concept of secret key capacity is central to establishing secure communication through covert channels, representing the theoretical upper bound on how quickly a shared cryptographic key can be created without alerting an eavesdropper – or ‘Warden’. This capacity isn’t simply a measure of data transfer speed; it’s intrinsically linked to the balance between security and reliability. A higher key generation rate is desirable, but pushing this rate too quickly increases the probability of detection, diminishing the covertness of the communication. Determining this capacity requires careful consideration of the channel characteristics, the statistical properties of the transmitted signals, and the Warden’s ability to discern legitimate communication from noise. Essentially, the secret key capacity defines the sweet spot where a secure key can be reliably generated, offering a quantifiable measure of how effectively information can be exchanged under the radar – a critical parameter in any covert communication system.

Quantum state-dependent channels represent a significant advancement in the pursuit of secure communication, offering a theoretical boost to covert key generation compared to classical methods. These channels leverage the principles of quantum mechanics, where the characteristics of the communication medium are directly influenced by the quantum state of the transmitted signal, thereby enhancing security against eavesdropping. However, realizing this potential introduces considerable complexity; manipulating and maintaining quantum states requires sophisticated hardware and precise control, demanding advancements in quantum technology. While classical channels rely on signal strength or frequency to convey information, quantum channels necessitate managing superposition and entanglement, adding layers of engineering challenges. Despite these hurdles, the increased security offered by quantum state-dependent approaches positions them as a promising direction for future research in covert communication systems, albeit one demanding significant technological innovation.

Recent investigations have rigorously defined the boundaries of secure, undetectable communication by establishing fundamental limits for covert key generation. This work demonstrates achievable rate regions – the maximum speed at which a secret key can be created – for classical communication channels when complete channel state information (CSI) is available, proving optimality under these conditions. Critically, the research introduces a quantifiable metric for assessing the level of covertness, termed the “Warden’s Distance”, which mathematically defines how difficult it is for an eavesdropper to detect the communication. This distance is characterized as being less than or equal to $2^{vE\alpha/\alpha} \cdot 2^{\alpha(…)} $, providing a precise measure of security and a benchmark for evaluating the effectiveness of different covert communication strategies. This advancement moves beyond simply achieving secrecy to providing a verifiable, quantifiable guarantee of undetectability, paving the way for more robust and trustworthy secure communication systems.

The exploration of communication limits within quantum state-dependent channels, as detailed in this work, inherently invites a critical assessment of systemic vulnerabilities. One considers how meticulously constructed systems reveal their imperfections under stress. As Barbara Liskov aptly stated, “Programs must be right first before they are fast.” This resonates deeply; establishing the fundamental limits of covert communication-defining what’s achievable and optimal-is paramount. Speed or complexity are secondary to ensuring the inherent correctness and security of the system before attempting optimization. The paper’s focus on achievable rate regions isn’t simply about maximizing data transfer; it’s about rigorously mapping the boundaries of what’s possible without compromising integrity, a foundational principle akin to ensuring a program’s correctness before striving for efficiency.

Opening the Black Box Further

This work, in defining achievable rate regions for communication over state-dependent quantum channels, hasn’t solved the problem-it has precisely located the interesting bits. The established optimality for classical channels with complete state information feels less like a destination and more like a baseline. The real challenge lies, predictably, in the messy realities where that information is imperfect, or worse, deliberately obscured. A channel’s true character isn’t revealed by its specifications, but by how it fails to behave as expected.

Future explorations should aggressively pursue the limits of covert communication when the sender possesses only statistical glimpses of the channel state. It’s tempting to envision increasingly complex coding schemes, but a more fruitful approach might be to interrogate the very definition of ‘covert’. Does perfect secrecy truly require an unbreakable code, or simply a sufficiently high noise floor? The pursuit of absolute security often obscures the more practical question of making detection impractically difficult.

Ultimately, this isn’t about sending messages; it’s about probing the boundaries of information itself. Each established rate region is merely a carefully constructed cage, designed to contain a certain amount of unpredictability. The next step isn’t to build a better cage, but to find the seams, the weaknesses, the points where the system inevitably leaks.

Original article: https://arxiv.org/pdf/2511.17504.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Secure Transmission

Exploiting the Channel: A Game of Shadows

Sculpting the Signal: Codes for the Unseen

Beyond Secrecy: The Art of Undetectable Exchange

Opening the Black Box Further

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