Silent Signals: Securing Quantum Communication with Coherent Light

Author: Denis Avetisyan

A new scheme leverages the principles of quantum mechanics and robust error correction to achieve secure communication over noisy channels.

The study demonstrates that achievable communication rates, optimized through smoothing parameters $\theta$, $\delta$, and $\epsilon$ under conditions of $\Delta = 0.05$ and a target error probability of $10^{-6}$, are directly linked to the system’s secret capacity-a relationship quantified for efficiency $\eta = 0.8$-revealing how performance is maximized within inherent limitations.

This work presents an efficient secret communication scheme for the bosonic wiretap channel, approaching optimal rates at low photon flow using pulse-position modulation and Reed-Solomon codes.

Secure communication demands increasingly sophisticated methods, yet many quantum key distribution protocols rely on complex hardware. This is addressed in ‘An Efficient Secret Communication Scheme for the Bosonic Wiretap Channel’, which introduces a computationally efficient protocol leveraging readily available lasers, direct detection, and robust error correction. The proposed scheme achieves an asymptotically optimal secret communication rate approaching the channel’s capacity limit in the low-photon-flow regime, while resisting coherent eavesdropping attacks. Could this practical approach unlock wider adoption of quantum-inspired secure communication technologies?

The Inherent Vulnerability of Communication

The modern world relies heavily on the secure transmission of information, from financial transactions and personal data to governmental and military communications. However, the very nature of communication – the propagation of signals through a medium – introduces inherent vulnerabilities to eavesdropping. Any signal transmitted through a shared channel, whether it be radio waves, fiber optic cables, or even physical couriers, is theoretically interceptible. This fundamental challenge isn’t simply a matter of technological limitations; it’s a consequence of the physics of information transfer. While encryption methods attempt to obscure the content of messages, the existence of the signal itself – its timing, strength, and even its presence – can reveal information to a determined observer. Consequently, achieving truly secure communication demands not only robust encryption but also innovative approaches to concealing the communication channel itself, a pursuit that drives ongoing research in areas like quantum cryptography and steganography.

The concept of a ‘wiretap channel’ serves as a foundational model for understanding communication vulnerabilities. This theoretical framework posits a scenario where a sender intends to transmit information to a legitimate receiver, Bob, but an eavesdropper, Eve, simultaneously attempts to intercept that communication. Importantly, the channel isn’t simply about a signal being ‘tapped’ – it’s a mathematical representation of a broadcast environment where Eve’s reception is statistically independent of Bob’s. This independence is crucial, as it allows researchers to quantify the degree to which Eve can gain information about the transmitted message without directly interfering with Bob’s ability to decode it. By analyzing the information accessible to both parties within this model, scientists can develop and evaluate strategies for secure communication that prioritize protecting the message from unauthorized access, even in the presence of a determined adversary.

Conventional encryption techniques, while historically effective, frequently depend on the sheer difficulty of solving complex mathematical problems – a security predicated on computational cost. This approach assumes that breaking the code requires resources exceeding those reasonably available to an attacker. However, the relentless progression of computing power, including the development of quantum computers capable of executing algorithms like Shor’s algorithm, poses a significant threat. These advancements progressively diminish the computational barrier, rendering previously secure encryption methods vulnerable to increasingly feasible attacks. Consequently, the long-term security of systems reliant solely on computational complexity is continually eroded, necessitating the exploration of alternative cryptographic approaches that are resistant to these evolving computational capabilities.

This scheme enables secure communication by encoding a message with a Reed-Solomon code, modulating it onto a quantum state via pulse-position modulation, and then decoding it through direct detection and extraction.

Establishing the Limits of Secure Transmission

Secrecy capacity represents the theoretical upper limit on the rate at which information can be transmitted between communicating parties, conventionally termed Bob and Eve, while maintaining confidentiality. This capacity, a fundamental benchmark in secure communication, is determined by the channel characteristics and noise levels. Specifically, it is quantified by the formula $ (1+η⋅\mathcal{E})⋅ln(1+η⋅\mathcal{E}) – (η⋅\mathcal{E})⋅ln(η⋅\mathcal{E}) – (1+(1-η)⋅\mathcal{E})⋅ln(1+(1-η)⋅\mathcal{E}) + ((1-η)⋅\mathcal{E})⋅ln((1-η)⋅\mathcal{E}) $, where η denotes the probability that the transmitted signal is a ‘0’, and $\mathcal{E}$ represents the channel noise. This calculation establishes the maximum achievable rate for secure communication given these parameters; any communication attempting to exceed this rate will inevitably leak information to a potential eavesdropper.

Randomness extractors are essential components in information-theoretic security schemes because practical sources of randomness are often imperfect and may contain biases or correlations. These extractors function by transforming a potentially weak random source – one with low min-entropy – into a shorter string of truly random bits. The core principle involves reducing the dependence on the weak source through carefully designed functions, effectively “distilling” the limited entropy into a highly random output. This process is crucial for achieving the secrecy capacity, as secure key generation and encryption rely on the availability of unbiased, unpredictable random numbers. Without effective randomness extraction, the security guarantees offered by information-theoretic principles cannot be realized in practical systems.

Randomness extractors utilize Quantum Min-Entropy as a metric to determine the amount of genuine randomness present in a source, even when that source appears partially random or biased. Quantum Min-Entropy, denoted as $H_{min}(X) = min_ρ \; -log_2(tr(ρP_X))$, represents the worst-case probability of obtaining any particular outcome from a random variable $X$, considering all possible quantum states ρ consistent with the observed distribution of $X$. This value directly bounds the length of a truly random string that can be extracted from the source with high probability. By quantifying randomness in this manner, extractors can guarantee the production of high-quality random bits, even from sources with limited or imperfect entropy, ensuring the security of cryptographic applications that rely on unpredictable inputs.

Finite Field Extractors represent a class of randomness extractors that improve upon traditional methods by performing operations within a finite field, denoted as $GF(q)$, where $q$ is a prime power. This approach allows for the efficient processing of weak random sources characterized by min-entropy, even when the entropy is significantly lower than the field size. By leveraging the algebraic properties of finite fields, these extractors can guarantee the extraction of high-quality random bits with a demonstrable security level, exceeding the capabilities of extractors limited to binary operations. This expanded functionality allows for the effective use of a wider variety of physical sources, including those exhibiting non-uniform or correlated randomness, and provides enhanced resistance to information leakage during the extraction process.

Modeling Reality: The Bosonic Wiretap Channel

The Bosonic Wiretap Channel represents a refinement of traditional wiretap channel models by explicitly incorporating the quantum characteristics of optical signals. Unlike classical models which treat signals as continuous waveforms, this channel utilizes $Coherent States$ as the input signal representation. Coherent states are a specific quantum state of the electromagnetic field, possessing a Poissonian photon number distribution and closely resembling classical light. This choice is motivated by the practical implementation of quantum communication systems where coherent light sources, such as lasers, are commonly employed. By using coherent states, the Bosonic Wiretap Channel provides a more accurate and realistic framework for analyzing the security of quantum communication protocols against eavesdropping attempts, particularly in scenarios involving photon loss.

The Pure-Loss Bosonic Wiretap Channel represents a significant refinement of theoretical quantum communication models by explicitly accounting for photon loss, a dominant impairment in practical fiber optic and free-space communication systems. This channel is mathematically defined by an amplitude attenuation factor applied to the transmitted coherent state, effectively reducing the signal strength. The impact of this loss is not merely a reduction in signal-to-noise ratio; it fundamentally alters the statistical properties of the received signal, necessitating security analyses that consider the altered distribution. The channel’s simplicity – characterized by a single loss parameter – allows for tractable analytical treatment while still providing a realistic representation of signal degradation experienced in real-world quantum key distribution (QKD) systems. The severity of loss is typically quantified by the channel transmittance, $T$, representing the fraction of photons successfully transmitted through the channel.

Direct detection is a crucial component in the Bosonic Wiretap Channel due to its practicality and efficiency in receiving quantum signals. Unlike homodyne or heterodyne detection which require precise measurement of the quadrature amplitudes or photon number, direct detection simply registers the presence of a signal exceeding a certain threshold. This simplifies the receiver design and reduces the complexity of the measurement process, making it more suitable for real-world implementations. Critically, this method does not necessitate knowledge of the precise number of photons received; the signal is determined by the cumulative effect of photon arrivals. While this comes at the cost of some information regarding the transmitted state, it offers a significant advantage in terms of implementation complexity and robustness against noise, particularly in lossy channels where photon counting becomes increasingly unreliable.

Security analysis of the Bosonic Wiretap Channel utilizes statistical bounds to quantify the probability of error in discerning the transmitted signal. Specifically, Bennett’s Inequality, the Chernoff Bound, and Hoeffding’s Inequality are employed to establish an upper limit on this error probability. These inequalities allow for the derivation of a bound of $ \le e^{-2n\theta^2} $, where ‘n’ represents the number of transmitted coherent states and ‘θ’ is a parameter related to the signal separation and channel noise. This bounding is critical for evaluating the channel’s capacity for secure communication by quantifying the difficulty an eavesdropper would face in correctly decoding the transmitted information.

Boosting Signal Integrity: Modulation and Error Correction

Pulse-Position Modulation, or PPM, presents a particularly effective strategy for transmitting information across the Bosonic Wiretap Channel by leveraging the timing of coherent state pulses. Instead of encoding data in the amplitude or phase of light, PPM encodes it in when a pulse arrives within a defined time window. This approach inherently resists certain types of eavesdropping attacks common in quantum communication, as any attempt to intercept and retransmit the pulses introduces timing errors. The robustness of PPM stems from its ability to maintain signal integrity even with some degree of noise or distortion; a slight shift in pulse timing is less disruptive than alterations to more delicate signal properties. By carefully controlling the spacing and duration of these pulses, a reliable and secure communication link can be established, offering a practical means of exploiting the unique properties of the Bosonic Wiretap Channel for confidential data transfer.

The inherent fragility of quantum communication channels necessitates robust error correction strategies, and Reed-Solomon codes provide a particularly effective solution. These codes operate by strategically adding redundant information to the original message, enabling the receiver to not only detect errors introduced during transmission – such as photon loss or noise – but also to reconstruct the original message with high fidelity. This is achieved through a mathematical process of encoding the data into a polynomial, transmitting the polynomial values, and then interpolating the original polynomial at the receiver, even if some values are lost or corrupted. The strength of Reed-Solomon codes lies in their ability to correct multiple errors within a single codeword, making them well-suited for the challenging conditions encountered in quantum key distribution and other secure communication protocols, ultimately bolstering the reliability and security of the transmitted information.

A crucial element within this communication scheme is the ‘Inverter’ – a component designed to strategically expand the initial message before it undergoes processing by the finite field extractor. This expansion isn’t arbitrary; it’s a calculated step necessary to ensure the extractor can function effectively and reliably decode the information. The finite field extractor, responsible for distilling the secret key, operates optimally when presented with a message of a specific, expanded length. Without the Inverter’s preparatory work, the extractor would be unable to accurately recover the intended message, potentially compromising the entire communication process. Essentially, the Inverter prepares the message, increasing its dimensionality to align with the requirements of the subsequent key recovery stage, thereby bolstering the security and reliability of the transmitted information.

The proposed communication scheme demonstrates a compelling approach to secure data transmission, achieving an information rate of $ (2η-1)⋅ℰ⋅ln(1/ℰ) $. This rate is particularly noteworthy as it asymptotically converges towards the theoretical limit of the secrecy capacity when dealing with a low number of photons. Importantly, the scheme’s security isn’t just about approaching this limit, but also about quantifiable bounds on potential information leakage; the difference between what is sent and what an eavesdropper might discern is mathematically constrained by $ Δ ≤ 1/2 √( |M| e^(-k ln b) + … + ϵ ) $. This bound establishes a ceiling on the vulnerability of the system, providing a rigorous measure of its resilience against attacks and confirming the feasibility of highly secure communication even in challenging quantum environments.

The pursuit of secure communication, as detailed in this scheme for the bosonic wiretap channel, isn’t about flawless transmission-it’s about managing inherent vulnerabilities. The authors demonstrate an asymptotically optimal rate, but the underlying principle remains consistent: information isn’t lost, merely obscured by noise and the ever-present threat of interception. This echoes a fundamental truth about systems built by humans – they aren’t designed for perfection, but for resilience. As Louis de Broglie observed, “It is tempting to think that the electron is a little billiard ball…but that is a false image.” The same holds true for communication channels; the idealized model rarely reflects the messy reality. Investors don’t learn from mistakes-they just find new ways to repeat them, and similarly, eavesdroppers will always seek new ways to penetrate even the most carefully constructed defenses. The efficiency gained through coherent pulses and error correction isn’t a solution, but a temporary delay of the inevitable.

Beyond the Signal

This pursuit of secure communication over bosonic channels, framed as a problem of signal extraction, reveals a deeper truth: it isn’t information people desire, but the illusion of control. The scheme’s efficiency at vanishing photon rates isn’t merely a technical achievement; it’s a testament to the human need to believe order can be imposed on noise. The mathematics describe a channel, but the motivation is an attempt to quiet the fundamental randomness of existence.

Future work will undoubtedly focus on practical implementations – minimizing error, increasing distance, and so on. But the truly interesting questions lie elsewhere. What happens when the eavesdropper isn’t trying to decode the message, but to disrupt the channel itself? This isn’t about cryptography, it’s about power. The efficiency gained through clever coding is a temporary reprieve, a localized victory against entropy. The real challenge isn’t transmitting a secret, but maintaining the belief in secrecy.

The reliance on Reed-Solomon codes and extractors suggests a faith in classical error correction. But quantum channels are not merely noisy classical channels. They introduce correlations, entanglements, and possibilities that classical tools struggle to address. Perhaps the future lies not in squeezing more information into the signal, but in embracing the inherent uncertainty, in finding security within the noise itself. It is a paradox: to control information, one must first accept its essential uncontrollability.

Original article: https://arxiv.org/pdf/2512.08623.pdf

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

The Inherent Vulnerability of Communication

Establishing the Limits of Secure Transmission

Modeling Reality: The Bosonic Wiretap Channel

Boosting Signal Integrity: Modulation and Error Correction

Beyond the Signal

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