⚛️ quantum physics

Secret Key Rate Limits in Coexisting Classical-Quantum Optical Links

This paper derives closed-form expressions to model non-linear interference from coexisting classical traffic in fiber-optic links, revealing that placing quantum key distribution channels in the upper E- or lower S-band yields higher secret key rates than the conventional O-band allocation.

Original authors: Lucas Alves Zischler, Amirhossein Ghazisaeidi, Antonio Mecozzi, Cristian Antonelli

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucas Alves Zischler, Amirhossein Ghazisaeidi, Antonio Mecozzi, Cristian Antonelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Sharing a Single-Lane Road

Imagine a fiber-optic cable is a super-fast highway for data. Usually, we have two types of traffic on this highway:

The Heavy Trucks (Classical Data): These are your Netflix streams, Zoom calls, and emails. They are loud, powerful, and move in huge groups.
The Ghost Bicycles (Quantum Data): These carry the "Secret Keys" needed to unlock your bank account or secure your messages. They are incredibly fragile, moving at the speed of light, and if they get bumped even slightly, they break.

The Problem: For years, to keep the Ghost Bicycles safe, we built them a separate, empty highway (a dedicated fiber). But that's expensive. This paper asks: Can we let the Ghost Bicycles ride in the same lane as the Heavy Trucks?

The answer is yes, but it's dangerous. The "wind" from the heavy trucks can knock the bicycles over.

The Villains: The "Wind" and the "Bumps"

When the Heavy Trucks (classical signals) zoom past, they create two main problems for the Ghost Bicycles (quantum signals):

The "Raman Wind" (Spontaneous Raman Scattering): Imagine the trucks are so fast they heat up the air around them. This heat creates a chaotic wind that blows sideways. In fiber optics, the light from the trucks scatters and creates a "noise wind" that blows into the quantum lane, confusing the delicate bicycles.
The "Four-Wave Mixing" Bumps (FWM): Sometimes, when three trucks pass each other at just the right speed and angle, they create a shockwave that hits a fourth truck (or the bicycle). It's like a ripple in a pond where three stones thrown in create a giant splash that hits a boat.

The Discovery: It's Not About Where You Think It Is

For a long time, engineers thought the safest place for the Ghost Bicycles was the O-band (around 1300nm).

The Old Logic: "Let's put the trucks in the C-band (1550nm) and the bikes in the O-band. They are far apart, so the trucks won't bother the bikes."
The New Discovery: The authors of this paper did the math and found a surprise. While the trucks are far away, the road itself (the fiber) is actually smoother and less bumpy in the E-band and S-band (a zone just above the C-band).

The Analogy:
Imagine the highway has a section with a lot of potholes (attenuation/loss) and a section with smooth asphalt.

Old Plan: Put the bikes in the "quiet" lane, but that lane has deep potholes that slow them down.
New Plan: Put the bikes in the "upper" lane (E/S-band). It's closer to the trucks, but the road is so smooth that the bikes can go faster and safer, provided we manage the wind correctly.

The Solution: The "Traffic Manager"

The paper provides a new "Traffic Manager" (a mathematical model) that calculates exactly how much wind and how many bumps the bikes will face based on:

How far the trucks are from the bikes.
How fast the trucks are going (power).
How long the road is (fiber length).

Key Findings from the Model:

The "Upper" Lane is Best: By moving the quantum signals to the upper E-band or lower S-band, we get a "Secret Key Rate" (the speed at which we can generate secure codes) that is higher than the traditional O-band.
The Trucks Can Move Too: If we move the trucks to the L-band (even further up the spectrum), the wind they blow down into the E/S-bands is much weaker. This means we can have more trucks and faster bikes without them crashing into each other.
Distance Matters: If the road is very long, the "wind" (noise) accumulates. The model shows that for long distances, the new E/S-band strategy is the only way to keep the connection alive.

Why Should You Care?

Cost: We don't need to lay down new, expensive cables for quantum security. We can use the existing internet cables we already have.
Security: As quantum computers get stronger, they will eventually break today's encryption. This research shows us how to build "Quantum-Safe" networks right now, using the infrastructure we already own.
Efficiency: It proves that we can pack more data and more security into the same physical space, making the internet faster and safer for everyone.

The Bottom Line

This paper is like a GPS update for the future of the internet. It tells us: "Stop putting your secure data in the quiet but bumpy lane. Move it to the smoother, slightly noisier lane, and you'll get there faster and safer."

By using this new map, we can finally mix our daily internet traffic with our ultra-secure quantum keys, making the digital world both cheaper to build and harder to hack.

1. Problem Statement

The paper addresses the critical challenge of deploying Quantum Key Distribution (QKD) over existing fiber-optic infrastructure alongside high-power classical data traffic (coexistence). While sharing fibers reduces deployment costs, classical signals generate significant noise that degrades weak quantum signals.

The primary impairments identified are:

Non-linear Interference: Specifically Spontaneous Raman Scattering (SpRS) and Four-Wave Mixing (FWM) generated by classical channels.
Linear Leakage: Unfiltered noise from classical transmitters and Rayleigh backscattering.
Impact: These noises increase the Quantum Bit Error Rate (QBER) in Discrete-Variable QKD (DV-QKD) and excess noise in Continuous-Variable QKD (CV-QKD), drastically reducing the Secret Key Rate (SKR).

A prevailing industry practice is to allocate QKD channels in the O-band (1260–1360 nm) and classical traffic in the C-band (1530–1565 nm) to maximize spectral separation. The authors question whether this is the optimal strategy given the complex interplay of fiber attenuation, non-linearities, and noise accumulation.

2. Methodology

The authors developed a comprehensive closed-form analytical model to evaluate interference in single-mode fibers (SMFs) for single-span systems.

Interference Modeling:
- Derived differential equations describing the evolution of noise power ( $P_{NLI}$ ) along the fiber length ( $z$ ) for both co-propagating and counter-propagating scenarios.
- Included terms for:
  - FWM: Modeled using phase-matching conditions and Manakov factors.
  - SpRS: Modeled with frequency-dependent efficiency ( $\eta_{ih}$ ).
  - Linear Leakage: Including Rayleigh backscattering.
- Provided closed-form solutions for total noise variance ( $P_{i}^{Int}$ ) by integrating these effects over the fiber length, accounting for fiber attenuation ( $\alpha$ ) and dispersion ( $\beta_2$ ).
- Introduced a linear averaging approximation for FWM to handle the rapid longitudinal oscillations caused by many channels, simplifying the calculation without significant loss of accuracy.
SKR Evaluation:
- Applied the noise model to two representative protocols:
  1. BB84 (DV-QKD): Calculated asymptotic SKR based on QBER derived from noise power spectral density (PSD).
  2. GMCS (CV-QKD): Calculated asymptotic SKR based on mutual information and the Holevo bound, normalized to shot-noise units.
- Analyzed the dependence of these rates on fiber length, channel spacing, classical signal power, and wavelength.
Simulation Parameters:
- Used realistic fiber parameters (refractive index, effective area, dispersion, non-linear coefficient) across O, E, S, C, L, and U bands.
- Simulated multi-band transmission scenarios (e.g., Classical in C-band vs. L-band; QKD in O, E, or S bands).

3. Key Contributions

Unified Analytical Framework: The paper provides the first closed-form expressions that simultaneously account for SpRS, FWM, linear leakage, and Rayleigh backscattering in a coexistence scenario, enabling rapid system design without heavy numerical simulations.
Optimal Wavelength Allocation Discovery: Contrary to the standard practice of placing QKD in the O-band, the study proves that placing quantum channels in the upper E-band (approx. 1360–1460 nm) or lower S-band yields significantly higher SKRs.
- Reasoning: While the O-band has low attenuation, the C-band classical traffic generates strong SpRS noise that propagates into the O-band. Moving the QKD channel slightly higher (E/S-band) reduces the SpRS impact from C-band pumps while maintaining lower attenuation than the L-band.
Impact of Classical Band Reallocation: The study demonstrates that moving classical traffic to the L-band further reduces noise in the E/S bands compared to C-band transmission, due to reduced non-linearities at longer wavelengths, without sacrificing classical capacity.
Protocol-Specific Scaling: Detailed analysis showing how DV-QKD (BB84) and CV-QKD (GMCS) scale differently with noise and loss, highlighting that GMCS is more robust to noise but still benefits from optimal spectral placement.

4. Key Results

Noise Evolution:
- FWM is dominant only within the transmission band and decays rapidly outside it.
- SpRS is the dominant noise source across the entire spectrum, particularly affecting shorter wavelengths when pumped by longer wavelengths (e.g., C-band pumping O/E bands).
- Counter-propagating noise (Rayleigh/SpRS) saturates after ~20 km, whereas co-propagating noise peaks at a specific distance ( $L_{Pk}$ ) before decaying.
SKR Optimization:
- Peak SKR Location: For a C-band loaded system, the maximum SKR occurs near 1400 nm (Upper E-band), not in the O-band.
- OH- Ion Impact: While the lower E-band suffers from water-peak (OH-) absorption, the upper E-band and lower S-band remain largely unaffected by OH- concentration and offer superior SKRs compared to the O-band.
- Bandwidth Effects: Allocating a continuous band for QKD (WDM QKD) smooths out localized noise dips (like water peaks), providing a more uniform aggregate SKR.
Classical Band Shift:
- Shifting classical traffic from C-band to L-band reduces SpRS noise in the E/S bands significantly.
- Shifting classical traffic to the S-band increases noise across the entire spectrum (including O-band) due to higher total power and non-linearities, penalizing QKD performance.

5. Significance

This work fundamentally shifts the design paradigm for classical-quantum coexistence:

Cost Reduction: It validates that dedicated fibers are not strictly necessary; QKD can be efficiently deployed on existing dark fiber or alongside live traffic by optimizing the spectral location.
Performance Gain: By moving QKD from the traditional O-band to the E/S-bands, network operators can achieve higher secret key rates without increasing hardware complexity.
Future-Proofing: The analytical model provides a tool for designing next-generation multi-band optical networks (O-E-S-C-L-U) where quantum and classical signals coexist dynamically.
Practical Implementation: The findings suggest that future commercial QKD systems should be engineered to operate in the upper E-band/lower S-band, and classical network planners should consider L-band allocation to minimize quantum channel interference.

In summary, the paper demonstrates that spectral engineering (specifically moving QKD to the E/S-bands and classical traffic to the L-band) is a more effective strategy for maximizing secure key rates than the traditional O/C-band separation.