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MIMO FSO Systems in Hybrid Quantum Noise Environments: SKR Analysis with One- and Two-way CV-QKD Protocols

This paper proposes and analyzes novel one- and two-way continuous-variable quantum key distribution (CV-QKD) protocols for multiple-input multiple-output (MIMO) free-space optical (FSO) systems, deriving secret key rate expressions that quantify performance gains against atmospheric turbulence, hybrid quantum noise, and collective Gaussian eavesdropping attacks.

Original authors: Sushil Kumar, Soumya P. Dash, George C. Alexandropoulos

Published 2026-02-23

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CC BY 4.0

Original authors: Sushil Kumar, Soumya P. Dash, George C. Alexandropoulos

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

Imagine you are trying to send a top-secret message to a friend across a vast, open field using a super-bright laser pointer. This is the world of Free-Space Optical (FSO) communication. It's fast, doesn't need cables, and is great for connecting things like satellites to the ground.

However, sending secrets through the air is tricky. The wind (atmospheric turbulence) can shake your laser beam, making it wobble or spread out so your friend misses it. Worse, a sneaky spy (let's call her Eve) might be hiding in the bushes, trying to intercept your message.

This paper is about building a super-secure, high-tech laser mail system that can handle the wind, the wobble, and the spy, even when the laws of physics get a bit messy (what the authors call "hybrid quantum noise").

Here is the breakdown of their solution, explained simply:

1. The Problem: The "Wobbly Laser" and the "Noisy Room"

Usually, scientists study these laser systems assuming there is only one laser and one receiver (like a single flashlight and one eye). But in the real world, the air is messy.

The Wind: It makes the beam spread out or miss the target entirely.
The Spy: Eve tries to listen in.
The Noise: Even if the air is calm, the equipment itself creates "static." The paper calls this Hybrid Quantum Noise. Think of it like trying to hear a whisper in a room where there is both a buzzing lightbulb (electronic noise) and people randomly shouting (photon noise). This makes the message fuzzy.

2. The Solution: The "Flashlight Array" (MIMO)

Instead of using just one flashlight, the authors propose using a MIMO system.

Analogy: Imagine instead of holding one flashlight, you are holding a grid of 16, 32, or even 64 flashlights all at once, and your friend has a matching grid of eyes to catch them.
Why it helps: If the wind blows one beam off course, the other 15 beams might still hit the target. It's like throwing a net instead of a single spear; you are much more likely to catch the fish (the secret key) even if conditions are bad.

3. The Secret Handshake: One-Way vs. Two-Way

The paper compares two ways to exchange the secret code:

The One-Way Protocol (The "Drop Box"):
Alice (the sender) shines her lasers at Bob (the receiver). Bob catches them and tries to decode the message.
- The Flaw: If the wind is too strong or the noise is too loud, Bob might lose the message. If Eve is listening, she might steal the key before Bob gets it.
The Two-Way Protocol (The "Ping-Pong Game"):
This is the paper's big innovation.
1. Bob starts: Bob shines his lasers at Alice first.
2. Alice responds: Alice catches Bob's light, mixes it with her own secret, and shines it back at Bob.
3. The Magic: Because the message traveled both ways, Alice and Bob can compare notes to cancel out the "wind" and the "noise."
- Analogy: Imagine you and a friend are trying to agree on a secret code while standing in a hurricane.
  - One-way: You shout a code. The wind changes your voice, and the friend hears gibberish.
  - Two-way: You shout a sound. The friend shouts it back. You both hear how the wind distorted the sound twice. By comparing the two distortions, you can mathematically "undo" the wind and figure out the original code. This makes the system much harder for the spy to crack.

4. The "Spy" and the "Math Shield"

The authors assume Eve is very smart. She uses a "Collective Gaussian Attack," which is a fancy way of saying she uses advanced quantum physics to try to steal the key without being noticed.

To stop her, the authors did some heavy math (using things called "Mutual Information" and "Holevo bounds").

Simple Translation: They calculated exactly how much "confusion" the noise and the wind create for the spy. They proved that by using the Two-Way Protocol with the Flashlight Array (MIMO), the confusion for the spy becomes so high that she can't figure out the key, even if she has a supercomputer.

5. The Results: Why This Matters

The paper ran simulations (computer tests) to see how well this works.

Distance: As you get further away, the signal usually dies. But with the MIMO array, the signal stays strong for much longer.
The Two-Way Advantage: The "Ping-Pong" method (Two-Way) consistently beat the "Drop Box" method (One-Way), especially when the weather was bad or the distance was far.
Scaling Up: The more flashlights (antennas) you add, the better it gets. It's like having more lanes on a highway; even if some lanes are blocked by wind, traffic keeps flowing.

The Bottom Line

This paper says: "Don't rely on a single laser beam to send secrets through the air. Use a whole army of lasers, and have the sender and receiver talk back and forth."

By doing this, you can send unbreakable secret keys (Quantum Key Distribution) over long distances, even when the atmosphere is turbulent and a super-spy is trying to listen in. It turns a shaky, risky connection into a robust, secure highway for the future of 6G and satellite internet.

1. Problem Statement

The paper addresses the challenge of securing Free-Space Optical (FSO) communications for future 6G networks, specifically focusing on Continuous-Variable Quantum Key Distribution (CV-QKD) in Multiple-Input Multiple-Output (MIMO) configurations.

Key challenges identified include:

Atmospheric Impairments: FSO channels suffer from beam spreading, pointing errors (due to mechanical vibrations/tracking), and turbulence-induced fading (scintillation), which degrade signal quality.
Hybrid Quantum Noise: Practical receivers face a combination of discrete photon-counting noise (Poisson noise from background radiation/shot noise) and continuous thermal/electronic noise (Gaussian noise). Existing studies often simplify this to pure Gaussian noise, which is inaccurate for low-light quantum receivers.
Security Threats: Legitimate users (Alice and Bob) face eavesdropping by an adversary (Eve) employing collective Gaussian attacks (e.g., entangling cloner attacks).
Protocol Limitations: Most existing MIMO CV-QKD studies focus on one-way protocols or simplified noise models. The performance of two-way protocols (bidirectional communication) under hybrid quantum noise in MIMO FSO systems remains largely unexplored.

2. Methodology

The authors propose a comprehensive analytical framework to evaluate the Secret Key Rate (SKR) of MIMO FSO CV-QKD systems under realistic conditions.

System Model:
- Channel: A MIMO FSO link with $N_T$ transmit and $N_R$ receive apertures.
- Channel Gain: Modeled using the field distribution of Gaussian beams, incorporating beam spreading, pointing errors (modeled via Weibull distribution), and turbulence-induced fading (modeled via Lognormal distribution).
- Noise Model: A Hybrid Quantum Noise model is adopted, defined as the convolution of a Poisson distribution (photon counting) and a Gaussian distribution (thermal/electronic noise).
- Attack Model: Eve performs a collective Gaussian attack using an entangling cloner (EPR state) to intercept the key exchange.
Protocols Analyzed:
1. One-Way Protocol: Alice sends Gaussian-modulated coherent states to Bob. Bob performs homodyne detection and Reverse Reconciliation (RR).
2. Two-Way Protocol: A bidirectional scheme where Bob first sends states to Alice. Alice then chooses an "ON" configuration (adding her own modulation to the received state and sending it back) or "OFF" configuration. The paper focuses on the ON configuration, which has shown superior performance in high-loss regimes.
Analytical Derivation:
- Mutual Information (MI): Due to the mathematical intractability of the exact differential entropy for the hybrid noise distribution (infinite summations), the authors derive tight upper and lower bounds for the mutual information between Alice and Bob.
- Holevo Information: The upper bound on the information accessible to Eve is calculated using the symplectic eigenvalues of the covariance matrices of the quantum states.
- SKR Formulation: The SKR is derived as $SKR = \beta I(A:B) - \chi(E:B)$ , where $\beta$ is reconciliation efficiency, $I(A:B)$ is the mutual information, and $\chi(E:B)$ is the Holevo bound.
- Asymptotic Analysis: High-modulation limits ( $V_s \gg V_0, \nu$ ) are analyzed to derive closed-form expressions and compare the performance of one-way vs. two-way protocols.

3. Key Contributions

Novel Mathematical Framework: Presentation of a rigorous transmissivity model for MIMO FSO channels that simultaneously accounts for beam spreading, pointing errors, turbulence fading, and hybrid quantum noise.
Protocol Proposal: Introduction and analysis of both one-way and two-way CV-QKD protocols specifically tailored for MIMO FSO systems under hybrid quantum noise.
Analytical Bounds: Derivation of novel closed-form and asymptotic expressions for the Secret Key Rate (SKR). Crucially, they provide tight bounds on mutual information to handle the complexity of the Poisson-Gaussian noise mixture.
Performance Comparison: Demonstration that the two-way protocol consistently outperforms the one-way protocol, particularly in low-transmissivity (high-loss) regimes.
MIMO Gain Quantification: Proof that the SKR gain of the two-way protocol over the one-way protocol is proportional to the rank of the MIMO channel ( $r_H$ ), validating the scalability of MIMO for quantum security.

4. Numerical Results

The paper validates the analytical framework through extensive numerical simulations with parameters typical for FSO systems (e.g., $\lambda = 1550$ nm, various MIMO orders like 4, 8, 16, 32).

Distance vs. SKR: SKR decays sharply over short distances and stabilizes at longer ranges. The two-way protocol consistently yields higher SKR than the one-way protocol. The performance gap is most significant at short ranges but remains positive over long distances.
MIMO Scaling: Increasing the MIMO order ( $N_T, N_R$ ) significantly improves SKR. The two-way protocol benefits more from MIMO scaling than the one-way protocol.
Turbulence Impact: The two-way protocol demonstrates greater resilience to atmospheric scintillation (log-irradiance variance $\sigma^2$ ) compared to the one-way protocol.
Detector Efficiency: SKR increases with detector efficiency ( $\eta$ ). The sensitivity of SKR to efficiency improvements is higher for larger MIMO configurations.
SNR Behavior: In low SNR regimes, the one-way protocol can occasionally outperform the two-way due to additive noise accumulation in the bidirectional path. However, as SNR increases, the two-way protocol's performance improves more steeply and saturates at a higher level.

5. Significance

This paper is significant for the advancement of quantum-secure 6G networks and satellite-ground communication for several reasons:

Realism: By incorporating hybrid quantum noise (Poisson + Gaussian), the analysis moves beyond idealized theoretical models to reflect practical receiver limitations.
Scalability: It proves that MIMO technology is not just a classical communication tool but a vital enabler for scaling quantum key distribution, offering substantial SKR gains.
Protocol Optimization: It establishes the two-way protocol as a superior choice for FSO CV-QKD, especially in environments with high path loss or turbulence, providing a roadmap for designing robust quantum communication links.
Theoretical Foundation: The derived bounds on mutual information and SKR provide a necessary theoretical tool for future research in non-Gaussian noise environments in quantum optics.

In conclusion, the study confirms that combining MIMO architectures with two-way CV-QKD protocols effectively mitigates the adverse effects of atmospheric turbulence and hybrid quantum noise, enabling secure, long-distance key exchange that is resilient against collective Gaussian attacks.