Entanglement distribution via satellite: an evaluation of competing protocols assuming realistic free-space optical channels

Technical Summary: Entanglement Distribution via Satellite

Problem Statement
The establishment of a global quantum network requires the high-rate, high-fidelity distribution of entangled resources between spatially separated nodes. While fiber-optic networks are limited by exponential decay in transmissivity with distance, satellite-based free-space optical (FSO) links offer a pathway to continental-scale communication (∼10³ km) by circumventing the need for intermediate repeaters. However, the efficacy of satellite-mediated entanglement distribution depends heavily on the network topology and the specific protocol employed.

Two primary configurations exist:

1. Relay Configuration: Entanglement is generated at a ground station (Alice), sent to a central satellite (Charlie), while a second entangled mode is generated at the destination station (Bob) and sent to Charlie. Charlie performs entanglement swapping (teleamplification) to establish entanglement between Alice and Bob.
2. Distribution Configuration: Entanglement is generated centrally at the satellite (Charlie) and distributed directly to both ground stations (Alice and Bob).

A critical technical challenge arises in atmospheric channels: uplink transmission (ground-to-satellite) suffers from significantly higher and more variable losses due to atmospheric turbulence in the lowest 20 km, whereas downlink transmission (satellite-to-ground) experiences lower, more stable losses. The paper investigates whether the superior scaling of the relay protocol (which theoretically scales with the loss of a single channel, $\eta$, rather than the product of two, $\eta^2$) can overcome the severe losses of the uplink compared to the distribution protocol's reliance on lower-loss downlinks.

Methodology
The authors evaluate these competing protocols across three distinct scenarios using both discrete-variable (DV) single-rail resources and continuous-variable (CV) two-mode squeezed vacuum (TMSV) resources. They also analyze the impact of probabilistic noiseless linear amplification (NLA) using first-order quantum scissors.

1. Symmetric Vacuum Channels (Triple-Satellite): A theoretical baseline where three Low-Earth Orbit (LEO) satellites communicate via symmetric, deterministic, pure-loss channels.
2. Symmetric Channels with Amplification: The same triple-satellite setup, but with Alice and Bob equipped with quantum scissors to perform distributed NLA (in the relay case) or local NLA (in the distribution case).
3. Asymmetric Atmospheric Channels (Ground-Satellite-Ground): A realistic scenario where two ground stations communicate with a central LEO satellite. The channels are modeled as stochastic, anisotropic fading channels. The authors simulate the propagation of Gaussian beams through the atmosphere using a split-step method with phase screens to generate probability density functions (PDFs) for channel transmissivity ($\eta$) for both uplink and downlink paths at various zenith angles ($0^\circ$ to $30^\circ$).

Performance is quantified by the rate of distillable entanglement, $R$, defined as the product of the success probability ($p$) and the maximum coherent information ($I_{AB}$) of the shared state.

Key Contributions and Results

• Unamplified Protocols (Vacuum):

  • In the symmetric triple-satellite network, the relay configuration is superior. The distribution rate scales as $R \sim O(\eta^2)$ for both DV and CV resources.
  • The distribution configuration fails to distribute entanglement when the single-channel transmissivity drops below $\eta \le 0.5$ (3 dB loss), as the channel becomes antidegradable.

• Amplified Protocols (Vacuum):

  • With the inclusion of NLA, the relay configuration (using distributed NLA) achieves a rate scaling linearly with channel loss ($R \sim O(\eta)$).
  • The distribution configuration (using local NLA at both receivers) scales quadratically with loss ($R \sim O(\eta^2)$) because success requires simultaneous heralding at two independent amplifiers.
  • In a vacuum environment, the relay protocol remains superior due to its better scaling.

• Amplified Protocols (Atmospheric Channels):

  • When applied to realistic ground-satellite-ground links, the dynamics shift. The uplink channels (required for the relay protocol) exhibit extreme fading, with transmissivity often dropping to $\sim 40$ dB, whereas downlink channels (used in the distribution protocol) are more stable with losses around $\sim 20$ dB.
  • DV vs. CV: The distribution protocol shows a significant advantage for DV resources over CV resources. While CV states can hold more "raw" entanglement, the truncation inherent in quantum scissors discards higher-order degrees of freedom, reducing the accessible entanglement for CV states more severely than for DV states.
  • Optimal Strategy: Despite the relay protocol's superior theoretical scaling ($O(\eta)$ vs $O(\eta^2)$), the extreme variability and high mean loss of the uplink channels render it less effective in practice. The distribution protocol (central generation + downlink distribution + local amplification) yields a more consistent and higher average rate of entanglement distribution.
  • Ensemble Effects: The authors note that channel fluctuations introduce effective noise. For the relay protocol, this significantly degrades the ensemble distillable entanglement, particularly for CV resources, where the CV relay protocol fails to produce a positive ensemble rate. The distribution protocol, relying on more stable downlinks, suffers negligible noise penalties from channel fluctuations.

Significance and Claims
The paper concludes that for satellite-mediated entanglement distribution between distant ground stations, the distribution configuration is the optimal strategy. Specifically, the authors recommend generating single-rail entangled Bell pairs centrally at the satellite and distributing them via atmospheric downlink channels, followed by amplification using first-order quantum scissors at the ground stations.

The significance of this work lies in its rigorous comparison of network topologies under realistic atmospheric conditions, demonstrating that while theoretical scaling laws favor the relay approach in idealized vacuum, the stochastic nature of atmospheric uplinks favors the distribution approach for practical implementation. The study also highlights the specific vulnerability of continuous-variable resources to the truncation effects of quantum amplifiers in high-loss environments, suggesting discrete-variable resources are preferable for this specific application.

The authors modestly note that their analysis assumes idealized detection and pure-loss channels (justified for night-time operation) and does not account for all practical non-Gaussian noise sources (e.g., thermal background, Doppler shifts, spatial mode mismatch), which would require further mitigation strategies like adaptive optics and spectral filtering in a real-world deployment.