Enhancing Long-distance Continuous-variable Quantum-key-distribution with an Error-correcting Relay

Noiseless linear amplifiers (NLAs) serve as an effective means to enable long-distance continuous-variable (CV) quantum key distribution (QKD), even under realistic conditions with non-unit reconciliation efficiency. Separately, unitary averaging has been suggested to mitigate some stochastic noise, including phase noise in continuous-variable states. In this work, we combine these two protocols to simultaneously compensate for thermal-loss effects and suppress phase noise, thereby enabling long-distance CV QKD that surpasses the repeaterless bound, the fundamental rate-distance limit, for repeaterless quantum communication systems.

💡 Research Summary

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This paper addresses two dominant practical impairments in continuous‑variable quantum key distribution (CV‑QKD): thermal‑loss in the optical channel and random phase noise affecting the transmitted quantum states. While noiseless linear amplifiers (NLAs) have been shown to extend the reach of CV‑QKD by compensating for loss, they are intrinsically probabilistic and only approximate ideal amplification for low‑amplitude signals. Conversely, unitary averaging (UA) has recently been proposed as a probabilistic method to suppress stochastic phase errors by sending a signal through multiple copies of the channel and heralding on vacuum detections of ancillary modes. The authors combine these two techniques into a hybrid protocol that first mitigates phase noise via UA and then compensates residual thermal loss using a distributed NLA based on quantum scissors.

The work begins with a detailed description of the entanglement‑based CV‑QKD scheme using a two‑mode squeezed vacuum (TMSV) state. The channel is modeled as a thermal‑loss channel with transmissivity η (derived from fiber loss 0.2 dB km⁻¹) and excess noise ε = 0.02. Phase noise is modeled as independent Gaussian fluctuations of the phase shifter with standard deviation σ. The authors derive the covariance matrices for the raw channel (Σ_PS) and for the UA‑processed channel (Σ_UA), and use the standard Gaussian‑state formulas to compute the mutual information I_AB and the Holevo bound χ_BE. The secret key rate (SKR) in the asymptotic regime is κ = β I_AB – χ_BE, with reconciliation efficiency β = 0.95.

In the UA analysis, Alice prepares the TMSV together with a vacuum ancilla, mixes one TMSV mode with the ancilla on a 50:50 beam‑splitter (encoding), sends both modes through the noisy channel, and Bob recombines them on a second 50:50 beam‑splitter (decoding). Successful vacuum detection on the ancilla heralds a phase‑reduced state. The authors evaluate both a single UA stage (N = 2 copies) and a double stage (N = 4 copies). Numerical results show that for moderate phase noise (σ = 0.1) the SKR without UA drops to zero around 190 km, whereas a single UA stage pushes the cutoff to ≈ 300 km (≈ 110 km improvement). For stronger phase noise (σ = 0.3) the UA‑only protocol extends the viable distance from a few kilometres to about 70 km when applied twice. These findings demonstrate that UA can dramatically mitigate phase noise, albeit at the cost of a success probability P_UA that multiplies the final key rate.

The NLA portion employs quantum scissors (k = 1) as a teleportation‑based noiseless linear amplifier. Alice and Bob each send a mode to an intermediate relay (Charlie), who interferes them on a 50:50 beam‑splitter and performs photon‑number‑resolving detection. A “click” event heralds successful amplification with gain g set by the beam‑splitter transmissivity. The success probability scales as √η, allowing the protocol to surpass the Pirandola‑Laurenza‑Ottaviani‑Banchi (PLOB) bound, which scales linearly with η. However, when phase noise is present, the effective gain and success probability degrade, limiting the NLA‑only SKR.

The hybrid UA‑NLA protocol first applies UA to suppress phase fluctuations, then uses the NLA to counteract thermal loss. The overall success probability is the product P_total = P_UA × P_NLA, and the final SKR is κ_hybrid = P_total