Continuous variable quantum key distribution through a fading channel with minimal information leakage

Abstract

Continuous variable quantum key distribution enables two parties to extract provably secure keys for cryptography by exploiting aspects of quantum mechanics. A central challenge in achieving QKD over long distances is overcoming optical loss, which limits the secret key rates and achievable transmission distances. Recent work has illustrated how to perform continuous variable quantum key distribution while eliminating information leakage to an eavesdropper in a pure-loss channel and minimizing information leakage in a noisy channel. In this work, we analyse this protocol under conditions where the quantum channel linking the sender and receiver is a fading channel, which can be used to model free-space links. We find that the protocol maintains strong performance over fading channels and outperforms comparable protocols.

The paper asks whether a previously proposed heralded squeezed-state CV-QKD protocol, designed to eliminate Eve’s information in pure-loss channels, remains effective over fading free-space links when active eavesdropping and finite-size effects are included. It compares the protocol with squeezed-state homodyne and coherent-state heterodyne schemes using covariance-matrix calculations, Gaussian entangling-cloner individual attacks, and Holevo bounds for collective attacks. Fading is represented through an effective transmissivity $T=\langle\sqrt{\eta}\rangle^2$ and quadrature-dependent added noise $\operatorname{Var}(\sqrt{\eta})(b_q-1)$. Setting the key-generating quadrature variance to shot noise, $b_x=1$, cancels this fading-induced contribution, but not excess noise. Numerical comparisons use excess noise $\xi=0.05$, squeezing capped at $10\,\mathrm{dB}$, and fading variance up to $0.01$. Under the individual-attack model, the heralded protocol retains positive asymptotic rates beyond $30\,\mathrm{dB}$ effective loss. With finite-size corrections at block size $N=10^{10}$ and reconciliation efficiency $\beta=0.95$, its loss range is approximately $18\,\mathrm{dB}$, versus approximately $14\,\mathrm{dB}$ and $10\,\mathrm{dB}$ for the squeezed-state and coherent-state benchmarks. The advantage is substantially reduced under collective attacks: fading still strongly limits rates, and the squeezed-state benchmark tolerates the greatest loss in the finite-size comparisons. The paper attributes this limitation to the Holevo quantity’s dependence on the full covariance matrix, including the conjugate quadrature. It also explicitly qualifies the individual-attack calculation as an effective Gaussian entangling-cloner model matching key-quadrature moments, rather than a general treatment of Eve’s information for the non-Gaussian fading ensemble.