Twin-Field Quantum Key Distribution with Fully Discrete Phase Randomization

Twin-field (TF) quantum key distribution (QKD) can overcome fundamental secret-key-rate bounds on point-to-point QKD links, allowing us to reach longer distances than ever before. Since its introduction, several TFQKD variants have been proposed, and some of them have already been implemented experimentally. Most of them assume that the users can emit weak coherent pulses with a continuous random phase. In practice, this assumption is often not satisfied, which could open up security loopholes in their implementations. To close this loophole, we propose and prove the security of a TFQKD variant that relies exclusively on discrete phase randomization. Remarkably, our results show that it can also provide higher secret-key rates than an equivalent continuous-phase-randomized protocol.

Figure 1

Secret key rate for our discrete-phase-randomized protocol at different values of $M$, in comparison to the fundamental bound for repeaterless QKD systems $-{\log }_2(1-\eta )$, where $\eta$ is the overall Alice-Bob transmissivity.

Figure 2

Comparison between the results of this work and those of Ref. [13], which uses continuous phase randomization in its test-mode emissions. For simplicity, to compute the results in Ref. [13], we assume that Alice’s and Bob’s test-mode rounds provide perfect estimates of the yield probabilities $Y_{nm}$ for $n+m\le 4$, while the rest are upper bounded by 1. This is an ideal scenario and, as shown in Ref. [13], the results will be slightly worse once one considers the imperfect estimates that result from the use of a finite set of decoy states, as we do for the results in this work.

Figure 3

Comparison between the value of some terms in our analysis for the ideal case $M\to \mathrm{\infty }$ and the analysis in Ref. [13]. (a) Upper bound on the phase-error rate, assuming a fixed value $\mu =0.06$. (b) Upper bound on the phase-error rate for the value of $\mu$ that optimizes the key rate in each analysis. (c) Value of $\mu$ that optimizes the key rate in each analysis. (d) Key mode gain for the value of $\mu$ that optimizes the key rate in each analysis.