Tight finite-key analysis for generalized high-dimensional quantum key distribution

Due to the capability of tolerating high error rate and generating more key bits per trial, high-dimensional quantum key distribution attracts wide interest. Despite great progresses in high-dimensional quantum key distribution, there are still some gaps between theory and experiment. One of these is that the security of the secret key heavily depends on the number of the emitted signals. So far, the existing security proofs are only suitable in the case with an infinite or unpractically large number of emitted signals. Here, by introducing the idea of “key classification” and developing relevant techniques based on the uncertainty relation for smooth entropies, we propose a tight finite-key analysis suitable for generalized high-dimensional quantum key distribution protocols. Benefitting from our theory, high-dimensional quantum key distribution protocols with finite resources become experimentally feasible.

Figure 1

Secret key rate $l/N$ vs sifted key length $N=n+(d+1)*m$ for the protocol when dimension $d=2$ (exactly six-state protocol). The solid curves show our results while the dash-dotted curves show the results given in Ref. [14]. The horizontal dashed lines represent the asymptotic rates for error rate $Q\in \{1\%,2.5\%,5\%\}$ (from top to bottom).

Figure 2

Secret key rate $l/N$ vs sifted key length $N=n+(d+1)*m$ for the protocol when dimension $d=3$. The solid curves show our results while the dash-dotted curves show the results given in Ref. [14]. The horizontal dashed lines represent the asymptotic rates for error rate $Q\in \{1\%,2.5\%,5\%\}$ (from top to bottom).

Figure 3

Secret key rate $l/N$ vs sifted key length $N=n+(d+1)*m$ for the protocol when dimension $d=17$. The solid curves show our results while the dash-dotted curves show the results given in Ref. [14]. The horizontal dashed lines represent the asymptotic rates for error rate $Q\in \{1\%,2.5\%,5\%\}$ (from top to bottom).