Improved finite-size key rates for discrete-modulated continuous variable quantum key distribution under coherent attacks

TL;DR

This work delivers a composable security proof for a discrete-modulated CVQKD protocol against coherent attacks in the finite-size regime, employing the generalized entropy accumulation theorem (GEAT) and conic optimization to obtain improved key rates. By using a four-state (4-PSK) modulation with discretised heterodyne detection and a novel affine min-tradeoff function, the method avoids virtual tomography and achieves positive keys at metropolitan distances with around $n\sim 10^8$ rounds. The combination of GEAT with conic optimisation substantially lowers the required block sizes compared to prior results (e.g., BPWFA23), while maintaining practical relevance under realistic channel conditions and error-correction efficiencies. The work also discusses practical implementation constraints, potential generalizations to other constellations, and directions for further improvement via Rényi GEAT and enhanced min-tradeoff optimization.

Abstract

Continuous variable quantum key distribution (CVQKD) with discrete modulation combines advantages of CVQKD, such as the implementability using readily available technologies, with advantages of discrete variable quantum key distribution, such as easier error correction procedures. We consider a prepare-and-measure CVQKD protocol, where Alice chooses from a set of four coherent states and Bob performs a heterodyne measurement, the result of which is discretised in both key and test rounds. We provide a security proof against coherent attacks in the finite-size regime, and compute the achievable key rate. To this end, we employ the generalised entropy accumulation theorem, as well as recent advances in conic optimisation, yielding improved key rates compared to previous works. At metropolitan distances, our method can provide positive key rates for the order of $10^8$ rounds.

Figures (6)

• Figure 1: Discretisations of phase space by Bob for parameter estimation (left) based on phases and amplitudes according to the parameters $\delta,\Delta$, and key generation (right) rounds which only require a binning according to the phase.
• Figure 2: Asymptotic secret keys for different values of the cutoff $N_c$, excess noise $\xi=1\%$ and error correction at the Shannon limit. The amplitude of the states was optimized at every distance, and $\Delta=5.0$, $\delta=2.0$.
• Figure 3: Finite secret key rate according to \ref{['eq:FiniteKeyRate']} for diverse block sizes $n$ under a cutoff $N_c=12$. The excess noise of the channel was taken as $\xi=1\%$, and the error correction at the Shannon limit. The amplitude of the states at every distance was taken to be the optimal one for $n\rightarrow \infty$, and we employed $N_c = 12$, $\Delta=5.0$, $\delta=2.0$ as well as $\epsilon=10^{-10}$, $\epsilon_\mathrm{PE}=10^{-10}$ and $\epsilon_\mathrm{PA}=10^{-10}$.
• Figure 4: Finite secret key rate for $n= 10^{10}$ rounds under diverse values of the error correction efficiency $f$. Any other parameters were taken to be the same as in Figure \ref{['fig:FRates']}.
• Figure 5: Comparison of the secret key rates reported in BPWFA23 and in our scenario under different values of $n$, together with $\xi = 1\%$, $N_c=12$ and $f=1\%$. For the curve derived under the GEAT, all other parameters were taken to be the same as in Figure \ref{['fig:FRates']}.

Theorems & Definitions (9)

• Remark 1
• Proposition 1
• Corollary 1
• Lemma 1
• Proposition 2
• Theorem 1
• Theorem 2
• Lemma 2
• Proposition 3