Discrete-modulation continuous-variable quantum key distribution with probabilistic amplitude shaping over a linear quantum channel

TL;DR

This protocol, beyond being easily reproducible in the laboratory, provides a way to closely approach the theoretical performance offered by GG02 and preserves the ability to assure an unconditional security level.

Abstract

The practical implementation difficulties arising from the Gaussian modulation of the GG02 protocol lead us to investigate the possibilities offered by the combination of probabilistic amplitude shaping technique and quadrature amplitude modulation formats in the context of continuous variable quantum key distribution systems. Our interest comes from the fact that quadrature amplitude modulation and probabilistic shaping can be implemented with current technologies and are widely used in classical telecom equipment. In this treatment, we assume to work in the scenario of a linear quantum channel and we analyze maximum achievable secure key rates, maximum reachable distances and the resilience to noise of our discrete-modulation based protocol with respect to GG02, which is taken as a benchmark. In particular, we deal with the infinite key size regime, consider a homodyne detection scheme, and analyze what happens for different cardinalities of the input alphabet at different distances, in the case of collective attacks and in the reverse reconciliation picture. We find that our protocol, beyond being easily reproducible in the laboratory, provides a way to closely approach the theoretical performance offered by GG02 and, at the same time, preserves the ability to assure an unconditional security level.

Figures (7)

• Figure 1: Prepare-and-measure version (top) and entanglement-based version (bottom) of our discrete modulated protocol.
• Figure 2: 16QAM constellation in classical (left) and quantum phase space (right): the symbols are spaced by $\Delta$ and centered at $(x_A,y_A)$ in the former case, spaced by $2 \Delta$ and placed in $(2 \sigma_0 x_A, 2 \sigma_0 y_A)$ in the latter.
• Figure 3: SKR for GG02 and for 4QAM, 16QAM, 64QAM with uniform and non-uniform input signaling in terms of the number of photons on Alice's side at $\unit[100]{km}$ in the case of collective attacks, homodyne detection, reverse reconciliation and no excess noise (left); corresponding values of $\mathrm{R}$ in the range $\unit[0.5\text{--}300]{km}$ (right).
• Figure 4: (Left) Maxima of SKR for the 64QAM with uniform and non-uniform input signaling in the range $\unit[0.5\text{--}300]{km}$ for different excess noise values $\xi$. GG02 curves are shown as dotted lines and are hardly distinguishable, as they are almost completely superimposed on the corresponding PAS curves. (Right) Corresponding values of $\mathrm{R}$ in the same range of distances and excess noise values; the dotted lines show the maximum achievable distances of GG02 for $\xi = 0.03$ (magenta) and $\xi = 0.05$ (light blue).
• Figure 5: $\bar{n}_{\rm max}$ and mean number photons support of SKR curves at Alice's side for 16QAM (left) and for 64QAM (right) with both uniform and non-uniform signaling and with respect to the GG02 case at different excess noise values (from top to bottom $\xi=0.01,0.03,0.05$).