High-order encoding schemes for floodlight quantum key distribution

Floodlight quantum key distribution (FL-QKD) has realized a 1.3 Gbit/s secret-key rate (SKR) over a 10-dB-loss channel against a frequency-domain collective attack [Quantum Sci. Technol. 3, 025007 (2018)]. It achieved this remarkable SKR by means of binary phase-shift keying (BPSK) of multiple optical modes. Moreover, it did so with available technology, and without space-division or wavelength-division multiplexing. In this paper we explore whether replacing FL-QKD's BPSK modulation with a high-order encoding can further increase that protocol's SKR. First, we show that going to $K$-ary phase-shift keying with $K=32$ doubles—from 2.0 to 4.5 Gbit/s—the theoretical prediction from [Phys. Rev. A 94, 012322 (2016)] for FL-QKD's BPSK SKR on a 50-km-long fiber link. Second, we show that $2d\times 2d$ quadrature amplitude modulation does not offer any SKR improvement beyond what its $d=1$ case—which is equivalent to quadrature phase-shift keying—provides.

Figure 1

Quantum channel setup for FL-QKD under frequency-domain collective attack. ASE: amplified spontaneous emission source. SPDC: spontaneous parametric downconverter. LO: local oscillator.

Figure 2

Signal constellation examples: (a) 8PSK; (b) $4\times 4$ QAM. In both cases the gray shading marks one-standard-deviation regions for Alice's receiver about the $\{{\overline{I}}_k+i{\overline{Q}}_k\}$—where ${\overline{I}}_k+i{\overline{Q}}_k$ is how Bob's $k\mathrm{th}$ transmitted symbol would appear, in a noise-free world, at the output of Alice's dual-homodyne receiver—and the red lines mark the boundaries of her minimum error-probability decision regions. Note that the eccentricity of the elliptical contours shown in (a) originates from the correlation between Alice's LO and the light she receives from Bob. That eccentricity has been exaggerated in (a) to emphasize the rotational symmetry, while in (b) circular contours are shown because, as demonstrated in Appendix pp2, the contours' eccentricity is negligible for parameter values of interest for FL-QKD.

Figure 3

SKR lower bounds for FL-QKD vs one-way path length $L$, with insets showing the optimized brightness $N_S$ of Alice's transmission to Bob. The assumed parameter values are given in the text. (a) ${\mathrm{SKR}}_{\mathrm{LB}}$ with KPSK for $1\le {log}_2\left(K\right)\le 5$. (b) ${\mathrm{SKR}}_{\mathrm{LB}}$ with $2d\times 2d$ square-lattice QAM for $1\le d\le 4$.