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Probabilistic amplitude shaping for continuous-variable quantum key distribution with discrete modulation over a wiretap channel

Michele N. Notarnicola, Stefano Olivares, Enrico Forestieri, Emanuele Parente, Luca Potì, Marco Secondini

TL;DR

The paper tackles the practicality gap in CV-QKD between the theoretically optimal Gaussian modulation and hardware limitations by introducing probabilistic amplitude shaping (PAS) for discrete QAM constellations. It presents a CV-QKD protocol using QAM with PAS, enabling MB-distributed symbol generation to better approximate Gaussian inputs, and analyzes key generation rates under a wiretap, pure-loss channel with lossless homodyne detection, comparing against PSK and the GG02 benchmark. Constellation design is guided by energy constraints and PAS parameters, with optimization of the Maxwell–Boltzmann parameter $\beta$ and constellation spacing $\Delta$ for QAM16 and QAM64. The work demonstrates that PAS-enhanced discrete modulation can approach GG02 performance while offering practical advantages for higher average powers and reconciliation efficiency in CV-QKD.

Abstract

To achieve the maximum information transfer and face a possible eavesdropper, the samples transmitted in continuous-variable quantum key distribution (CV-QKD) protocols are to be drawn from a continuous Gaussian distribution. As a matter of fact, in practical implementations the transmitter has a finite (power) dynamics and the Gaussian sampling can be only approximated. This requires the quantum protocols to operate at small powers. In this paper, we show that a suitable probabilistic amplitude shaping of a finite set of symbols allows to approximate at will the optimal channel capacity also for increasing average powers. We investigate the feasibility of this approach in the framework of CV-QKD, propose a protocol employing discrete quadrature amplitude modulation assisted with probabilistic amplitude shaping, and we perform the key generation rate analysis assuming a wiretap channel and lossless homodyne detection.

Probabilistic amplitude shaping for continuous-variable quantum key distribution with discrete modulation over a wiretap channel

TL;DR

The paper tackles the practicality gap in CV-QKD between the theoretically optimal Gaussian modulation and hardware limitations by introducing probabilistic amplitude shaping (PAS) for discrete QAM constellations. It presents a CV-QKD protocol using QAM with PAS, enabling MB-distributed symbol generation to better approximate Gaussian inputs, and analyzes key generation rates under a wiretap, pure-loss channel with lossless homodyne detection, comparing against PSK and the GG02 benchmark. Constellation design is guided by energy constraints and PAS parameters, with optimization of the Maxwell–Boltzmann parameter and constellation spacing for QAM16 and QAM64. The work demonstrates that PAS-enhanced discrete modulation can approach GG02 performance while offering practical advantages for higher average powers and reconciliation efficiency in CV-QKD.

Abstract

To achieve the maximum information transfer and face a possible eavesdropper, the samples transmitted in continuous-variable quantum key distribution (CV-QKD) protocols are to be drawn from a continuous Gaussian distribution. As a matter of fact, in practical implementations the transmitter has a finite (power) dynamics and the Gaussian sampling can be only approximated. This requires the quantum protocols to operate at small powers. In this paper, we show that a suitable probabilistic amplitude shaping of a finite set of symbols allows to approximate at will the optimal channel capacity also for increasing average powers. We investigate the feasibility of this approach in the framework of CV-QKD, propose a protocol employing discrete quadrature amplitude modulation assisted with probabilistic amplitude shaping, and we perform the key generation rate analysis assuming a wiretap channel and lossless homodyne detection.
Paper Structure (4 sections, 1 equation, 2 figures)

This paper contains 4 sections, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. CV-QKD with QAM discrete modulation and PAS
  3. Modulation stage
  4. Choice of the constellation parameters

Figures (2)

  • Figure 1: Schematic representation of the CV-QKD protocol discussed in the paper for a discrete modulation format. Alice generates symbols $z=x_{A}, y_{A}$ by samping either a uniform distribution $\mathcal{P}(z)$ (case ${\mathrm{I}}$) or a Maxwell-Boltzmann distribution $\mathcal{M}_\beta(z)$ (case ${\mathrm{II}}$), encodes them onto $\mathinner{|{x_{A}+iy_{A}}\rangle}$ and sends them to Bob through an untrusted pure-loss channel. Bob investigates the channel by performing a homodyne measurement of $q/p$, chosen at random. In this scenario, Eve only collects the fraction of the signals lost during the propagation through the channel.
  • Figure 2: The QAM16 constellation ($M=4$), represented in both the (classical) complex space of coherent amplitudes and the (quantum) phase space.