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Security bounds for unidimensional discrete-modulated CV-QKD: a Gaussian extremality approach

John A. Mora Rodríguez, Maron F. Anka, Leonardo J. Pereira, Micael A. Dias, Alexandre B. Tacla

Abstract

Unidimensional (1D) Gaussian-modulated continuous-variable quantum key distribution protocols have been proposed as a way to simplify implementation and reduce costs through single-quadrature modulation, requiring only one modulator while maintaining compatibility with standard optical infrastructure. Here, we determine security bounds for 1D discrete-modulated protocol under the Gaussian extremality assumption by extending the method of Ghorai et al. [Phys. Rev. X 9, 021059 (2019)]. We establish the appropriate symmetry arguments to extend the method to the 1D discrete-modulated case, define the physicality zone in which the protocol is allowed to operate, and prove security against collective attacks in the asymptotic regime via semidefinite programming. Our analysis for uniformly distributed coherent states reveals a fundamental limitation: the Gaussian extremality assumption systematically overestimates Eve's information with increasing constellation size, yielding bounds so conservative that secure key extraction becomes impossible for constellations larger than four states, even under ideal conditions. This overestimation worsens with excess noise and restricts viable modulation amplitudes to impractically small values. Unlike two-dimensional (2D) protocols, where Gaussian extremality improves with constellation size, 1D protocols lack the growing phase-space isotropy required for the approximation to remain tight as the constellation grows. Our results expose these limitations and highlight the necessity of alternative methods or optimized non-uniform constellation designs for this class of protocols.

Security bounds for unidimensional discrete-modulated CV-QKD: a Gaussian extremality approach

Abstract

Unidimensional (1D) Gaussian-modulated continuous-variable quantum key distribution protocols have been proposed as a way to simplify implementation and reduce costs through single-quadrature modulation, requiring only one modulator while maintaining compatibility with standard optical infrastructure. Here, we determine security bounds for 1D discrete-modulated protocol under the Gaussian extremality assumption by extending the method of Ghorai et al. [Phys. Rev. X 9, 021059 (2019)]. We establish the appropriate symmetry arguments to extend the method to the 1D discrete-modulated case, define the physicality zone in which the protocol is allowed to operate, and prove security against collective attacks in the asymptotic regime via semidefinite programming. Our analysis for uniformly distributed coherent states reveals a fundamental limitation: the Gaussian extremality assumption systematically overestimates Eve's information with increasing constellation size, yielding bounds so conservative that secure key extraction becomes impossible for constellations larger than four states, even under ideal conditions. This overestimation worsens with excess noise and restricts viable modulation amplitudes to impractically small values. Unlike two-dimensional (2D) protocols, where Gaussian extremality improves with constellation size, 1D protocols lack the growing phase-space isotropy required for the approximation to remain tight as the constellation grows. Our results expose these limitations and highlight the necessity of alternative methods or optimized non-uniform constellation designs for this class of protocols.
Paper Structure (11 sections, 1 theorem, 59 equations, 7 figures)

This paper contains 11 sections, 1 theorem, 59 equations, 7 figures.

Table of Contents

  1. Introduction
  2. 1D Gaussian-modulated protocol
  3. 1D discrete-modulated protocol
  4. Gaussian extremality and symmetries assumptions
  5. Physicality region
  6. Lower bound for SKR via SDP
  7. Correlation parameters
  8. Purification and SDP definition
  9. Numerical results
  10. Conclusion
  11. Physicality region

Key Result

Proposition 1

Within the physicality region, $f(C_q, C_p^{max})$ is an even function, that is,

Figures (7)

  • Figure 1: The physicality region comprises the purple zone and its borders, i.e., the curves for the functions $C_p^-$ (red dashed curve) and $C_p^+$ (blue solid curve). The space beyond this region is the unphysical zone, where the protocol is not allowed to operate. Parameters: $\alpha_0 = 0.1$, $d = 10$ km, and $\xi = 0$ for 2-states (left) and 4-states (right) constellations.
  • Figure 2: Maximum Holevo information as a function of the correlation $C_q$ with $\alpha_0 = 0.1$, $d = 10$ km, $\xi = 0$ for 2 (blue bottom curve), 4 (orange middle curve), and 6-states (green upper curve) constellations.
  • Figure 3: (a) Mutual and (b) Holevo information for the 2-, 4-, and 6-state constellations for the pure-loss (solid curves) and SDP (dashed curves) models with variance of $V = \{1.04, 1.10, 1.18\}$, respectively. Parameters: $\alpha_0=0.1$, $\xi=0$.
  • Figure 4: Comparison between the 1D Gaussian-modulated CV-QKD protocol (solid curves) with the pure-loss (dashed curves) and SDP models for the discrete-modulated protocol with 2- and 4-state constellations. The dotted (dash-dotted) curve represents the SDP model with (without) correlation estimation in $p$. The variance is $V = 1.04 (1.1)$ for the 2(4)-state cases and their respective Gaussian curves. Parameters: $\alpha_0 = 0.1$, $\xi = 0$, and $\beta = 1$.
  • Figure 5: Comparison between the 1D Gaussian-modulated CV-QKD protocol (solid curves) with the SDP model for the discrete-modulated protocol with 4-states constellation for $\xi = 0$ (orange curves) and $\xi = 0.005$ (blue curves) with $\beta = 1$. The dotted (dash-dotted) curve represents the SDP model with (without) correlation estimation in $p$. The variance is $V = 1.04 (1.1)$ for the 2(4)-states cases and their respective Gaussian curves.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof