Finite-size quantum key distribution rates from Rényi entropies using conic optimization

TL;DR

This work introduces a fast, general conic-optimization framework for finite-size QKD security based on Rényi entropies within the MEAT paradigm. By performing facial reduction on the sandwiched Rényi relative entropy cone and defining dedicated RényiQKD and FastRényiQKD cones, the authors enable stable, scalable optimization of the conditional Rényi entropy across protocols. They provide explicit conic formulations, barrier functions, and gradient/Hessian calculus, and implement the approach in Julia (ConicQKD.jl) to demonstrate improvements for BB84, MUBs, and DMCV QKD. The results show tighter finite-size key-rate bounds and better numerical performance, enabling practical, long-distance QKD with general security guarantees without resorting to ad-hoc or depolarizing-noise tricks.

Abstract

Finite-size general security proofs for quantum key distribution based on Rényi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional Rényi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.

Figures (5)

• Figure 1: Finite secret key rate for qubit BB84 protocol using the FastRényiQKD cone for a different number of rounds $n$. All curves consider $v=0.97$, and $f=1.16$. The probability $p^K$ was optimized for each point according to a coarse-grained tuning, while the Rényi parameter $\alpha$ was optimized numerically.
• Figure 2: Finite secret key rate for qubit BB84 protocol using the FastRényiQKD cone (solid lines), compared with the results showcased in kamin25MEATsecurity (dashed lines) for different numbers of rounds $n$. All curves consider $v=0.97$, and $f=1.16$. The probability $p^K$ was optimized for each point according to a coarse-grained tuning, while the Rényi parameter $\alpha$ was optimized numerically. For a fair comparison, we set the security parameters $\varepsilon_\mathrm{PA} = \varepsilon_\mathrm{EC} = \frac{1}{2}10^{-80}$ and $\delta \to 0$ to match a fixed-length implementation from kamin25MEATsecurity.
• Figure 3: Finite secret key generation rate for the MUB protocol using the FastRényiQKD \ref{['eq: fast qkd cone']} cone with $d=5$, $m=6$, $v = 0.9$, $f = 1.16$, $p^\mathrm{K}=0.5$, and different values for the number of rounds $n$ and the Rényi parameter $\alpha$.
• Figure 4: Finite secret key generation rate for the MUB protocol using the RényiQKD \ref{['eq: qkd cone']} and FastRényiQKD \ref{['eq: fast qkd cone']} cones with $d=5$, $m=6$, $v = 0.9$, $f = 1.16$, $p^\mathrm{K}=0.5$, $n = 10^9$, and Rényi parameter $\alpha$ varying from $1.1$ to $2$.
• Figure 5: Secret key generation rates for the DMCV protocol using the FastRényiQKD cone \ref{['eq: fast qkd cone']} for different values of the number of rounds $n$, using $N_c = 10$, $\Delta = 4.0$ and $\Delta_s = 1.5$ and error correction at the Shannon limit. $\alpha$, $p^\mathrm{K}$ and the coherent state amplitude were optimized for each point according to a coarse-grained tuning.

Theorems & Definitions (33)

• Proposition 3.1: Bretagnolle-Huber–Carol inequality
• Theorem 3.2
• Corollary 3.3
• Proposition A.1
• Remark 1
• Proposition A.2
• Proposition B.1
• proof
• Proposition C.1
• proof