# Security of continuous-variable quantum key distribution via a Gaussian de Finetti reduction

## TL;DR

This paper introduces a new Gaussian de Finetti reduction method for continuous-variable quantum key distribution, proving security against general attacks by focusing on Gaussian collective attacks, thus confirming their optimality.

## Contribution

It develops a novel Gaussian de Finetti reduction leveraging $U(n)$ invariance and generalized $SU(2,2)$ coherent states, enabling security proofs against general attacks.

## Key findings

- Security against general attacks reduces to security against Gaussian collective attacks.
- The new reduction confirms Gaussian attacks are optimal for these protocols.
- Provides a rigorous foundation for the security of continuous-variable QKD.

## Abstract

Establishing the security of continuous-variable quantum key distribution against general attacks in a realistic finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. Indeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on de Finetti reductions only provide security for unrealistically large block lengths. We address this problem here by considering a new type of Gaussian de Finetti reduction, that exploits the invariance of some continuous-variable protocols under the action of the unitary group $U(n)$ (instead of the symmetric group $S_n$ as in usual de Finetti theorems), and by introducing generalized $SU(2,2)$ coherent states. Our reduction shows that it is sufficient to prove the security of these protocols against Gaussian collective attacks in order to obtain security against general attacks, thereby confirming rigorously the widely held belief that Gaussian attacks are indeed optimal against such protocols.