A de Finetti representation theorem for infinite dimensional quantum systems and applications to quantum cryptography

TL;DR

This paper extends the quantum de Finetti theorem to infinite dimensional systems under verifiable conditions, enabling security proofs for quantum cryptography protocols like QKD with realistic states.

Contribution

It generalizes the de Finetti theorem to infinite dimensions, facilitating security analysis of quantum cryptography with practical, unbounded systems.

Findings

The theorem applies to infinite-dimensional quantum states under certain conditions.

It supports security proofs for QKD with weak coherent and Gaussian states.

The results are experimentally verifiable and relevant for real-world quantum cryptography.

Abstract

According to the quantum de Finetti theorem, if the state of an N-partite system is invariant under permutations of the subsystems then it can be approximated by a state where almost all subsystems are identical copies of each other, provided N is sufficiently large compared to the dimension of the subsystems. The de Finetti theorem has various applications in physics and information theory, where it is for instance used to prove the security of quantum cryptographic schemes. Here, we extend de Finetti's theorem, showing that the approximation also holds for infinite dimensional systems, as long as the state satisfies certain experimentally verifiable conditions. This is relevant for applications such as quantum key distribution (QKD), where it is often hard - or even impossible - to bound the dimension of the information carriers (which may be corrupted by an adversary). In particular, our result can be applied to prove the security of QKD based on weak coherent states or Gaussian states against general attacks.