De Finetti Theorems for Quantum Conditional Probability Distributions with Symmetry

TL;DR

This paper introduces two new de Finetti theorems for quantum conditional probability distributions, aiding the analysis of device-independent quantum key distribution by reducing complex behaviors to simpler iid models.

Contribution

The paper presents novel de Finetti theorems for quantum conditional distributions and explores their application in restricting attacker strategies in DIQKD protocols.

Findings

Two new de Finetti theorems relate quantum conditional distributions to iid convex combinations.

One theorem can enforce restrictions on attackers in DIQKD.

Strengthening restrictions to collective attacks remains challenging.

Abstract

The aim of device-independent quantum key distribution (DIQKD) is to study protocols that allow the generation of a secret shared key between two parties under minimal assumptions on the devices that produce the key. These devices are merely modeled as black boxes and mathematically described as conditional probability distributions. A major obstacle in the analysis of DIQKD protocols is the huge space of possible black box behaviors. De Finetti theorems can help to overcome this problem by reducing the analysis to black boxes that have an iid structure. Here we show two new de Finetti theorems that relate conditional probability distributions in the quantum set to de Finetti distributions (convex combinations of iid distributions), that are themselves in the quantum set. We also show how one of these de Finetti theorems can be used to enforce some restrictions onto the attacker of a DIQKD protocol. Finally we observe that some desirable strengthenings of this restriction, for instance to collective attacks only, are not straightforwardly possible.