  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.
