  Non-locality is a powerful resource for various communication and information
theoretic tasks, e.g., to establish a secret key between two parties, or to
reduce the communication complexity of distributed computing. Typically, the
more non-local a system is, the more useful it is as a resource for such tasks.
We address the issue of non-locality distillation, i.e., whether it is possible
to create a strongly non-local system by local operations on several weakly
non-local ones. More specifically, we consider a setting where non-local
systems can be realized via measurements on underlying shared quantum states.
The hardest instances for non-locality distillation are the isotropic quantum
systems: if a certain isotropic system can be distilled, then all systems of
the same non-locality can be distilled as well. The main result of this paper
is that non-locality cannot be distilled from such isotropic quantum systems.
Our results are based on the theory of cross norms defined over the tensor
product of certain Banach spaces. In particular, we introduce a
single-parameter family of cross norms, which is used to construct a hierarchy
of convex sets that are closed under local operations. This hierarchy
interpolates between the set of local systems and an approximation to the set
of quantum systems.
