The matrix rank is robust for approximations, as for every matrix, there is an epsilon-ball of elements having the same or larger rank. It is known that this statement is false for multipartite tensors. In particular, tensors exhibit a gap between their tensor rank and their border rank. The same behavior also applies to tensor network decompositions, for example, tensor networks with a geometry containing a loop.

In this talk, we show that gaps between rank and border rank also occur for positive and invariant tensor decompositions. We present examples of nonnegative tensors and multipartite positive semidefinite matrices with a gap for several notions of positive and invariant tensor (network) decompositions. Moreover, we show a correspondence between certain types of quantum correlation scenarios and constraints in positive ranks. This allows showing that certain sets of multipartite probability distributions generated from local measurements on a tensor network state are not closed. Hence, testing the membership of these quantum correlation scenarios is impossible in finite time.