  A causal structure is a description of the functional dependencies between
random variables. A distribution is compatible with a given causal structure if
it can be realized by a process respecting these dependencies. Deciding whether
a distribution is compatible with a structure is a practically and
fundamentally relevant, yet very difficult problem. Only recently has a general
class of algorithms been proposed: These so-called inflation techniques
associate to any causal structure a hierarchy of increasingly strict
compatibility tests, where each test can be formulated as a computationally
efficient convex optimization problem. Remarkably, it has been shown that in
the classical case, this hierarchy is complete in the sense that each
non-compatible distribution will be detected at some level of the hierarchy. An
inflation hierarchy has also been formulated for causal structures that allow
for the observed classical random variables to arise from measurements on
quantum states - however, no proof of completeness of this quantum inflation
hierarchy has been supplied. In this paper, we construct a first version of the
quantum inflation hierarchy that is provably convergent. From a technical point
of view, convergence proofs are built on de Finetti Theorems, which show that
certain symmetries (which can be imposed in convex optimization problems) imply
independence of random variables (which is not directly a convex constraint). A
main technical ingredient to our proof is a Quantum de Finetti Theorem that
holds for general tensor products of $C^*$-algebras, generalizing previous work
that was restricted to minimal tensor products.
