  As first noted by Rafael Sorkin, there is a limit to quantum interference.
The interference pattern formed in a multi-slit experiment is a function of the
interference patterns formed between pairs of slits, there are no genuinely new
features resulting from considering three slits instead of two. Sorkin has
introduced a hierarchy of mathematically conceivable higher-order interference
behaviours, where classical theory lies at the first level of this hierarchy
and quantum theory theory at the second. Informally, the order in this
hierarchy corresponds to the number of slits on which the interference pattern
has an irreducible dependence. Many authors have wondered why quantum
interference is limited to the second level of this hierarchy. Does the
existence of higher-order interference violate some natural physical principle
that we believe should be fundamental? In the current work we show that such
principles can be found which limit interference behaviour to second-order, or
"quantum-like", interference, but that do not restrict us to the entire quantum
formalism. We work within the operational framework of generalised
probabilistic theories, and prove that any theory satisfying Causality, Purity
Preservation, Pure Sharpness, and Purification---four principles that formalise
the fundamental character of purity in nature---exhibits at most second-order
interference. Hence these theories are, at least conceptually, very "close" to
quantum theory. Along the way we show that systems in such theories correspond
to Euclidean Jordan algebras. Hence, they are self-dual and, moreover,
multi-slit experiments in such theories are described by pure projectors.
