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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 theinterference patterns formed between pairs of slits, there are no genuinely newfeatures resulting from considering three slits instead of two. Sorkin hasintroduced a hierarchy of mathematically conceivable higher-order interferencebehaviours, where classical theory lies at the first level of this hierarchyand quantum theory theory at the second. Informally, the order in thishierarchy corresponds to the number of slits on which the interference patternhas an irreducible dependence. Many authors have wondered why quantuminterference is limited to the second level of this hierarchy. Does theexistence of higher-order interference violate some natural physical principlethat we believe should be fundamental? In the current work we show that suchprinciples can be found which limit interference behaviour to second-order, or"quantum-like", interference, but that do not restrict us to the entire quantumformalism. We work within the operational framework of generalisedprobabilistic theories, and prove that any theory satisfying Causality, PurityPreservation, Pure Sharpness, and Purification---four principles that formalisethe fundamental character of purity in nature---exhibits at most second-orderinterference. Hence these theories are, at least conceptually, very "close" toquantum theory. Along the way we show that systems in such theories correspondto Euclidean Jordan algebras. Hence, they are self-dual and, moreover,multi-slit experiments in such theories are described by pure projectors.

Ruling out higher-order interference from purity principles