  Can quantum theory be seen as a special case of a more general probabilistic
theory, similarly as classical theory is a special case of the quantum one? We
study here the class of generalized probabilistic theories defined by the order
of interference they exhibit as proposed by Sorkin. A simple operational
argument shows that the theories require higher-order tensors as a
representation of physical states. For the third-order interference we derive
an explicit theory of "density cubes" and show that quantum theory, i.e. theory
of density matrices, is naturally embedded in it. We derive the genuine
non-quantum class of states and non-trivial dynamics for the case of
three-level system and show how one can construct the states of higher
dimensions. Additionally to genuine third-order interference, the density cubes
are shown to violate the Leggett-Garg inequality beyond the quantum Tsirelson
bound for temporal correlations.
