  In this article, I use an operational formulation of the Choi-Jamio\l{}kowski
isomorphism to explore an approach to quantum mechanics in which the state is
not the fundamental object. I first situate this project in the context of
generalized probabilistic theories and argue that this framework may be
understood as a means of drawing conclusions about the intratheoretic causal
structure of quantum mechanics which are independent of any specific
ontological picture. I then give an operational formulation of the
Choi-Jamio\l{}kowski isomorphism and show that in an operational theory which
exhibits this isomorphism, several features of the theory which are usually
regarded as properties of the quantum state can be derived from constraints on
non-local correlations. This demonstrates that there is no need to postulate
states to be the bearers of these properties, since they can be understood as
consequences of a fundamental equivalence between multipartite and temporal
correlations.
