Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories

In any probabilistic theory, we say that a bipartite state. on a composite system AB steers its marginal state omega(B) if, for any decomposition of omega(B) as a mixture omega(B) = Sigma(i)p(i)beta(i) of states beta(i) on B, there exists an observable {a(i)} on A such that the conditional states omega(B vertical bar ai) ai are exactly the states beta(i). This is always so for pure bipartite states in quantum mechanics, a fact first observed by Schrodinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical.