# Oxford Quantum Talks Archive

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

*Perimeter Institute*

Quantum Physics and Logic 2010, May 2010, University of Oxford

In any probabilistic theory, we may say a bipartite state omega on a composite system AB steers its marginal state omega^B if, for any decomposition of omega^B as a mixture omega^B = sum_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|a_i} 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 is steered by some bipartite state on two copies of that system, of a composite amounts to the homogeneity of the cone of unnormalized states. 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.

[video] [streaming video]