University of Oxford Logo University of OxfordDepartment of Computer Science - Home
On Facebook
Facebook
Follow us on twitter
Twitter
Linked in
Linked in
Flickr
Flickr
Google plus
Google plus
Digg
Digg
Pinterest
Pinterest
Stumble Upon
Stumble Upon

Oxford Quantum Talks Archive

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

Howard Barnum, 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]