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Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories | | Howard Barnum
; Carl Phillipp Gaebler
; Alexander Wilce
; | | Date: |
30 Dec 2009 | | Abstract: | In any probabilistic theory, we may say a bipartite state on a composite
system AB steers its marginal state (on, say, system B) if, for any
decomposition of the marginal as a mixture, with probabilities p_i, of states
b_i of B, there exists an observable a_i on A such that the states of B
conditional on getting outcome a_i on A, are exactly the states b_i, and the
probabilities of outcomes a_i are p_i. This is always so for pure bipartite
states in quantum mechanics, a fact first observed by Schroedinger 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. | | Source: | arXiv, 0912.5532 | | Services: | Forum | Review | PDF | Favorites | |
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