8,711 research outputs found
Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables
We define what it means for a state in a convex cone of states on a space of
observables to be generalized-entangled relative to a subspace of the
observables, in a general ordered linear spaces framework for operational
theories. This extends the notion of ordinary entanglement in quantum
information theory to a much more general framework. Some important special
cases are described, in which the distinguished observables are subspaces of
the observables of a quantum system, leading to results like the identification
of generalized unentangled states with Lie-group-theoretic coherent states when
the special observables form an irreducibly represented Lie algebra. Some open
problems, including that of generalizing the semigroup of local operations with
classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J.
Theor. Phy
Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach
We study the continuity of an abstract generalization of the maximum-entropy
inference - a maximizer. It is defined as a right-inverse of a linear map
restricted to a convex body which uniquely maximizes on each fiber of the
linear map a continuous function on the convex body. Using convex geometry we
prove, amongst others, the existence of discontinuities of the maximizer at
limits of extremal points not being extremal points themselves and apply the
result to quantum correlations. Further, we use numerical range methods in the
case of quantum inference which refers to two observables. One result is a
complete characterization of points of discontinuity for matrices.Comment: 27 page
On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets
The Krein-Milman theorem (1940) states that every convex compact subset of a
Hausdorfflocally convex topological space, is the closed convex hull of its
extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the
general framework of -convexity. Under general conditions on the class of
functions , the Krein-Milman-Ky Fan theorem asserts then, that every
compact -convex subset of a Hausdorff space, is the -convex hull of
its -extremal points. We prove in this paper that, in the metrizable case
the situation is rather better. Indeed, we can replace the set of
-extremal points by the smaller subset of -exposed points. We
establish under general conditions on the class of functions , that every
-convex compact metrizable subset of a Hausdorff space, is the
-convex hull of its -exposed points. As a consequence we obtain
that each convex weak compact metrizable (resp. convex weak compact
metrizable) subset of a Banach space (resp. of a dual Banach space), is the
closed convex hull of its exposed points (resp. the weak closed convex hull
of its weak exposed points). This result fails in general for compact
-convex subsets that are not metrizable
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