2,741 research outputs found
Pedersen ideals of tensor products of nonunital C*-algebras
We show that positive elements of a Pedersen ideal of a tensor product can be
approximated in a particularly strong sense by sums of tensor products of
positive elements. This has a range of applications to the structure of tracial
cones and related topics, such as the Cuntz-Pedersen space or the Cuntz
semigroup. For example, we determine the cone of lower semicontinuous traces of
a tensor product in terms of the traces of the tensor factors, in an arbitrary
C*-tensor norm. We show that the positive elements of a Pedersen ideal are
sometimes stable under Cuntz equivalence. We generalize a result of Pedersen's
by showing that certain classes of completely positive maps take a Pedersen
ideal into a Pedersen ideal. We provide theorems that in many cases compute the
Cuntz semigroup of a tensor product.Comment: circulated as preprint 2017
On the lattice structure of probability spaces in quantum mechanics
Let C be the set of all possible quantum states. We study the convex subsets
of C with attention focused on the lattice theoretical structure of these
convex subsets and, as a result, find a framework capable of unifying several
aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent
principle. We also encounter links with entanglement witnesses, which leads to
a new separability criteria expressed in lattice language. We also provide an
extension of a separability criteria based on convex polytopes to the infinite
dimensional case and show that it reveals interesting facets concerning the
geometrical structure of the convex subsets. It is seen that the above
mentioned framework is also capable of generalization to any statistical theory
via the so-called convex operational models' approach. In particular, we show
how to extend the geometrical structure underlying entanglement to any
statistical model, an extension which may be useful for studying correlations
in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416
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
- …