42,934 research outputs found
On the axiomatization of convex subsets of Banach spaces
We prove that any convex-like structure in the sense of Nate Brown is
affinely and isometrically isomorphic to a closed convex subset of a Banach
space. This answers an open question of Brown. As an intermediate step, we
identify Brown's algebraic axioms as equivalent to certain well-known axioms of
abstract convexity. We conclude with a new characterization of convex subsets
of Banach spaces.Comment: 8 pages, 1 figure. v3: added post-publication note on missing
reference with partly overlapping materia
Characterizing finite-dimensional quantum behavior
We study and extend the semidefinite programming (SDP) hierarchies introduced
in [Phys. Rev. Lett. 115, 020501] for the characterization of the statistical
correlations arising from finite dimensional quantum systems. First, we
introduce the dimension-constrained noncommutative polynomial optimization
(NPO) paradigm, where a number of polynomial inequalities are defined and
optimization is conducted over all feasible operator representations of bounded
dimensionality. Important problems in device independent and semi-device
independent quantum information science can be formulated (or almost
formulated) in this framework. We present effective SDP hierarchies to attack
the general dimension-constrained NPO problem (and related ones) and prove
their asymptotic convergence. To illustrate the power of these relaxations, we
use them to derive new dimension witnesses for temporal and Bell-type
correlation scenarios, and also to bound the probability of success of quantum
random access codes.Comment: 17 page
Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
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