2,916 research outputs found
The simplest causal inequalities and their violation
In a scenario where two parties share, act on and exchange some physical
resource, the assumption that the parties' actions are ordered according to a
definite causal structure yields constraints on the possible correlations that
can be established. We show that the set of correlations that are compatible
with a definite causal order forms a polytope, whose facets define causal
inequalities. We fully characterize this causal polytope in the simplest case
of bipartite correlations with binary inputs and outputs. We find two families
of nonequivalent causal inequalities; both can be violated in the recently
introduced framework of process matrices, which extends the standard quantum
formalism by relaxing the implicit assumption of a fixed causal structure. Our
work paves the way to a more systematic investigation of causal inequalities in
a theory-independent way, and of their violation within the framework of
process matrices.Comment: 7 + 4 pages, 2 figure
Looking for symmetric Bell inequalities
Finding all Bell inequalities for a given number of parties, measurement
settings, and measurement outcomes is in general a computationally hard task.
We show that all Bell inequalities which are symmetric under the exchange of
parties can be found by examining a symmetrized polytope which is simpler than
the full Bell polytope. As an illustration of our method, we generate 238885
new Bell inequalities and 1085 new Svetlichny inequalities. We find, in
particular, facet inequalities for Bell experiments involving two parties and
two measurement settings that are not of the
Collins-Gisin-Linden-Massar-Popescu type.Comment: Joined the associated website as an ancillary file, 17 pages, 1
figure, 1 tabl
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Moment-angle manifolds and Panov's problem
We answer a problem posed by Panov, which is to describe the relationship
between the wedge summands in a homotopy decomposition of the moment-angle
complex corresponding to a disjoint union of k points and the connected sum
factors in a diffeomorphism decomposition of the moment-angle manifold
corresponding to the simple polytope obtained by making k vertex cuts on a
standard d-simplex. This establishes a bridge between two very different
approaches to moment-angle manifolds.Comment: In form accepted by International Mathematics Research Notices 201
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