19,266 research outputs found
On the foundations and extremal structure of the holographic entropy cone
The holographic entropy cone (HEC) is a polyhedral cone first introduced in
the study of a class of quantum entropy inequalities. It admits a
graph-theoretic description in terms of minimum cuts in weighted graphs, a
characterization which naturally generalizes the cut function for complete
graphs. Unfortunately, no complete facet or extreme-ray representation of the
HEC is known. In this work, starting from a purely graph-theoretic perspective,
we develop a theoretical and computational foundation for the HEC. The paper is
self-contained, giving new proofs of known results and proving several new
results as well. These are also used to develop two systematic approaches for
finding the facets and extreme rays of the HEC, which we illustrate by
recomputing the HEC on terminals and improving its graph description. Some
interesting open problems are stated throughout.Comment: 32 pages, 2 figures, 3 tables. Revised to expand the description of
the connection to quantum information theory, including additional reference
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Generating facets for the cut polytope of a graph by triangular elimination
The cut polytope of a graph arises in many fields. Although much is known
about facets of the cut polytope of the complete graph, very little is known
for general graphs. The study of Bell inequalities in quantum information
science requires knowledge of the facets of the cut polytope of the complete
bipartite graph or, more generally, the complete k-partite graph. Lifting is a
central tool to prove certain inequalities are facet inducing for the cut
polytope. In this paper we introduce a lifting operation, named triangular
elimination, applicable to the cut polytope of a wide range of graphs.
Triangular elimination is a specific combination of zero-lifting and
Fourier-Motzkin elimination using the triangle inequality. We prove sufficient
conditions for the triangular elimination of facet inducing inequalities to be
facet inducing. The proof is based on a variation of the lifting lemma adapted
to general graphs. The result can be used to derive facet inducing inequalities
of the cut polytope of various graphs from those of the complete graph. We also
investigate the symmetry of facet inducing inequalities of the cut polytope of
the complete bipartite graph derived by triangular elimination.Comment: 19 pages, 1 figure; filled details of the proof of Theorem 4, made
many other small change
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