45 research outputs found
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Quantum Correlations in the Minimal Scenario
In the minimal scenario of quantum correlations, two parties can choose from
two observables with two possible outcomes each. Probabilities are specified by
four marginals and four correlations. The resulting four-dimensional convex
body of correlations, denoted , is fundamental for quantum
information theory. It is here studied through the lens of convex algebraic
geometry. We review and systematize what is known and add many details,
visualizations, and complete proofs. A new result is that is
isomorphic to its polar dual. The boundary of consists of
three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds
of exposed extreme points. These share all basic properties with the usual
maximally CHSH-violating correlations. These patches are separated by cubic
surfaces of non-exposed extreme points. We provide a trigonometric
parametrization of all extreme points, along with their exposing Tsirelson
inequalities and quantum models. All non-classical extreme points (exposed or
not) are self-testing, i.e., realized by an essentially unique quantum model.
Two principles, which are specific to the minimal scenario, allow a quick and
complete overview: The first is the pushout transformation, the application of
the sine function to each coordinate. This transforms the classical polytope
exactly into the correlation body , also identifying the boundary
structures. The second principle, self-duality, reveals the polar dual, i.e.,
the set of all Tsirelson inequalities satisfied by all quantum correlations.
The convex body includes the classical correlations, a cross
polytope, and is contained in the no-signaling body, a 4-cube. These polytopes
are dual to each other, and the linear transformation realizing this duality
also identifies with its dual.Comment: We also discuss the sets of correlations achieved with fixed Hilbert
space dimension, fixed state or fixed observable
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
Characterisation of Spherical Splits
We investigate the properties of collections of linear bipartitions of points embedded into , which we call collections of affine splits. Our main concern is characterising the collections generated when the points are embedded into ; that is, when the collection of splits is spherical. We find that maximal systems of splits occur for points embedded in general position or general position in for affine and spherical splits, respectively. Furthermore, we explore the connection of such systems with oriented matroids and show that a maximal collection of spherical splits map to the topes of a uniform, acyclic oriented matroid of rank 4, which is a uniform matroid polytope. Additionally, we introduce the graphs associated with collections of splits and show that maximal collections of spherical splits induce maximal planar graphs and, hence, the simplicial 3-polytopes. Finally, we introduce some methodologies for generating either the hyperplanes corresponding to a split system on an arbitrary embedding of points through a linear programming approach or generating the polytope given an abstract system of splits by utilising the properties of matroid polytopes. Establishing a solid theory for understanding spherical split systems provides a basis for not only combinatorial–geometric investigations, but also the development of bioinformatic tools for investigating non-tree-like evolutionary histories in a three-dimensional manner