45 research outputs found

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Quantum Correlations in the Minimal Scenario

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    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 Q\mathcal{Q}, 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 Q\mathcal{Q} is isomorphic to its polar dual. The boundary of Q\mathcal{Q} 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 Q\mathcal{Q}, 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 Q\mathcal{Q} 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 Q\mathcal{Q} with its dual.Comment: We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observable

    EUROCOMB 21 Book of extended abstracts

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    Discrete Geometry (hybrid meeting)

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    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

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    We investigate the properties of collections of linear bipartitions of points embedded into R3\R^3, which we call collections of affine splits. Our main concern is characterising the collections generated when the points are embedded into S2S^2; 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 S2S^2 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

    A Toroidal Maxwell-Cremona-Delaunay Correspondence

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    Advances in Discrete Differential Geometry

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    Differential Geometr

    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

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