×

On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids. (English) Zbl 1304.52019

Summary: Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in two- and three-dimensional Euclidean spaces, which focuses on the set of extreme points and the action of affine transformations on it. Regarding the above-mentioned axiomatization of quantum physics, our result corresponds to the case of simplest (2-level) quantum system. We also discuss a possible extension to higher dimensions. In the second part, towards generalizations of the framework of general probabilistic theories and several existing results including ones in the first part from the case of compact and finite-dimensional physical systems as in most of the literature to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
81P16 Quantum state spaces, operational and probabilistic concepts
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Matoušek, J., (Lectures on Discrete Geometry. Lectures on Discrete Geometry, GTM, vol. 212 (2002), Springer) · Zbl 0999.52006
[2] Sturmfels, B., Gröbner Bases and Convex Polytopes (1995), American Mathematical Society
[3] Araki, H., (Einfuhrung in die Axiomatische Quantenfeldtheorie, I, II. Einfuhrung in die Axiomatische Quantenfeldtheorie, I, II, Lecture Note Distributed By Swiss Federal Institute of Technology (1962)), See also · Zbl 0505.46053
[4] Gudder, S. P., Stochastic Method in Quantum Mechanics (1979), Dover · Zbl 0439.46047
[5] Holevo, A. S., Probabilistic and Statistical Aspects of Quantum Theory (1982), North-Holland: North-Holland Amsterdam · Zbl 0497.46053
[7] Mackey, G., Mathematical Foundations of Quantum Mechanics (1963), Dover · Zbl 0114.44002
[8] Babai, L., Symmetry groups of vertex-transitive polytopes, Geom. Dedicata, 6, 331-337 (1977) · Zbl 0388.05025
[9] Davies, E. B., Symmetries of compact convex sets, Quart. J. Math. Oxford Ser. (2), 25, 323-328 (1974) · Zbl 0298.22018
[12] Masanes, L.; Mueller, M. P., A derivation of quantum theory from physical requirements, New J. Phys., 13, 063001 (2011) · Zbl 1448.81027
[13] Kimura, G.; Nuida, K.; Imai, H., Distinguishability measures and entropies for general probabilistic theories, Rep. Math. Phys., 66, 175-206 (2010) · Zbl 1225.81028
[14] Alfsen, E. M.; Shultz, F. W., State spaces of Jordan algebras, Acta Math., 140, 155-190 (1978) · Zbl 0397.46066
[15] Iochum, B.; Shultz, F. W., Normal state spaces of Jordan and von Neumann algebras, J. Funct. Anal., 50, 317-328 (1983) · Zbl 0507.46055
[16] Kimura, G., The Bloch vector for \(N\)-level systems, Phys. Lett. A, 314, 339 (2003) · Zbl 1052.81117
[19] Grünbaum, B., (Convex Polytopes. Convex Polytopes, GTM, vol. 221 (2003), Springer)
[20] Hofmann, K. H.; Morris, S. A., The Structure of Compact Groups (1998), Walter de Gruyter
[21] Danzer, L.; Laugwitz, D.; Lenz, H., Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden, Arch. Math. (Basel), 8, 214-219 (1957) · Zbl 0078.35803
[22] Schur, I., Über gruppen periodischer substitutionen, Sitzungsber. Preuss. Akad. Wiss., 619-627 (1911) · JFM 42.0155.01
[23] Lam, T. Y., (A First Course in Noncommutative Rings. A First Course in Noncommutative Rings, GTM, vol. 131 (2001), Springer) · Zbl 0980.16001
[24] Kargapolov, M. I., On a problem of O. Ju. Šmidt, Sibirsk. Mat. Žh., 4, 232-235 (1963) · Zbl 0144.26002
[25] Hall, P.; Kulatilaka, C. R., A property of locally finite groups, J. Lond. Math. Soc., 39, 235-239 (1964) · Zbl 0136.27903
[26] (Hart, K. P.; Nagata, J.; Vaughan, J. E., Encyclopedia of General Topology (2004), Elsevier) · Zbl 1059.54001
[27] Schaefer, H. H.; Wolff, M. P., (Topological Vector Spaces. Topological Vector Spaces, GTM, vol. 3 (1999), Springer) · Zbl 0983.46002
[28] Nuida, K.; Kimura, G.; Miyadera, T., Optimal observables for minimum-error state discrimination in general probabilistic theories, J. Math. Phys., 51, 093505 (2010) · Zbl 1309.81024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.