TY - JOUR

T1 - On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids

AU - Kimura, G.

AU - Nuida, K.

N1 - Funding Information:
A part of this paper was presented at 14th Workshop on Quantum Information Processing (QIP 2011), Singapore, January 10–14, 2011. This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 20700017 and No. 22740079 ), The Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan . Moreover, the authors thank the anonymous referees in the current and the previous submissions of the paper for their precious comments.

PY - 2014/12

Y1 - 2014/12

N2 - 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.

AB - 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.

KW - 2-level quantum system

KW - Convex set

KW - Ellipsoid

KW - Vertex-transitivity

UR - http://www.scopus.com/inward/record.url?scp=84904333408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904333408&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2014.06.004

DO - 10.1016/j.geomphys.2014.06.004

M3 - Article

AN - SCOPUS:84904333408

SN - 0393-0440

VL - 86

SP - 1

EP - 18

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

ER -