Wigner's Theorem and geometry of extreme positive maps

TL;DR

This paper explores the structure of symmetry transformations on quantum states, providing a convex geometric perspective and a variant of Wigner's theorem to classify extreme positive maps.

Contribution

It offers a new convex geometric approach to characterize extreme positive maps, extending Wigner's theorem to a broader context.

Findings

Characterization of extreme points in the convex set of symmetry maps

A variant of Wigner's theorem for convex sets of transformations

Insights into the classification of positive maps

Abstract

We consider transformation maps on the space of states which are symmetries in the sense of Wigner. Due to the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body, to be able to contribute to the classification of positive maps. Our study provides a variant of Wigner's theorem originally proved for ray transformations in Hilbert spaces.