Positive maps, states, entanglement and all that; some old and new problems

TL;DR

This paper introduces a new approach to characterizing positive maps in quantum theory, focusing on facial structures and their role in understanding states and entanglement, with detailed analysis for low-dimensional systems.

Contribution

It presents a novel facial-structure-based framework for classifying positive maps and clarifies differences between 2D and 3D cases, including explicit decompositions.

Findings

Explicit form of positive map decomposition for 2D systems

Uniqueness of extremal positive map decomposition in 2D

Differences in positive map structures between 2D and 3D systems

Abstract

We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence between Schroedinger's and Heisenberg's pictures is reviewed in this more general setting. Furthermore, we discuss in detail the structure of positive maps for two and three dimensional systems. In particular, the explicit form of decomposition of a positive map and the uniqueness of this decomposition for extremal positive maps for 2 dimensional case are described. The difference of the structure of positive maps between 2 dimensional and 3 dimensional cases is clarified. The resulting characterization of positive maps is applied to the study of quantum correlations and entanglement