A few algebraic problems in the theory of quantum entanglement

TL;DR

This paper uses algebraic methods to solve fundamental problems in quantum entanglement theory, providing new characterizations of entangled states and mapping cones with both theoretical and computational approaches.

Contribution

It introduces two general theorems on convex sets of quantum maps and characterizes positive-partial-transpose entangled states of minimum rank, advancing algebraic approaches in quantum entanglement.

Findings

Characterization of mapping cones as an analogue of the positive maps criterion

Full characterization of positive-partial-transpose entangled states of minimum rank

Solutions to problems involving higher rank numerical ranges and separable states

Abstract

We present solutions to a set of problems that arise in quantum entanglement theory, whose common trait is the use of algebraic methods. The backbone of the thesis consists of two general theorems, pertaining to specific convex sets of quantum maps. They are complemented by solutions of many more specific problems of rather technical character. These problems concerned questions such as the higher rank numerical ranges, the lengths of separable states, the solutions of compression equations, the completely entangled subspaces, and maximally entangled states. The solutions were obtained partly by hand, and partly by using the Groebner basis toolset available in computer algebra systems. The main results, which are those of general nature, consist of a characterization of mapping cones, which is a full analogue of the positive maps criterion for separability, as well as of a full characterization of positive-partial-transpose entangled states of minimum rank. The thesis also contains an introduction to the theory of quantum entanglement and to basic algebraic geometry, whose methods have been used