Uncertainty relations and joint numerical ranges

TL;DR

This paper explores the geometry of joint numerical ranges in quantum mechanics, linking noncommutativity with uncertainty relations and phase transitions, and discusses existing results, generalizations, and applications.

Contribution

It provides a comprehensive overview of joint numerical range theory, including generalizations and applications in quantum physics, especially in uncertainty relations and phase transitions.

Findings

JNR geometry becomes complex with higher dimensions

Connections established between JNR and uncertainty principles

Applications in understanding quantum phase transitions

Abstract

Noncommutativity lies in the heart of quantum theory and provides rich set of interesting questions in physics and mathematics. In this work I present some of them through the concept of Joint Numerical Range (JNR) - the set of simultaneously attainable averages of quantum observables, which in general need not commute. Full description of JNR geometry quickly becomes complex as the dimensionality of quantum system grows. In this thesis I present already existing results in the theory of JNR, generalizations of the formalism and applications of JNR in the theory of phase transitions and uncertainty relations.