Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

TL;DR

This paper surveys the geometric formalism of quantum mechanics, analyzing its applications to finite-dimensional systems and exploring benefits in understanding entanglement witnesses.

Contribution

It provides a comprehensive review of differential geometric approaches to quantum mechanics and demonstrates their practical advantages in analyzing quantum systems.

Findings

Geometric formulation offers new insights into quantum state analysis.

Application to qubits, qutrits, and two-qubit systems illustrates practical benefits.

Enhanced understanding of entanglement witnesses through geometric methods.

Abstract

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.