This book is the result of research by Cláudio Henrique Von Ehnert, Samuel Soares de Souza Santos and Eduardo Corrêa dos Santos, exploring advanced themes in quantum theory. Divided into four chapters, it addresses topological invariants in non-Hermitian systems, such as the Chern number, the degree of winding and the Zak number, as well as models such as Kitaev and SSH. It also discusses experimental connections, topological phases and dissipative properties. Finally, it explores entanglement in many-particle systems, including measurements such as von Neumann entropy and negativity, integrating theory, simulation and practice.