This book is a collection of lecture notes contributions from a summer school on decoherence entanglement entropy and a workshop on MPS matrix product states DMRG density matrix renormalization group Subjects of the summer school include introduction to MPS black holes qubits and octonions weak measurement entanglement measures and separability generalized Bell inequalities among others Subjects of the workshop are dedicated to MPS and DMRG Applications to strongly correlated systems and integrable systems are also mentioned Contributions to this book are prepared in a self contained manner so that a reader with a modest background in quantum information and quantum computing may understand the subjects