Entanglement Between Noncomplementary Parts of Many-Body Systems

This thesis investigates the structure and behaviour of entanglement, the purely quantum mechanical part of correlations, in many-body systems, employing both numerical and analytical techniques at the interface of condensed matter theory and quantum information theory. Entanglement can be seen as a precious resource which, for example, enables the noiseless and instant transmission of quantum information, provided the communicating parties share a sufficient "amount" of it. Furthermore, measures of entanglement of a quantum mechanical state are perceived as useful probes of collective properties of many-body systems. For instance, certain measures are capable of detecting and classifying ground-state phases and, particularly, transition (or critical) points separating such phases. Chapters 2 and 3 focus on entanglement in many-body systems and its use as a potential resource for communication protocols. They address the questions of how a substantial amount of entanglement can be established between distant subsystems, and how efficiently this entanglement could be "harvested" by way of measurements. The subsequent chapters 4 and 5 are devoted to universality of entanglement between large collections of particles undergoing a quantum phase transition, where, despite the enormous complexity of these systems, collective properties including entanglement no longer depend crucially on the microscopic details.

Introduction.- Exploiting Quench Dynamics in Spin Chains for Distant Entanglement and Quantum Communication.- Extraction of Pure Entangled States from Many-Body Systems by Distant Local Projections.- Scaling of Negativity of Separating Blocks in Spin Chains and Critically.-Universality of the Negativity in the Lipkin-Mechkov-Glick Model.- Conclusions and Outlook.- A. Diagonalisation of the XX Model.- B. Factorisation of the Fermionic Correlation Functions.- C. Time Dependence of the Reduced Density Operator Following Quench.- D. Density Matrix Renormalisation Group Algorithm.- E. Proof of Williamson’s Theorem.- F. Partial Transposition in Continuous Variable Systems.- G. Gaussian Wigner Representation of Bosonic Vacuum.- H. Ground State Covariance Matrix of a Quadtratic Hamiltonean.- I. Bipartitie Entanglement of Gaussian States.- J. Density Matrix Spectra of Bosonic Gaussian States.- K. Bosonisation of the LMG Hamiltonian.- Bibliography