Many-body entanglement in gapped quantum systems : representation, classification, and application

Abstract

Entanglement is a special form of quantum correlation that exists among quantum particles and it has been realized that surprising things can happen when a large number of particles are entangled together. For example, topological orders emerge in condensed matter systems where the constituent 1023 particles are entangled in a nontrivial way; moreover, quantum computers, which can perform certain tasks significantly faster than classical computers, are made possible by the existence of entanglement among a large number of particles. However, a systematic understanding of entanglement in many-body systems is missing, leaving open the questions of what kinds of many-body entanglement exist, where to find them and what they can be used for. In this thesis, I present my work towards a more systematic understanding of many-body entanglement in systems where the particles interact with each other locally and the ground state of the system is separated from the excited states by a finite energy gap. Under such physically realistic locality and gap constraints, I am able to obtain more understanding concerning the efficient representation of many-body entangled states, the classification of such states according to their universal properties and the application of such states in quantum computation. More specifically, this thesis is focused on the tensor network representation of many-body entangled states and studies how the tensors in the representation reflect the universal properties of the states. An algorithm is presented to extract the universal properties from the tensors and certain symmetry constraints are found necessary for the tensors to represent states with nontrivial topological order. Classification of gapped quantum states is then carried out based on this representation. An operational procedure relating states with the same universal properties is established which is then applied to systems in one and higher dimensions. This leads not only to the discovery of new quantum phases but also to a more systematic understanding of them. A more complete understanding of possible many-body entanglement structures enables us to design an experimentally more feasible many-body entangled state for application in measurement-based quantum computation. Moreover, the framework of measurement-based quantum computation is generalized from spin to fermion systems leading to new possibilities for experimental realization.