Multipartite entanglement: properties, measures and application

The fundamental laws of Nature are quantum mechanical. In recent years this fact has been recognized as an opportunity to store and process information in ways that go beyond, e.g., the standard (or, as typically referred to, classical) computers, based on a binary and exclusive logic. Indeed, the extension of the theory of information and computation to the quantum case provides a number of promising applications, ranging from a possible exponential speed-up in solving computational problems (with respect to classical computers) and an efficient simulation of quantum systems, to unconditionally secure cryptographic communication schemes. Quantum entanglement, i.e. a stronger-than-classical correlation between subsystems, appears to be a key element in this sense. Therefore, it is necessary to study its properties and, recognizing its full status as a physical resource, to quantify it. We are interested in multipartite entanglement, i.e. when more than two subsystems are involved. Multipartite entanglement is fundamental in many contexts, from many-body physics, to distributed quantum information tasks, to one-way quantum computing. Multipartite entanglement measures serve to quantify the entanglement content of a quantum state. In the multipartite context, many axiomatically defined measures lack a clear physical/operational meaning. This is especially true for measures which are related to the so-called local invariants, i.e. state dependent quantities which remain constant under local operations. We will identify common and specific properties of such measures and relate them to useful tasks. In particular, the quantification of entanglement for the ubiquitous, yet simply defined, class of graph states, will serve to understand better both the measures and the class of states itself. Finally, one fundamental property of multipartite entanglement is that of monogamy, i.e. a system can not be highly entangled with more than one other system at the same time. Monogamy leads naturally to define multipartite entanglement measures in terms of residual correlations. We will establish strong monogamy constraints and define multipartite measures by means of them. Our goals can thus be summarized as: to study properties of some multipartite entanglement measures known in literature and define new ones, emphasizing the usefulness of entanglement to realize tasks (e.g. teleportation); to quantify multipartite entanglement for a specific yet relevant class of multipartite states: graph states; to investigate the monogamy property of multipartite entanglement, i.e. the bounds on how entanglement can be distributed among multiple parties.