Entanglement in continuous-variable systems: recent advances and current perspectives

Abstract

We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.