ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

6 Introduction
Unruh effect was analytically characterized for two parties sharing a two-mode
Gaussian state in an inertial frame, in the cases of either one or both observers
undergoing uniform acceleration. Within the monogamy framework, we precisely
explained the loss of entanglement as a redistribution of the inertial entanglement
into multipartite quantum correlations among accessible and unaccessible modes
from a non-inertial perspective.
This Dissertation is organized as follows.
Part I is devoted to introducing the main ingredients of our analysis, entanglement
on one side, and Gaussian states on the other. In particular, Chapter
1 contains the basics of entanglement theory: how to quantify quantum information,
the separability problem, different entanglement measures, and a discussion
on entanglement sharing. In Chapter 2 we give a self-contained introduction to
phase-space and symplectic methods in the study of Gaussian states of infinitedimensional
bosonic systems, we discuss the covariance matrix formalism, and we
provide a classification of pure and mixed Gaussian states according to the various
standard forms that the associated covariance matrices can take.
We collect in Part II all results concerning bipartite entanglement of Gaussian
states with two or more modes. In Chapter 3 we illustrate the machinery of
bipartite entanglement qualification and quantification in Gaussian states. The
massive Chapter 4 contains our specific results on two-mode Gaussian states, including
the existence of extremally (minimally and maximally) entangled states at
given degrees of mixedness, and the different orderings induced on entangled states
by different measures of entanglement. In Chapter 5 we describe the unitary (and
therefore reversible) localization of bipartite multimode entanglement to a bipartite
two-mode entanglement in fully symmetric and bisymmetric multimode Gaussian
states, and its scaling with the number of modes.
Multipartite entanglement of Gaussian states is the topic of Part III. In Chapter
6 we present our crucial advances in the understanding of entanglement sharing
in multimode Gaussian states, including the proof of the monogamy inequality
on distributed entanglement for all Gaussian states. Multipartite entanglement of
three-mode Gaussian states is analyzed in Chapter 7 by discussing the structural
properties of such states, and the main consequences of the monogamy inequality,
such as the quantification of genuine tripartite entanglement, and the promiscuous
nature of entanglement sharing in Gaussian states with symmetry properties. Chapter
8 deals with the remarkable property of multipartite entanglement in Gaussian
states (as opposed to low-dimensional systems), to coexist to an arbitrary extent
with bipartite entanglement, in simple families of states of at least four modes,
within the holding of the monogamy inequality.
In Part IV we show how to engineer multimode Gaussian resources with optical
means. Chapter 9 contains schemes for the production of extremally entangled twomode
states, as well as experimental results on the production, characterization and
manipulation of two-mode entanglement with a novel optical setup. In Chapter 10
we provide a systematic investigation on the preparation of several families of threeand
four-mode Gaussian states with peculiar entanglement properties, providing
efficient schemes. Chapter 11 deals instead with the general instance of pure Nmode
Gaussian states, in which case for the relevant family of ‘block-diagonal’
states an optimal state engineering recipe is proposed, which enables to connect
generic entanglement to operationally meaningful resources.