ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Introduction 5
Gaussian states. Implications of this analysis include, in particular, the definition
of the ‘residual contangle’ as the first bona fide measure of genuine multipartite
(specifically, tripartite) entanglement in a continuous variable setting, a complete
quantitative analysis of multipartite entanglement in the paradigmatic instance of
three-mode Gaussian states, the discovery of the promiscuous nature of entanglement
sharing in symmetric ‘GHZ/W ’ Gaussian states, and the demonstration of
the possible coexistence of unlimited bipartite and multipartite entanglement in
states of at least four modes.
Our investigation was not confined to theoretical and structural aspects of entanglement
only. Along parallel lines, we got interested on one hand in how to
produce bipartite and/or multipartite entanglement in the lab with efficient means,
and on the other hand in how to optimally employ such entanglement for practical
applications, endowing the entanglement itself with an operational interpretation.
In the case of two-mode Gaussian states, we joined an experiment concerning production,
characterization and manipulation of entanglement in the context of quantum
optics. In the three- and four-mode instances, we proposed several schemes to
efficiently engineer family of Gaussian states with relevant entanglement properties.
In general, we devised an optimal scheme to produce generic pure N-mode Gaussian
states in a standard form not encoding direct correlations between position and momentum
operators (and so encompassing all the instances of multimode Gaussian
states commonly employed in practical implementations); such an analysis allows
to interpret entanglement in this subclass of Gaussian states entirely in terms of the
two-point correlations between any pair of modes. In this respect, one theoretical
result of direct interest for the characterization of entanglement in Gaussian states,
is the qualitative and quantitative equivalence we established between the presence
of bipartite (multipartite) entanglement in two-mode (N-mode) fully symmetric
Gaussian states shared as resources for a two-party teleportation experiment (Nparty
teleportation network), and the maximal fidelity of the protocol, optimized
over local single-mode unitary operations performed on the shared resource. In the
special case of three-mode, pure GHZ/W states, this optimal fidelity is a monotonically
increasing function of the residual contangle (which quantifies genuine tripartite
entanglement), providing the latter with a strong operational significance.
Based on this equivalence, we presented a proposal to experimentally verify the
promiscuous sharing structure of tripartite Gaussian entanglement in such states
in terms of the success of two-party and three-party teleportation experiments.
Telecloning with three-mode Gaussian resources was also thoroughly investigated.
We finally considered two applications of the Gaussian machinery to the companion
areas of condensed matter/statistical mechanics, and relativity theory. Concerning
the former, we studied entanglement distribution in ground states of translationally
invariant many-body harmonic lattice systems endowed with a Gaussian
‘valence bond’ structure. We characterized the range of correlations in such harmonic
models, connecting it to the degree of entanglement in a smaller Gaussian
structure, named ‘building block’, which enters in the valence bond construction.
We also discussed the experimental production of Gaussian valence bond states of
an arbitrary number of modes, and their usefulness for multiparty telecloning of
coherent states. On the other hand, in a relativistic setting we studied the distribution
of entanglement between modes of a free scalar field from the perspective
of observers in relative acceleration. The degradation of entanglement due to the