ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Entanglement of Gaussian states: what next?
Conclusion and Outlook
The centrality of Gaussian states in CV quantum information is motivated not only
by their peculiar structural properties which make their description amenable of an
analytical analysis, but also by the ability to produce, manipulate and detect such
states with remarkable accuracy in realistic, experimental settings.
The scope of this Dissertation has been almost entirely theoretical. We provided
important advances for what concerns the structural and informational characterization
of bipartite entanglement, and the definition and quantification of multipartite
entanglement in Gaussian states. State engineering prescriptions and several
applications to diverse fields (quantum communication, quantum optics, manybody
physics, relativity) were discussed as well. We are not going here to list again
the individual and numerous results obtained in all those contexts — retrievable
in Refs. [GA2—GA20] and in the previous Parts of this Dissertation — to avoid
unnecessary repetitions with the front matter. We will try instead to frame our
results into a broader perspective, with the aim of providing an as self-contained as
possible outlook of the current directions of the CV quantum information research,
with and beyond Gaussian states.
For reasons of space and time, we cannot discuss in sufficient detail all the
additional proposals and experimental demonstrations concerning on one hand the
state engineering of two-, three- and in general N-mode Gaussian states, and on
the other hand the use of such states as resources for the realization of quantum
information protocols, which were not covered by the present Dissertation. Excellent
review papers are already available for what concerns both the optical state
engineering of multiphoton quantum states of discrete and CV systems [65], and
the implementations of quantum information and communication with continuous
variables [235, 40].
Let us just mention that, from a practical point of view, Gaussian resources
have been widely used to implement paradigmatic protocols of CV quantum information,
such as two-party and multiparty teleportation [39, 89, 236, 277, 182]
(see Chapter 12), and quantum key distribution [102, 160, 103]; they have been
proposed for achieving one-way quantum computation with CV generalizations of
cluster states [155], and in the multiparty setting they have been proven useful
to solve Byzantine agreement [161]. Gaussian states are currently considered key
resources to realize light-matter interfaced quantum communication networks. It
has been experimentally demonstrated how a coherent state of light can be stored
onto an atomic memory [130], and teleported to a distant atomic ensemble via a
hybrid light-matter two-mode entangled Gaussian resource [216].
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