ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

4 Introduction
states and in the presence of mixedness. While important insights have been gained
on these issues in the context of qubit systems (two-level quantum systems traditionally
employed as the main logical units for quantum computing and quantum
information in general), a less satisfactory understanding has been achieved until
recent times on higher-dimensional systems, as the structure of entangled states
in Hilbert spaces of high dimensionality scales exhibiting a formidable degree of
complexity.
However, and quite remarkably, in systems endowed with infinite-dimensional
Hilbert spaces (where entanglement can arise between degrees of freedom with continuous
spectra), recent advances have been recorded for what concerns the understanding
and the quantification of the entanglement properties of a restricted class
of states, the so-called Gaussian states. Gaussian states distinctively stand out of
the infinite variety of continuous variable systems, because on one hand they allow
a clean framework for the analytical study of the structure of non-local correlations,
and on the other hand they are of great practical relevance in applications to
quantum optics and quantum information. Two-mode and multimode coherent and
squeezed Gaussian states are indeed key resources, producible and manipulatable
in the lab with a high degree of control, for a plethora of two-party and multi-party
quantum communication protocols, ranging from deterministic teleportation and
secure key distribution, to quantum data storage and cluster computation.
This PhD Dissertation collects my personal contributions to the understanding,
qualification, quantification, structure, production, operational interpretation, and
applications of entanglement in Gaussian states of continuous variable systems. Let
us briefly mention some of the most important results, the majority of which have
appeared in Refs. [GA2—GA20].
In the first place we enriched the well-established theory of bipartite entanglement
in two-mode Gaussian states, providing new physically insightful connections
between the entanglement and the degrees of information associated with
the global system and its subsystems. We thus showed that the negativity (an
entanglement monotone) can be accurately qualified and quantitatively estimated
in those states by direct purity measurements. We also proved that different entanglement
quantifiers (negativities and Gaussian entanglement measures) induce
inequivalent orderings on the set of entangled, nonsymmetric two-mode Gaussian
states. We then extended our scope to investigate multimode, bipartite entanglement
in N-mode Gaussian states endowed with some symmetry constraints, and
its scaling with the number of the modes; this study was enabled by our central
observation that entanglement in such states is unitarily localizable to an effective
two-mode entanglement. We could thus extend the validity of the necessary
and sufficient positive-partial-transposition condition for separability to bisymmetric
Gaussian states of an arbitrary number of modes, and exactly quantify the block
entanglement between different mode partitions, revealing signatures of a genuine
multipartite entanglement arising among all modes. Under these premises, we developed
ex novo a theory of multipartite entanglement for Gaussian states, based
on the crucial fact that entanglement cannot be freely shared and its distribution
is constrained to a monogamy inequality, which we proved to hold for all (pure
and mixed) N-mode Gaussian states distributed among N parties. To this aim, we
introduced new continuous variable entanglement monotones, namely (Gaussian)
‘contangle’ and Gaussian tangle, for the quantification of entanglement sharing in