ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Introduction
About eighty years after their inception, quantum mechanics and quantum theory
are still an endless source of new and precious knowledge on the physical world
and at the same time keep evolving in their mathematical structures, conceptual
foundations, and intellectual and cultural implications. This is one of the reasons
why quantum physics is still so specially fascinating to all those that approach it
for the first time and never ceases to be so for those that are professionally involved
with it. In particular, since the early nineties of the last century and in the last
ten-fifteen years, a quiet revolution has taken place in the quantum arena. This
revolution has progressively indicated and clarified that aspects once thought to be
problematic, such as quantum non-separability and “spooky” actions at a distance,
are actually not only the origin of paradoxes but rather some of the key ingredients
that are allowing a deeper understanding of quantum mechanics, its applications
to new and exciting fields of research (such as quantum information and quantum
computation), and tremendous progress in the development of its mathematical and
conceptual foundations. Among the key elements of the current re-foundation of
quantum theory, entanglement certainly plays a very important role, also because it
is a concept that can be mathematically qualified and quantified in a way that allows
it to provide new and general characterizations of quantum properties, operations,
and states.
The existence of entangled states, stemming directly from the superposition
principle, can be regarded as a founding feature, or better “the characteristic trait”
(according to Schrödinger) of quantum mechanics itself. Entanglement arises when
the state of two or more subsystems of a compound quantum system cannot be
factorized into pure local states of the subsystems. The subsystems thus share
quantum correlations which can be stronger than any classical correlation. Quantum
information science was born upon the key observation that the exploitation
of such nonclassical correlations enables encoding, processing and distribution of
information in ways impossible, or very inefficient, with classical means. Hence
the possibility of implementing entangled resources resulted in futuristic proposals
(quantum teleportation, quantum cryptography, quantum computation, ...) which
are now made, to a certain extent, into reality. On a broader perspective, it is now
recognized that entanglement plays a fundamental role in the physics of many-body
systems, in particular in critical phenomena like quantum phase transitions, and in
the description of the interactions between complex systems at the quantum scale.
Despite its prominent role in the physics of microscopic but also macroscopic
systems, it still stands as an open issue to achieve a conclusive characterization
and quantification of bipartite entanglement for mixed states, and especially to
provide a definition and interpretation of multipartite entanglement both for pure
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