ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 1
Characterizing entanglement
According to Erwin Schrödinger, quantum entanglement is not “one but rather the
characteristic trait of quantum mechanics, the one that enforces its entire departure
from classical lines of thought” [201]. Entanglement has been widely recognized as
a fundamental aspect of quantum theory, stemming directly from the superposition
principle and quantum non-factorizability. Remarkably, it is now also acknowledged
as a fundamental physical resource, much on the same status as energy and
entropy, and as a key factor in the realization of information processes otherwise
impossible to implement on classical systems. Thus the degree of entanglement and
information are the crucial features of a quantum state from the point of view of
Quantum Information Theory [163, 111]. Indeed, the search for proper mathematical
frameworks to quantify such features in general (mixed) quantum states cannot
be yet considered accomplished. In view of such considerations, it is clear that the
full understanding of the relationships between the quantum correlations contained
in a multipartite state and the global and local (i.e. referring to the reduced states
of the subsystems) degrees of information of the state, is of critical importance.
In particular, it would represent a relevant step towards the clarification of the
nature of quantum correlations and, possibly, of the distinction between quantum
and classical correlations in mixed quantum states [112, 13, 101].
We open this Chapter with a discussion about the interpretation and measures
of information in quantum systems. We then move to a detailed discussion about
quantum entanglement, its definition, qualification and quantification in the bipartite
and, to some extent, in the multipartite setting. Special attention will be
devoted to the fundamental property of entanglement to be distributed in a so-called
“monogamous” way, and the implications of such feature for the characterization of
many-body entanglement sharing.
1.1. Information contained in a quantum state
Both discrete-variable and continuous-variable systems, endowed respectively with
a finite-dimensional and a infinite-dimensional Hilbert space, can be efficiently employed
in quantum information theory for the encoding, manipulation and transmission
of information. In this context, it is natural to question how much information
a quantum state contains.
Suppose we have prepared a physical system in a certain state and we would
like to test the system somehow, for instance with a measurement. Before performing
the experimental verification, we can only predict the probabilities p1, . . . , pN
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