ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 6
Gaussian entanglement sharing
One of the main challenges in fundamental quantum theory, as well as in quantum
information and computation sciences, lies in the characterization and quantification
of bipartite entanglement for mixed states, and in the definition and interpretation
of multipartite entanglement both for pure states and in the presence
of mixedness [163, 111]. More intriguingly, a quantitative, physically significant,
characterization of the entanglement of states shared by many parties can be attempted:
this approach, introduced in a seminal paper by Coffman, Kundu and
Wootters (CKW) [59], has lead to the discovery of so-called “monogamy inequalities”
[see Eq. (1.45)], constraining the maximal entanglement distributed among
different internal partitions of a multiparty system. Such inequalities are uprising
as one of the fundamental guidelines on which proper multipartite entanglement
measures have to be built [GA12].
While important progresses have been gained on these issues in the context of
qubit systems (as reviewed in Sec. 1.4), a less satisfactory understanding had been
achieved until recent times on higher-dimensional systems, associated to Hilbert
spaces with an increasingly complex structure. However, and quite remarkably,
in infinite-dimensional Hilbert spaces of CV systems, important progresses have
been obtained in the understanding of the (bipartite) entanglement properties of
the fundamental class of Gaussian states, as it clearly emerges, we hope, from the
previous Parts of this Dissertation.
Building on these insights, we have performed the first analysis of multipartite
entanglement sharing in a CV scenario. This has resulted, in particular, in the first
(and unique to date) mathematically and physically bona fide measure of genuine
tripartite entanglement for arbitrary three-mode Gaussian states [GA10, GA11], in
a proof of the monogamy inequality on distributed entanglement for all Gaussian
states [GA15], and in the demonstration of the promiscuous sharing structure of
multipartite entanglement in Gaussian states [GA10], which arises in three-mode
symmetric states [GA11, GA16] and can be unlimited in states of more than three
modes [GA19].
These and related results are the subject of the present Part of this Dissertation.
We begin in this Chapter by introducing our novel entanglement monotones
(contangle, Gaussian contangle and Gaussian tangle) apt to quantify distributed
Gaussian entanglement, thus generalizing to the CV setting the tangle [59] defined
for systems of two qubits by Eq. (1.48).
Motivated by the analysis of the block entanglement hierarchy and its scaling
structure in fully symmetric Gaussian states (see Sec. 5.2) we will proceed by establishing
a monogamy constraint on the entanglement distribution in such states.
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