ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

22 1. Characterizing entanglement
extensions. This is currently under our investigation, but no significant
progress has been achieved yet.
1.3.3.3. Entanglement-induced ordering of states. Let us remark that having so many
entanglement measures (of which only a small portion has been recalled here) means
in particular that different orderings are induced on the set of entangled states. One
can show that any two LOCC-monotone entanglement measures can only impose
the same ordering on the set of entangled states, if they are actually exactly the
same measure [255]. Therefore there exist in general pairs of states ϱA and ϱB
such that E ′ (ϱA) > E ′ (ϱB) and E ′′ (ϱA) < E ′′ (ϱB), according to two different
entanglement monotones E ′ (ϱ) and E ′′ (ϱ) (see Sec. 4.5 for an explicit analysis in
the case of two-mode Gaussian states [GA7]). Given the wide range of tasks that
exploit entanglement [163], one might understand that the motivations behind the
definitions of entanglement as ‘that property which is exploited in such protocols’
are manifold. This means that situations will almost certainly arise where a state ϱA
is better than another state ϱB for achieving one task, but for achieving a different
task ϱB is better than ϱA. Consequently, the fact that using a task-based approach
to quantifying entanglement will certainly not lead to a single unified perspective,
is somehow expected.
In this respect, it is important to know that (in finite-dimensional Hilbert
spaces) all bipartite entangled states are useful for quantum information processing
[151]. For a long time the quantum information community has used a ‘negative’
characterization of the term entanglement, essentially defining entangled states as
those that cannot be created by LOCC alone [188]. However, remarkably, it has
been recently shown that for any non-separable state ϱ according to Def. 2, one
can find another state σ whose teleportation fidelity may be enhanced if ϱ is also
present 4 [151, 150, 35]. This is interesting as it allows us to positively characterize
non-separable states as those possessing a useful resource that is not present in
separable states. The synonymous use of the terms non-separable and entangled is
hence justified.
1.4. Multipartite entanglement sharing and monogamy constraints
It is a central trait of quantum information theory that there exist limitations to
the free sharing of quantum correlations among multiple parties. Such monogamy
constraints have been introduced in a landmark paper by Coffman, Kundu and
Wootters, who derived a quantitative inequality expressing a trade-off between the
couplewise and the genuine tripartite entanglement for states of three qubits [59].
Since then, a lot of efforts have been devoted to the investigation of distributed entanglement
in multipartite quantum systems. In this Section, based on Ref. [GA12],
we report in a unifying framework a bird’s eye view of the most relevant results
that have been established so far on entanglement sharing. We will take off from
the domain of N qubits, graze qudits (i.e. d-dimensional quantum systems), and
drop the premises for the fully continuous-variable analysis of entanglement sharing
in Gaussian states which will presented in Part III.
4 We have independently achieved a somehow similar operational interpretation for (generally
multipartite) continuous-variable entanglement of symmetric Gaussian states in terms of optimal
teleportation fidelity [GA9], as will be discussed in Sec. 12.2.