ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

5.2. Quantification and scaling of entanglement in fully symmetric states 105
grows and, as a consequence, the OLE increases with N. In the limit N → ∞, the
N × N entanglement diverges while the 1 × 1 one vanishes. This exactly holds both
for pure and mixed states, although the global degree of mixedness produces the
typical behavior that tends to reduce the total entanglement of the state.
5.2.3. Discussion
We have shown that bisymmetric (pure or mixed) multimode Gaussian states,
whose structural properties are introduced in Sec. 2.4.3, can be reduced by local
symplectic operations to the tensor product of a correlated two-mode Gaussian
state and of uncorrelated thermal states (the latter being obviously irrelevant as
far as the correlation properties of the multimode Gaussian state are concerned).
As a consequence, all the entanglement of bisymmetric multimode Gaussian states
of arbitrary M × N bipartitions is unitarily localizable in a single (arbitrary) pair
of modes shared by the two parties. Such a useful reduction to two-mode Gaussian
states is somehow similar to the one holding for states with fully degenerate
symplectic spectra [29, 92], encompassing the relevant instance of pure states, for
which all the symplectic eigenvalues are equal to 1 (see Sec. 2.4.2.1). The present
result allows to extend the PPT criterion as a necessary and sufficient condition
for separability to all bisymmetric multimode Gaussian states of arbitrary M × N
bipartitions (as shown in Sec. 3.1.1), and to quantify their entanglement [GA4, GA5].
Notice that, in the general bisymmetric instance addressed in this Chapter, the
possibility of performing a two-mode reduction is crucially partition-dependent.
However, as we have explicitly shown, in the case of fully symmetric states all the
possible bipartitions can be analyzed and compared, yielding remarkable insight
into the structure of the multimode block entanglement of Gaussian states. This
leads finally to the determination of the maximum, or optimal localizable entanglement
that can be unitarily concentrated on a single pair of modes.
It is important to notice that the multipartite entanglement in the considered
class of multimode Gaussian states can be produced and detected [236, 240], and
also, by virtue of the present analysis, reversibly localized by all-optical means.
Moreover, the multipartite entanglement allows for a reliable (i.e. with fidelity
F > Fcl, where Fcl = 1/2 is the classical threshold, see Chapter 12) quantum
teleportation between any two parties with the assistance of the remaining others
[236]. The connection between entanglement in the symmetric Gaussian resource
states and optimal teleportation-network fidelity has been clarified in [GA9], and
will be discussed in Sec. 12.2.
More generally, the present Chapter has the important role of bridging between
the two central parts of this Dissertation, the one dealing with bipartite entanglement
on one hand, and the one dealing with multipartite entanglement on the
other hand. We have characterized entanglement in multimode Gaussian states by
reducing it to a two-mode problem. By comparing the equivalent two-mode entanglements
in the different bipartitions we have unambiguously shown that genuine
multipartite entanglement is present in the studied Gaussian states. It is now time
to analyze in more detail the sharing phenomenon responsible for the distribution
of entanglement from a bipartite, two-mode form, to a genuine multipartite manifestation
in N-mode Gaussian states, under and beyond symmetry constraints.