ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

Conclusion and Outlook 263
increased (distilled) by resorting to Gaussian operations only [78, 205, 90]. Similarly,
for universal one-way quantum computation using Gaussian cluster states, a
single-mode non-Gaussian measurement is required [155].
There is indeed a fundamental motivation for investigating entanglement in
non-Gaussian states, as the extremality of Gaussian states imposes that they are
the minimally entangled states among all states of CV systems with given second
moments [269]. Experimentally, it has been recently demonstrated [172] that a twomode
squeezed Gaussian state can be “de-Gaussified” by coherent subtraction of a
single photon, resulting in a mixed non-Gaussian state whose non-local properties
and entanglement degree are enhanced (enabling a better efficiency for teleporting
coherent states [132]). Theoretically, the characterization of even bipartite entanglement
(let alone multipartite) in non-Gaussian states stands as a formidable task.
One immediate observation is that any (non-Gaussian) multimode state with a
CM corresponding to an entangled Gaussian state is itself entangled too [235, 269].
Therefore, most of the results presented in this Dissertation may serve to detect entanglement
in a broader class of states of infinite-dimensional Hilbert spaces. They
are, however, all sufficient conditions on entanglement based on the second moments
only of the canonical operators. As such, for arbitrary non-Gaussian states,
they are in general very inefficient — meaning that most entangled non-Gaussian
states fail to be detected by these criteria. The description of non-Gaussian states
requires indeed (an infinite set of) high order statistical moments: as an obvious
consequence, also an inseparability criterion for these states should involve
high order correlations. Recently, some separability criteria based on hierarchies
of conditions involving higher moments of the canonical operators have been introduced
to provide a sharper detection of inseparability in generic non-Gaussian
states [3, 162, 114, 157].
In particular, Shchukin and Vogel [214] introduced an elegant and unifying
approach to separability based on the PPT requirement, that is constructed in
the form of an infinite series of inequalities, and includes as special cases all the
above cited results (including the conditions on second moments [70, 218] qualifying
separability in Gaussian states, see Sec. 3.1.1), thus demonstrating the important
role of PPT in building a strong criterion for the detection of entanglement. The
conditions by Shchukin and Vogel can be applied to distinguish between the several
separability classes in a multipartite CV system [215]. To this aim, entanglement
witnesses are useful as well [125].
The efficiency of some of the above-mentioned inseparability criteria based on
higher order moments, for detecting bipartite entanglement in the non-Gaussian
family of squeezed number states of two-mode radiation fields, has been recently
evaluated [64]. We mention a further interesting approach to non-Gaussian entanglement
reported by McHugh et al. [154], who showed that entanglement of
multiphoton squeezed states is completely characterized by observing that with respect
to a new set of modes, those non-Gaussian states actually assume Gaussian
character.
Future perspectives
Many open issues and unanswered intriguing questions naturally arise when peeping
out of the parental house of Gaussian states. There is always the risk of being
trapped in the infinite mathematical complexity of the CV Hilbert space losing the