ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

192 11. Efficient production of pure N-mode Gaussian states
the typical entanglement of pure Gaussian states under “thermodynamical” statespace
measures as computable along the lines of Ref. [209]. This would prove the
optimality and generality of our scheme in an operational way, which is indeed more
useful for practical applications.
11.4. Generic versus typical entanglement: are off-block-diagonal correlations
relevant?
The structural properties of pure N-mode Gaussian states under local operations
have been addressed in Sec. 2.4.2 and completely characterized in Appendix A
[GA18]. Here [GA14], block-diagonal states (i.e. with no correlations between position
and momentum operators) have been in particular proven to possess generic
entanglement in the sense of Sec. 11.2.1, and their standard form covariances (determining
any form of entanglement in such states) have been physically understood
in terms of two-body correlations. It is thus quite natural to question if
the N(N − 3)/2 additional parameters encoded in ˆq-ˆp correlations for non-blockdiagonal
pure states, have a definite impact or not on the bipartite and multipartite
entanglement.
At present, usual CV protocols are devised, even in multimode settings (see
Chapter 12), to make use of states without any ˆq-ˆp correlations. In such cases, the
economical (relying on N(N − 1)/2 parameters) “block-diagonal state engineering”
scheme detailed in Fig. 11.1 is clearly the optimal general strategy for the production
of entangled resources. However, theoretical considerations strongly suggest that
states with σqp = 0 [adopting the notation of Eq. (A.4)] might have remarkable
potential for improved quantum-informational applications. In fact, considering as
just mentioned the thermodynamical entanglement framework of Gaussian states
[209], one can define natural averages either on the whole set of pure Gaussian
states, or restricting to states with σqp = 0. Well, numerics unambiguously show
[GA18] that the (thermodynamically-averaged) “generic” entanglement (under any
bipartition) of Gaussian states without ˆq-ˆp correlations (like the ones considered in
Sec. 11.2.1) is systematically lower than the “typical” entanglement of completely
general pure Gaussian states, with this behavior getting more and more manifest
as the total number of modes N increases (clearly, according to Sec. 2.4.2, this
discrepancy only arises for N > 3). In a way, the full entanglement potential of
Gaussian states is diminished by the restriction to block-diagonal states.
On the other hand, the comparison between the average entanglement generated
in randomizing processes based on the engineering scheme of Fig. 11.2, and
the block-diagonal one of Fig. 11.1, is under current investigation as well. If the
general scheme of Fig. 11.2, based on beam-splitters and seraphiques, turned out
to be out-performing the simpler ones (like the one of Fig. 11.1, based on beamsplitters
only) in terms of entanglement generation — as expected in view of the
argument above — this would provide us with a formidable motivation to explore
novel CV protocols capable of adequately exploiting ˆq-ˆp correlated resources.