ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

262 Conclusion and Outlook
Gaussian states also play a prominent role in many-body physics, being ground
and thermal states of harmonic lattice Hamiltonians [11]. Entanglement entropy
scaling in these systems has been shown to follow an area law [187, 60]. Thermodynamical
concepts have also been applied to the characterization of Gaussian
entanglement: recently, a “microcanonical” measure over the second moments of
pure Gaussian states under an energy constraint has been introduced [209] (see also
Sec. 11.4), and employed to investigate the statistical properties of the bipartite entanglement
in such states. Under that measure, the distribution of entanglement
concentrates around a finite value at the thermodynamical limit and, in general,
the typical entanglement of Gaussian states with maximal energy E is not close to
the maximum allowed by E.
A rather recent field of research concerns the investigation of Gaussian states in
a relativistic setting [4, GA20], as we have seen in Chapter 14. Within the general
framework of relativistic quantum information [179], such studies are of relevance
to understand the phenomenon of information loss through a black hole horizon
[GA21], and more generally to gain some knowledge on the structure of the curved
spacetime [25, 14].
In a non-relativistic framework, the investigation of the structure of entanglement
in hybrid CV-qubit systems is not only of conceptual importance, but it is
relevant for applications as well. From the monogamy point of view (see Chapter 6),
some interesting hints come from a recent study of the ground-state entanglement in
highly connected systems made of harmonic oscillators and spin-1/2 systems [83].
On a more practical ground, we should at least mention a proposal for a quantum
optical implementation of hybrid quantum computation, where qubit degrees
of freedom for computation are combined with Gaussian modes for communication
[241], and a suggested scheme of hybrid quantum repeaters for long-distance
distribution of quantum entanglement based on dispersive interactions between coherent
light with large average photon number and single, far-detuned atoms or
semiconductor impurities in optical cavities [136]. A hybrid CV memory realized
by indirect interactions between different Gaussian modes, mediated by qubits,
has been recently shown to have very appealing features compared to pure-qubit
quantum registers [175].
It seems fitting to conclude this overview by commenting on the intriguing
possibility of observing CV (Gaussian) entanglement at the interface between microscopic
and macroscopic scales. In this context, it is encouraging that the existence
of optomechanical entanglement between a macroscopic movable mirror and
a cavity field has been theoretically demonstrated and predicted to be quite robust
in realistic experimental situations, up to temperatures well in reach of current
cryogenic technologies [256].
Entanglement of non-Gaussian states: a new arena
The infinite-dimensional quantum world, however, is not confined to Gaussian
states. In fact, some recent results demonstrate that basically the current state-ofthe-art
in the theoretical understanding and experimental control of CV entanglement
is strongly pushing towards the boundaries of the oasis of Gaussian states and
Gaussian operations. For instance, the entanglement of Gaussian states cannot be