ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

206 12. Multiparty quantum communication with Gaussian resources
12.2.3. Operational interpretation of tripartite Gaussian entanglement and how
to experimentally investigate its sharing structure
12.2.3.1. Entanglement of teleportation and residual contangle. Let us focus, for the
following discussion, on the case N = 3, i.e. on three-mode states shared as resources
for a three-party teleportation network. This protocol is a basic, natural
candidate to operationally investigate the sharing structure of CV entanglement in
three-mode symmetric Gaussian states.
A first theoretical question that arises is to compare the tripartite entanglement
of teleportation Eq. (12.26), which possesses a strong operational motivation, and
the tripartite residual (Gaussian) contangle Eq. (7.36) (defined in Sec. 7.2.2), which
is endowed with a clear physical interpretation in the framework of entanglement
sharing and is built on solid mathematical foundations (being an entanglement
monotone under Gaussian LOCC, see Sec. 7.2.2.1). Remarkably, in the case of
pure three-mode shared resources — i.e. CV GHZ/W states, obtained by setting
n1 = n2 = 1 in Eqs. (12.17,12.18), see Sec. 7.3.1 and Fig. 10.3 — the two measures
are completely equivalent, being monotonically increasing functions of each other.
Namely, from Eq. (7.40),
G res
τ (σ GHZ/W
s ) = log 2 2 √ 2ET − (ET + 1) E2 T + 1
(ET − 1) 1
−
ET (ET + 4) + 1 2 log2
E2 T + 1
ET (ET + 4) + 1 ,
(12.28)
where ET ≡ E (3)
T in Eq. (12.26). Let us moreover recall that Gres τ coincides with
the true residual contangle (globally minimized in principle over all, including non-
Gaussian, decompositions), Eq. (7.35), in these states (see Sec. 7.3.1). The residual
(Gaussian) contangle is thus enriched of an interesting meaning as a resource enabling
a better-than-classical three-party teleportation experiment, while no operational
interpretations are presently known for the three-way residual tangle quantifying
tripartite entanglement sharing in qubit systems [59] (see Sec. 1.4.3).
We remark that in the tripartite instance, the optimal teleportation-network
fidelity of Eq. (12.25) (N = 3) achieves indeed its global maximum over all possible
Gaussian POVMs performed on the shared resource, as can be confirmed with the
methods of Ref. [183].
12.2.3.2. The power of promiscuity in symmetric three-mode resources. The relationship
between optimal teleportation fidelity and residual (Gaussian) contangle, embodied
by Eq. (12.28), entails that there is a ‘unique’ kind of three-party CV entanglement
in pure symmetric three-mode Gaussian states (alias CV finite-squeezing
GHZ/W states, introduced in Sec. 7.3.1), which merges at least three (usually
inequivalent) properties: those of being maximally genuinely tripartite entangled,
maximally bipartite entangled in any two-mode reduction, and ‘maximally efficient’
(in the sense of the optimal fidelity) for three-mode teleportation networks. Recall
that the first two properties, taken together, label such entanglement as promiscuous,
as discussed in Sec. 7.3.3. These features add up to the property of tripartite
GHZ/W Gaussian states of being maximally robust against decoherence effects
among all three-mode Gaussian states, as shown in Sec. 7.4.1.
All this theoretical evidence strongly promotes GHZ/W states, experimentally
realizable with current optical technology [8, 34] (see Sec. 10.1.2.1), as paradigmatic
candidates for the encoding and transmission of CV quantum information and in