ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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8.2. Entanglement in partially symmetric four-mode Gaussian states 153
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GΤ res Σ
Figure 8.3. Residual multipartite entanglement Gτ res (σ) [see Eq. (8.5)],
which in the regime of large squeezing a is completely distributed in the form
of genuine four-partite quantum correlations. The four-partite entanglement
is monotonically increasing with increasing squeezing a, and diverges as a
approaches infinity. The multimode Gaussian state σ constructed with an
arbitrarily large degree of squeezing a, consequently, exhibits a coexistence
of unlimited multipartite and pairwise bipartite entanglement in the form of
EPR correlations. In systems of many qubits, and even in Gaussian states of
CV systems with a number of modes smaller than four (see Chapter 7), such
an unlimited and unconstrained promiscuous distribution of entanglement is
strictly forbidden.
Numerical investigations in the space of all pure three-mode Gaussian states seem
to confirm that the upper bound of Eq. (8.12) is actually sharp (meaning that the
three-mode contangle is globally minimized on the state γ p ), but this statement
can be left here as a conjecture since it is not required for our subsequent analysis.
The upper bound Gτ bound (σ 1|2|3) is always nonnegative (as an obvious consequence
of monogamy, see Sec. 7.2.1), moreover it is decreasing with increasing
squeezing a, and vanishes in the limit a → ∞, as shown in Fig. 8.2. Therefore,
in the regime of increasingly high a, eventually approaching infinity, any form of
tripartite entanglement among any three modes in the state σ is negligible (exactly
vanishing in the limit). As a crucial consequence, in that regime the residual entanglement
Gτ res (σ) determined by Eq. (8.5) is all stored in four-mode quantum
correlations and quantifies the genuine four-partite entanglement.
8.2.3.3. Genuine four-partite entanglement: promiscuous beyond limits. We finally
observe that Gτ res (σ), Eq. (8.5), is an increasing function of a for any value of s
(see Fig. 8.3), and it diverges in the limit a → ∞. This proves that the class of pure
four-mode Gaussian states with CM σ given by Eq. (8.1) exhibits genuine fourpartite
entanglement which grows unboundedly with increasing squeezing a and,
simultaneously, possesses pairwise bipartite entanglement in the mixed two-mode
reduced states of modes {1, 2} and {3, 4}, that increases unboundedly as well with