ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

144 7. Tripartite entanglement in three-mode Gaussian states
squeezing dB
7
6
5
4
3
2
1
0
Promiscuous Sharing
Fully Inseparable
Fully Separable
BiSeparable
Tripartite Bound Entangled
0 1 2 3 4 5 6 7
noise dB
Figure 7.6. Summary of separability and entanglement properties of threemode
squeezed thermal states, or noisy GHZ/W states, in the space of the
two parameters n and s. The separability is classified according to the scheme
of Sec. 7.1.1. Above the dotted line the states are fully inseparable (Class
1); below the solid line they are fully separable (Class 5). In the narrow
intermediate region, noisy GHZ/W states are three-mode biseparable (Class
4), i.e. they exhibit tripartite bound entanglement. The relations defining the
boundaries for the different regions are given in Eqs. (7.54–7.56). In the fully
inseparable region, the residual (Gaussian) contangle Eq. (7.57) is depicted
as a contour plot, growing with increasing darkness from Gres τ = 0 (along
the dotted line) to Gres τ ≈ 1.9 (at n = 0 dB, s = 7 dB). On the left side
of the dashed line, whose expression is given by Eq. (7.59), not only genuine
tripartite entanglement is present, but also each reduced two-mode bipartition
is entangled. In this region, Gres τ is strictly larger than in the region where the
two-mode reductions are separable. This evidences the promiscuous sharing
structure of multipartite CV entanglement in symmetric, even mixed, threemode
Gaussian states.
states are summarized, as functions of the parameters n and s expressed in decibels.
20 Explicitly, one finds that for
n ≥ √ 3 , (7.58)
the entanglement sharing can never be promiscuous, as the reduced two-mode entanglement
is zero for any (even arbitrarily large) squeezing s. Otherwise, applying
PPT criterion, one finds that for sufficiently high squeezing bipartite entanglement
20 The noise expressed in decibels (dB) is obtained from the covariance matrix elements via
the formula Nij(dB) = 10 log 10 (σij).