ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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a3
7.2. Distributed entanglement and genuine tripartite quantum correlations 133
3
2
1
1
2
a2 a2
3
0
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1
1.5
0.5
G Τ res
Figure 7.3. Three-dimensional plot of the residual Gaussian contangle
Gres τ (σp ) in pure three-mode Gaussian states σp , determined by the three
local mixednesses al, l = 1, 2, 3. One of the local mixednesses is kept fixed
(a1 = 2). The remaining ones vary constrained by the triangle inequality
(7.22), as depicted in Fig. 7.1(b). The explicit expression of Gres τ is given by
Eq. (7.38). See text for further details.
to one of the two modes 2 or 3 (the result is the same by symmetry), while for
a2 = a3 > a1 mode 1 becomes the probe mode.
7.2.3.1. Residual contangle and distillability of mixed states. For generic mixed threemode
Gaussian states, a quite cumbersome analytical expression for the 1|2 and 1|3
Gaussian contangles may be written, which explicitly solves the minimization over
the angle θ in Eq. (4.64). On the other hand, the optimization appearing in the
computation of the 1|(23) bipartite Gaussian contangle [see Eq. (6.13)] has to be
solved only numerically. However, exploiting techniques like the unitary localization
of entanglement described in Chapter 5, and results like that of Eq. (6.12), closed
expressions for the residual Gaussian contangle can be found as well in relevant
classes of mixed three-mode Gaussian states endowed with some symmetry constraints.
Interesting examples of these states and the investigation of their physical
properties will be discussed in Sec. 7.3.
As an additional remark, let us recall that, although the entanglement of Gaussian
states is always distillable with respect to 1×N bipartitions [265] (see Sec. 3.1.1),
they can exhibit bound entanglement in 1×1×1 tripartitions [94]. In this case, the