ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

132 7. Tripartite entanglement in three-mode Gaussian states
Therefore, we have established the following [GA10].
➢ Monotonicity of the residual Gaussian contangle under Gaussian LOCC.
The residual Gaussian contangle Gres τ is a proper and computable measure
of genuine multipartite (specifically, tripartite) entanglement in three-mode
Gaussian states, being an entanglement monotone under Gaussian LOCC.
It is worth noting that the minimum in Eq. (7.36), that at first sight might
appear a redundant (or artificial) requirement, is physically meaningful and mathematically
necessary. In fact, if one chooses to fix a reference partition, or to take
e.g. the maximum (and not the minimum) over all possible mode permutations in
Eq. (7.36), the resulting “measure” is not monotone under Gaussian LOCC and
thus is definitely not a measure of tripartite entanglement.
7.2.3. Tripartite entanglement of pure three-mode Gaussian states
We now work out in detail an explicit application, by describing the complete
procedure to determine the genuine tripartite entanglement in a pure three-mode
Gaussian state with a completely general (not necessarily in standard form) CM
σp , as presented in Ref. [GA11].
(i) Determine the local purities: The state is globally pure (Det σp = 1). The
only quantities needed for the computation of the tripartite entanglement
are therefore the three local mixednesses al, defined by Eq. (7.5), of the
single-mode reduced states σl, l = 1, 2, 3 [see Eq. (2.20)]. Notice that
the global CM σp needs not to be in the standard form of Eq. (7.19),
as the single-mode determinants are local symplectic invariants. From
an experimental point of view, the parameters al can be extracted from
the CM using the homodyne tomographic reconstruction of the state [62];
or they can be directly measured with the aid of single photon detectors
[87, 263].
(ii) Find the minimum: From Eq. (7.37), the minimum in the definition (7.36)
of the residual Gaussian contangle Gres τ is attained in the partition where
the bipartite entanglements are decomposed choosing as probe mode l the
one in the single-mode reduced state of smallest local mixedness al ≡ amin.
(iii) Check range and compute: Given the mode with smallest local mixedness
amin (say, for instance, mode 1) and the parameters s and d defined in
Eqs. (7.25, 7.26), if amin = 1 then mode 1 is uncorrelated from the others:
Gres τ = 0. If, instead, amin > 1 then
G res
τ (σ p ) = arcsinh 2
a2
min − 1 − Q(amin, s, d) , (7.38)
with Q ≡ G 1|2
τ + G 1|3
τ defined by Eqs. (7.28, 7.29). Note that if d <
−(a2 min−1)/4s then G1|2 τ = 0. Instead, if d > (a2 min−1)/4s then G1|3 τ = 0.
Otherwise, all terms in Eq. (7.36) are nonvanishing.
The residual Gaussian contangle Eq. (7.36) in generic pure three-mode Gaussian
states is plotted in Fig. 7.3 as a function of a2 and a3, at constant a1 = 2. For fixed
a1, it is interesting to notice that Gres τ is maximal for a2 = a3, i.e. for bisymmetric
states (see Fig. 5.1). Notice also how the residual Gaussian contangle of these
bisymmetric pure states has a cusp for a1 = a2 = a3. In fact, from Eq. (7.37),
for a2 = a3 < a1 the minimum in Eq. (7.36) is attained decomposing with respect