ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

7.3. Sharing structure of tripartite entanglement: promiscuous Gaussian states 135
For any finite squeezing (equivalently, any finite local mixedness a), however,
the above entanglement sharing study leads ourselves to re-baptize these states as
“CV GHZ/W states” [GA10, GA11, GA16], and denote their CM by σ GHZ/W
s .
The residual Gaussian contangle of GHZ/W states with finite squeezing takes
the simple form (see Sec. 6.2.2)
G res
τ (σ GHZ/W
s ) = arcsinh 2
a2 − 1
− 1
2 log2
3a2 − 1 − √ 9a4 − 10a2
+ 1
(7.40)
.
2
It is straightforward to see that Gres τ (σ GHZ/W
s ) is nonvanishing as soon as a > 1.
Therefore, the GHZ/W states belong to the class of fully inseparable three-mode
states [94, 236, 235, 240] (Class 1, see Sec. 7.1.1). We finally recall that in a
GHZ/W state the residual Gaussian contangle Gres τ Eq. (7.36) coincides with the
true residual contangle E 1|2|3
τ Eq. (7.35). This property clearly holds because the
Gaussian pure-state decomposition is the optimal one in every bipartition, due to
the fact that the global three-mode state is pure and the reduced two-mode states
are symmetric (see Sec. 4.2.2).
7.3.2. T states with zero reduced bipartite entanglement
The peculiar nature of entanglement sharing in CV GHZ/W states is further confirmed
by the following observation. If one requires maximization of the 1 × 2
bipartite Gaussian contangle G i|(jk)
τ under the constraint of separability of all the
reduced two-mode states (like it happens in the GHZ state of three qubits), one
finds a class of symmetric mixed states characterized by being three-mode Gaussian
states of partial minimum uncertainty (see Sec. 2.2.2.2). They are in fact characterized
by having their smallest symplectic eigenvalue equal to 1, and represent
thus the three-mode generalization of two-mode symmetric GLEMS (introduced in
Sec. 4.3.3.1).
We will name these states T states, with T standing for tripartite entanglement
only [GA10, GA11, GA16]. They are described by a CM σT s of the form Eq. (2.60),
with α = a2, ε = diag{e + , e− } and
e + = a2 − 5 + 9a2 (a2 − 2) + 25
,
4a
e − = 5 − 9a2 + 9a2 (a2 − 2) + 25
12a
. (7.41)
The T states, like the GHZ/W states, are determined only by the local mixedness
a, are fully separable for a = 1, and fully inseparable for a > 1. The residual
Gaussian contangle Eq. (7.36) can be analytically computed for these mixed states
as a function of a. First of all one notices that, due to the complete symmetry
of the state, each mode can be chosen indifferently to be the reference one in
Eq. (7.36). Being the 1 × 1 entanglements all zero by construction, Gres τ = G i|(jk)
τ .
The 1 × 2 bipartite Gaussian contangle can be in turn obtained exploiting the
unitary localization procedure (see Chapter 5 and Fig. 5.1). Let us choose mode
1 as the probe mode and combine modes 2 and 3 at a 50:50 beam-splitter, a local
unitary operation with respect to the bipartition 1|(23) that defines the transformed