ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

180 10. Tripartite and four-partite state engineering
where the suffix “0” refers to the vacuum. Then one combines the three modes in a
tritter
ˆB123 ≡ ˆ B23(π/4) · ˆ B12(arccos 1/3) , (10.12)
where the action of an ideal (phase-free) beam-splitter operation ˆ Bij on a pair of
modes i and j is defined by Eq. (2.25).
The output of the tritter yields a CM of the form Eq. (2.60) with
1 2r1 −2r2 α = diag e + 2e
3
,
1 2r1 −2r2 ε = diag e − e
3
,
1 −2r1 2r2 e + 2e
3
, (10.13)
1 −2r1 2r2 e − e
3
. (10.14)
This resulting pure and fully symmetric three-mode Gaussian state, obtained in
general with differently squeezed inputs r1 = r2, is locally equivalent to the state
prepared with all initial squeezings equal to the average ¯r = (r1 + r2)/2 (this
will be discussed in more detail in connection with teleportation experiments, see
Sec. 12.2).
The CM described by Eqs. (10.13,10.14) represents a CV GHZ/W state. It can
be in fact transformed, by local symplectic operations, into the standard form CM
given by Eq. (7.39), with
a = 1
4 cosh [2 (r1 + r2)] + 5 . (10.15)
3
The preparation scheme of CV GHZ/W states is depicted in Fig. 10.3. It has been
experimentally implemented [8], and the fully inseparability of the produced states
has been verified through the violation of the separability inequalities derived in
Ref. [240]. Very recently, the production of strongly entangled GHZ/W states has
also been demonstrated by using a novel optical parametric oscillator, based on
concurrent χ (2) nonlinearities [34].
10.1.2.2. Noisy GHZ/W states. Noisy GHZ/W states, whose entanglement has been
characterized in Sec. 7.4.2, can be obtained as GHZ/W states generated from
(Gaussian) thermal states: one starts with three single-mode squeezed thermal
states (with average photon number ¯n = [n − 1]/2) and combine them through
a tritter Eq. (10.12), with the same procedure described in Fig. 10.3 for n = 1.
The initial single, separable, modes are thus described by the following operators
in Heisenberg picture,
ˆq1 = √ ne r ˆq 0 1 , ˆp1 = √ ne −r ˆp 0 1 , (10.16)
ˆq2,3 = √ ne −r ˆq 0 2,3 , ˆp2,3 = √ ne r ˆp 0 2,3 . (10.17)
Defining s ≡ e2r , at the output of the tritter one obtains a CM of the form
Eq. (2.60), with
2 n(s + 2)
α = diag
,
3s
n(2s2 2 n(s − 1)
ε = diag
,
3s
+ 1)
,
3s
−
(10.18)
n(s2
− 1)
.
3s
(10.19)
This resulting CM is locally equivalent to the standard form of Eq. (7.52), with
a = n√ 2s 4 + 5s 2 + 2
3s
. (10.20)