ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

178 10. Tripartite and four-partite state engineering
a3
5
4
3
2
1
1 2 3 4 5
a2
Figure 10.2. Plot of 100 000 randomly generated pure three-mode Gaussian
states, described by their single-mode mixednesses a2 and a3, at fixed a1 = 2.
The states are produced by simulated applications of the allotment operator
with random beam-splitter transmittivities s and t, and span the whole physical
range of parameters allowed by Ineq. (7.17). A comparison of this plot
with Fig. 7.1(b) may be instructive. See text for further details.
Then, substituting Eq. (10.9) in σ p
out yields a reparametrization of the output state
in terms of a1 (which is given), s and t. Now solve (numerically) the system of
nonlinear equations {Det σ2 = a 2 2, Det σ3 = a 2 3} in the variables s and t. Finally,
substitute back the obtained values of the two transmittivities in Eq. (10.9), to
have the desired triple {m, s, t} as functions of the local mixednesses {a1, a2, a3}
characterizing the target state.
We have therefore demonstrated the following [GA16].
➢ State engineering of pure three-mode Gaussian states. An arbitrary pure
three-mode Gaussian state, with a CM locally equivalent to the standard form
of Eq. (7.19), can be produced with the current experimental technology by
linear quantum optics, employing the allotment box — a passive redistribution
of two-mode entanglement among three modes — with exactly tuned amounts
of input two-mode squeezing and beam-splitter properties, without any free
parameter left.
A pictorial test of this procedure is shown in Fig. 10.2, where at a given local
mixedness of mode 1 (a1 = 2), several runs of the allotment operator have been
simulated with randomized beam-splitter transmittivities s and t. Starting from a
two-mode squeezed input with m given by Eq. (10.9), tensor a vacuum, the resulting
output states are plotted in the space of a2 and a3. By comparing Fig. 10.2 with
Fig. 7.1(b), one clearly sees that the randomly generated states distribute towards