ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

(out)
(in)
TRITTER
10.1. Optical production of three-mode Gaussian states 179
BS
1:2
momentumsqueezed
(r 1 ) positionsqueezed
(r 2 )
BS
1:1
positionsqueezed
(r 2 )
Figure 10.3. Scheme to produce CV GHZ/W states, as proposed in Ref. [236]
and implemented in Ref. [8]. Three independently squeezed beams, one in
momentum and two in position, are combined through a double beam-splitter
(tritter). The output yields a pure, symmetric, fully inseparable three-mode
Gaussian state, also known as CV GHZ/W state.
a complete fill of the physical region emerging from the triangle inequality (7.17),
thus confirming the generality of our scheme.
10.1.2. Tripartite state engineering handbook and simplified schemes
Having a generalization of the “allotment” for the production of arbitrary mixed
three-mode Gaussian states turns out to be a quite involved task. However, for
many classes of tripartite states introduced in Chapter 7, efficient state engineering
schemes can be devised. Also in special instances of pure states, depending in general
on less than three parameters, cheaper recipes than the general one in terms of
the allotment box are available. We will now complement the entanglement analysis
of Secs. 7.3 and 7.4 with such practical proposals, as presented in Ref. [GA16].
10.1.2.1. CV GHZ/W states. Several schemes have been proposed to produce what
we call finite-squeezing GHZ/W states of continuous variables, i.e. fully symmetric
pure three-mode Gaussian states with promiscuous entanglement sharing (see
Sec. 7.3.1). In particular, as discussed by van Loock and Braunstein [236], these
states can be produced by mixing three squeezed beams through a double beamsplitter,
or tritter [36]. One starts with mode 1 squeezed in momentum, and modes
2 and 3 squeezed in position. In Heisenberg picture:
ˆq1 = e r1 ˆq 0 1 , ˆp1 = e −r1 ˆp 0 1 , (10.10)
ˆq2,3 = e −r2 ˆq 0 2,3 , ˆp2,3 = e r2 ˆp 0 2,3 , (10.11)