ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

10.1. Optical production of three-mode Gaussian states 177
Â123
BS (t)
BS 2:1
BS (s)
two-mode squeezed (r) vacuum
Figure 10.1. Scheme to produce generic pure three-mode Gaussian states. A
two-mode squeezed state and a single-mode vacuum are combined by the “allotment”
operator Â123, which is a sequence of three beam-splitters, Eq. (10.4).
The output yields a generic pure Gaussian state of modes 1 (●), 2 (■), and 3
(▲), whose CM depends on the initial squeezing factor m = cosh(2r) and on
two beam-splitter transmittivities s and t.
in phase space). The elements of the CM σ p
out, not reported here for brevity, are
functions of the three parameters
m ∈ [1, ∞), s ∈ [0, 1], t ∈ [0, 1] , (10.8)
being respectively related to the initial squeezing in the two-mode squeezed state
of modes 1 and 2, and two beam-splitter transmittivities (the transmittivity of
the third beam-splitter is fixed). In fact, by letting these three parameters vary in
their respective domain, the presented procedure allows for the creation of arbitrary
three-mode pure Gaussian states (up to local unitaries), with any possible triple of
local mixednesses {a1, a2, a3} ranging in the physical region defined by the triangle
inequality (7.17).
This can be shown as follows. Once identified σ p
out with the block form of
Eq. (2.20) (for N = 3), one can solve analytically the equation Det σ1 = a 2 1 to find
m(a1, s, t) = t[t(s−1)2 +s−1]+ √ a 2 1 (st+t−1)2 +4s(t−1)t(2t−1)(2st−1)
(st+t−1) 2 . (10.9)