ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

190 11. Efficient production of pure N-mode Gaussian states
11.3. Economical state engineering of arbitrary pure Gaussian states?
Borrowing the ideas leading to the state engineering of block-diagonal pure Gaussian
states [GA14], see Fig. 11.1, we propose here a scheme [GA18], involving (N 2 −
2N) independent optical elements, to produce more general N-mode pure Gaussian
states, encoding correlations between positions and momentum operators as
well. To this aim, we introduce ‘counter-beam-splitter’ transformations, named
“seraphiques”, which, recovering the phase space ordering of Sec. 2.1, act on two
modes j and k as
⎛ √ √
τ 0 0 1 − τ
⎜
√ √
Cj,k(τ) = ⎜ 0
√
τ − 1 − τ 0
⎝
√
0 1 − τ τ 0
− √ ⎞
⎟
⎠ , (11.2)
√
1 − τ 0 0 τ
where the amplitude τ is related to an angle θ in phase space by τ = cos 2 θ.
Such operations can be obtained from usual beam-splitters Bj,k(τ), Eq. (2.26), by
applying a π/2 phase shifter Pk on only one of the two considered modes. Pk
is a local rotation mapping, in Heisenberg picture, ˆqk ↦→ −ˆpk and ˆpk ↦→ ˆqk. In
phase space, one has Cj,k(τ) = P T k Bj,k(τ)Pk. Notice that, even though Cj,k(τ) is
equal to the product of single-mode operations and beam-splitters, this does not
mean that such a transformation is “equivalent” to a beam-splitter in terms of state
generation. In fact, the local operations do not commute with the beam-splitters,
so that a product of the kind Bj,k(τ ′ )Cj,k(τ ′′ ) cannot be written as Bj,k(τ)Sl for
some local operation Sl and τ.
The state engineering scheme runs along exactly the same lines as the one for
the block-diagonal states, Sec. 11.2.3, the only modification being that for each
pair of modes except the last one (N − 1, N), a beam-splitter transformation is
followed by a seraphique. In more detail (see Fig. 11.2): first of all (step 1), one
squeezes mode 1 of an amount s, and mode 2 of an amount 1/s (i.e. one squeezes
the first mode in one quadrature and the second, of the same amount, in the
orthogonal quadrature); then one lets the two modes interfere at a 50 : 50 beamsplitter.
One has so created a two-mode squeezed state between modes 1 and 2,
which corresponds to the Schmidt form of the pure Gaussian state with respect
to the 1 × (N − 1) bipartition. The second step basically corresponds to a redistribution
of the initial two-mode entanglement among all modes; this task can
be obtained by letting each additional k mode (k = 3 . . . N) interact step-by-step
with all the previous l ones (l = 2 . . . k − 1), via beam-splitters and seraphiques
(which are in turn combinations of beam-splitters and phase shifters). It is easy
to see that this scheme is implemented with minimal resources. Namely, the state
engineering process is characterized by one squeezing degree (step 1), plus N − 2
individual squeezings, together with N−2 i=1 i = (N − 1)(N − 2)/2 beam-splitter
transmittivities, and [ N−2 i=1 i] − 1 = N(N − 3)/2 seraphique amplitudes, which
amount to a total of (N 2 −2N) quantities, exactly the ones parametrizing a generic
pure Gaussian state of N ≥ 3 modes up to local symplectic operations, Eq. (2.56).
While this scheme (Fig. 11.2) is surely more general than the one for blockdiagonal
states (Fig. 11.1), as it enables to efficiently create a broader class of
pure Gaussian states for N > 3, we will leave it as an open question to check
if it is general enough to produce all pure N-mode Gaussian states up to local
unitaries. Verifying this analytically leads to pretty intractable expressions already