ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

34 2. Gaussian states: structural properties
The CM of N-mode coherent states (including the vacuum) is instead the 2N ×
2N identity matrix.
2.2.2. Symplectic operations
A major role in the theoretical and experimental manipulation of Gaussian states
is played by unitary operations which preserve the Gaussian character of the states
on which they act. Such operations are all those generated by Hamiltonian terms at
most quadratic in the field operators. As a consequence of the Stone-Von Neumann
theorem, the so-called metaplectic representation [219] entails that any such unitary
operation at the Hilbert space level corresponds, in phase space, to a symplectic
transformation, i.e. to a linear transformation S which preserves the symplectic
form Ω, so that
S T ΩS = Ω . (2.23)
Symplectic transformations on a 2N-dimensional phase space form the (real) symplectic
group Sp (2N,) [10]. Such transformations act linearly on first moments and
by congruence on CMs, σ ↦→ SσS T . Eq. (2.23) implies Det S = 1, ∀ S ∈ Sp (2N,).
Ideal beam-splitters, phase shifters and squeezers are all described by some kind of
symplectic transformation (see e.g. [GA8]). For instance, the two-mode squeezing
operator Eq. (2.21) corresponds to the symplectic transformation
Si,j(r) =
⎛
⎜
⎝
cosh r 0 sinh r 0
0 cosh r 0 − sinh r
sinh r 0 cosh r 0
0 − sinh r 0 cosh r
⎞
⎟
⎠ , (2.24)
where the matrix is understood to act on the couple of modes i and j. In this way,
the two-mode squeezed state, Eq. (2.22), can be obtained as σ sq
i,j (r) = Si,j(r)S T i,j (r)
exploiting the fact that the CM of the two-mode vacuum state is the 4 × 4 identity
matrix.
Another common symplectic operation is the ideal (phase-free) beam-splitter,
whose action ˆ Bi,j on a pair of modes i and j is defined as
âi ˆBi,j(θ)
→ âi cos θ + âj sin θ
:
, (2.25)
âj → âi sin θ − âj cos θ
with âl being the annihilation operator of mode k. A beam-splitter with transmittivity
τ corresponds to a rotation of θ = arccos √ τ in phase space (θ = π/4
amounts to a balanced 50:50 beam-splitter, τ = 1/2), described by a symplectic
transformation
⎛ √ √ ⎞
τ 0 1 − τ 0
⎜
√ √
Bi,j(τ) = ⎜ √
0 τ 0 1 − τ ⎟
⎝
√ ⎟
1 − τ
√
0 − τ 0 ⎠ . (2.26)
√
0 1 − τ 0 − τ
Single-mode symplectic operations are easily retrieved as well, being just combinations
of planar (orthogonal) rotations and of single-mode squeezings of the
form
Sj(r) = diag ( e r , e −r ) , (2.27)
acting on mode j, for r > 0. In this respect, let us mention that the two-mode
squeezed state Eq. (2.22) can also be obtained indirectly, by individually squeezing
two single modes i and j in orthogonal quadratures, and by letting them interfere