ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

172 9. Two-mode Gaussian states in the lab
Figure 9.9. Normalized noise variances at 3.5 MHz of the rotated modes
while scanning the local oscillator phase for an angle of the plate of 0.3 ◦ ,
before and after the non-local operation. The homodyne detections are inquadrature.
After this operation, squeezing is observed on orthogonal quadra-
tures.
In this instance one finds for the partially transposed symplectic eigenvalue ˜ν−
0.46 (corresponding to a logarithmic negativity between A1 and A2 much lower than
the previous value: EN = 1.13), whereas the smallest eigenvalues read λ1 = λ2
0.40. The entanglement can thus be raised via passive operations to the optimal
value EN = 1.32, corresponding to ˜ν− = √ λ1λ2.
The passive transformation capable of optimizing the entanglement is easily
found, according to the theoretical analysis of Sec. 9.2.2. If O is the rotation
diagonalizing the 2 × 2 symmetric matrix α ′′ defined in Eq. (9.10), then the transformation
K ≡ B T (1/2)( ⊕ O)B(1/2) turns the CM σ(ρ = 0.3 ◦ ) into ¯σ(ρ = 0.3 ◦ ),
which is diagonal in the orthogonal quadratures A∓ and in a symmetric standard
form in the quadratures A1,2. The entanglement of such a matrix is therefore optimal
under passive operations. The optimal symplectic operation K consists in a
‘phase-shift’ of the rotated modes A1,2. In the experimental practice, such an operation
can be readily performed on co-propagating, orthogonally polarized beams
[219]. The minimal combination of waveplates needed can be shown to consist in
three waveplates: two λ/4 waveplates denoted Q and one λ/2 waveplate denoted H.
When using any combination of these three plates, the state can be put back into
standard form which will maximize the entanglement. Fig. 9.9 gives the normalized
noise variances before and after this operation. In agreement with the theory, the