ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

2.2. Mathematical description of Gaussian states 35
at a 50:50 beam-splitter. The total transformation realizes what we can call a
“twin-beam box”,
Ti,j(r) = Bi,j(1/2) · (Si(r) ⊕ Sj(−r)) , (2.28)
which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the
identity matrix), results in the production of a pure two-mode squeezed Gaussian
state with CM exactly equal to Ti,j(r)T T i,j (r) ≡ σsq i,j (r) from Eq. (2.22).
In general, symplectic transformations in phase space are generated by exponentiation
of matrices written as JΩ, where J is antisymmetric [10]. Such generators
can be symmetric or antisymmetric. The operations Bij(τ), Eq. (2.26), generated
by antisymmetric operators are orthogonal and, acting by congruence on the CM σ,
preserve the value of Tr σ. Since Tr σ gives the contribution of the second moments
to the average of the Hamiltonian
k â†
kâk, these transformations are said to be
passive (they belong to the compact subgroup of Sp (2N,)). Instead, operations
Si,j(r), Eq. (2.24), generated by symmetric operators, are not orthogonal and do
not preserve Tr σ (they belong to the non-compact subgroup of Sp (2N,)). This
mathematical difference between squeezers and phase-space rotations accounts, in a
quite elegant way, for the difference between active (energy consuming) and passive
(energy preserving) optical transformations [268].
⊕N
Let us remark that local symplectic operations belong to the group Sp (2,) .
They correspond, on the Hilbert space level, to tensor products of unitary transformations,
each acting on the space of a single mode. It is useful to notice that the
determinants of each 2×2 submatrix of a N-mode CM, Eq. (2.20), are all invariant
⊕N 6 under local symplectic operations S ∈ Sp (2,) . This mathematical property
reflects the physical requirement that both marginal informational properties, and
correlations between the various individual subsystems, cannot be altered by local
operations only.
2.2.2.1. Symplectic eigenvalues and invariants. A crucial symplectic transformation is
the one realizing the decomposition of a Gaussian state in normal modes. Through
this decomposition, thanks to Williamson theorem [267], the CM of a N-mode
Gaussian state can always be written in the so-called Williamson normal, or diagonal
form
σ = S T νS , (2.29)
where S ∈ Sp (2N,R) and ν is the CM
ν =
N
νk 0
k=1
0 νk
, (2.30)
corresponding to a tensor product state with a diagonal density matrix ϱ⊗ given
by
ϱ⊗ =
∞
2 νk − 1
|n〉kk〈n| ,
νk + 1 νk + 1
(2.31)
k
n=0
where |n〉k denotes the number state of order n in the Fock space Hk. In the
Williamson form, each mode with frequency ωk is a Gaussian state in thermal
6 The invariance of the off-diagonal terms Det εi,j follows from Binet’s formula for the determinant
of a matrix [18], plus the fact that any symplectic transformation S has Det S = 1.