ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

30 2. Gaussian states: structural properties
where Ω is the symplectic form
N
Ω = ω , ω =
k=1
0 1
−1 0
. (2.8)
The space Hk is spanned by the Fock basis {|n〉k} of eigenstates of the number
operator ˆnk = â †
k âk, representing the Hamiltonian of the non-interacting mode via
Eq. (2.2). The Hamiltonian of any mode is bounded from below, thus ensuring the
stability of the system, so that for any mode a vacuum state |0〉k ∈ Hk exists, for
which âk|0〉k = 0. The vacuum state of the global Hilbert space will be denoted by
|0〉 =
k |0〉k. In the single-mode Hilbert space Hk, the eigenstates of âk constitute
the important set of coherent states [259], which is overcomplete in Hk. Coherent
states result from applying the single-mode Weyl displacement operator ˆ Dk to the
vacuum |0〉k, |α〉k = ˆ Dk(α)|0〉k, where
ˆDk(α) = e αâ†
k −α∗ âk , (2.9)
and the coherent amplitude α ∈ satisfies âk|α〉k = α|α〉k. In terms of the Fock
basis of mode k a coherent state reads
1 −
|α〉k = e 2 |α|2
∞ αn √ |n〉k . (2.10)
n!
Tensor products of coherent states of different modes are obtained by applying the
N-mode Weyl operators ˆ Dξ to the global vacuum |0〉. For future convenience, we
define the operators ˆ Dξ in terms of the canonical operators ˆ R,
n=1
ˆDξ = e i ˆ R T Ωξ , with ξ ∈ 2N . (2.11)
One has then |ξ〉 = ˆ Dξ|0〉, which entails âk|ξ〉 = (ξk + iξk+1)|ξ〉.
2.1.1. Quantum phase-space picture
The states of a CV system are the set of positive trace-class operators {ϱ} on the
Hilbert space H , Eq. (2.1). However, the complete description of any quantum
state ϱ of such an infinite-dimensional system can be provided by one of its s-ordered
characteristic functions [16]
χs(ξ) = Tr [ϱ ˆ Dξ] e sξ2 /2 , (2.12)
with ξ ∈ 2N , · standing for the Euclidean norm of 2N . The vector ξ belongs
to the real 2N-dimensional space Γ = ( 2N , Ω), which is called phase space, in
analogy with classical Hamiltonian dynamics. One can see from the definition of
the characteristic functions that in the phase space picture, the tensor product
structure is replaced by a direct sum structure, so that the N-mode phase space is
Γ =
k Γk, where Γk = ( 2 , ω) is the local phase space associated with mode k.
The family of characteristic functions is in turn related, via complex Fourier
transform, to the quasi-probability distributions Ws, which constitute another set
of complete descriptions of the quantum states
Ws(ξ) = 1
π2
κχs(κ) e iκTΩξ 2N
d . (2.13)
2N
As well known, there exist states for which the function Ws is not a regular probability
distribution for any s, because it can in general be singular or assume negative