ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

which, together with Inequality (7.1), implies
7.1. Three-mode Gaussian states 123
∆ij = Det σij + 1 , ∀ i, j : i = j . (7.4)
The last equation shows that any reduced two-mode state of a pure three-mode
Gaussian state saturates the partial uncertainty relation Eq. (7.1). The states
endowed with such a partial minimal uncertainty (namely, with their smallest symplectic
eigenvalue equal to 1) are states of minimal negativity for given global and
local purities, alias GLEMS (Gaussian least entangled mixed states) [GA2, GA3],
introduced in Sec. 4.3.3.1. In general, by invoking the phase-space Schmidt decomposition
(see Sec. 2.4.2.1) [116, 29, 92], it immediately follows that any (N −1)-mode
reduced state of a N-mode pure Gaussian state is a mixed state of partial minimum
uncertainty (a sort of generalized GLEMS), with N −2 symplectic eigenvalues
fixed to 1 and only one, in general, greater than 1 — shortly, with symplectic rank
ℵ = 1, see Sec. 2.2.2.2 — thus saturating Eq. (2.35). This argument is resumed in
Appendix A.2.
In fact, our simple proof, straightforwardly derived in terms of symplectic invariants,
provides some further insight into the structure of CMs characterizing
three-mode Gaussian states. What matters to our aims, is that the standard form
CM of Gaussian states is completely determined by their global and local invariants,
as discussed in Sec. 2.4. Therefore, because of Eq. (7.3), the entanglement
between any pair of modes embedded in a three-mode pure Gaussian state is fully
determined by the local invariants Det σl, for l = 1, 2, 3, whatever proper measure
we choose to quantify it. Furthermore, the entanglement of a σ i|(jk) bipartition of
a pure three-mode state is determined by the entropy of one of the reduced states,
i.e. , once again, by the quantity Det σi. Thus, the three local symplectic invariants
Det σ1, Det σ2 and Det σ3 fully determine the entanglement of any bipartition of
a pure three-mode Gaussian state. We will show that they suffice to determine as
well the genuine tripartite entanglement encoded in the state [GA11].
For ease of notation, in the following we will denote by al the local single-mode
symplectic eigenvalue associated to mode l with CM σl,
al ≡ Det σl . (7.5)
Eq. (2.37) shows that the quantities al are simply related to the purities of the
reduced single-mode states, i.e. the local purities µl, by the relation
µl = 1
al
. (7.6)
Since the set {al}, l = 1, 2, 3, fully determines the entanglement of any of the
1 × 2 and 1 × 1 bipartitions of the state, it is important to determine the range
of the allowed values for such quantities. This is required in order to provide
a complete quantitative characterization of the entanglement of three-mode pure
Gaussian states. To this aim, let us focus on the reduced two-mode CM σ12 and
let us bring it (by local unitaries) in standard form [70, 218], so that Eq. (2.20) is
recast in the form
σl = diag{al, al} , l = 1, 2 ;
ε12 = diag{c12, d12} , (7.7)
where c12 and d12 are the intermodal covariances, and, as we will show below,
can be evaluated independently in pure three-mode Gaussian states. Notice that