ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

52 3. Characterizing entanglement of Gaussian states
entangled Gaussian states, whose entanglement is undistillable, have been proven
to exist already in the instance NA = NB = 2 [265].
We have demonstrated the existence of “bisymmetric” (NA +NB)-mode Gaussian
states for which PPT is again equivalent to separability [GA5]. In view of
the invariance of PPT criterion under local unitary transformations (which can be
appreciated by the definition of partial transpose at the Hilbert space level) and
considering the results proved in Sec. 2.4.3 on the unitary localization of bisymmetric
Gaussian states, see Eq. (2.65), it is immediate to verity that the following
property holds [GA5].
➢ PPT criterion for bisymmetric multimode Gaussian states. For generic
NA × NB bipartitions, the positivity of the partial transpose (PPT) is a
necessary and sufficient condition for the separability of bisymmetric (NA +
NB)-mode mixed Gaussian states of the form Eq. (2.63). In the case of
fully symmetric mixed Gaussian states, Eq. (2.60), of an arbitrary number of
modes, PPT is equivalent to separability across any global bipartition of the
modes.
This statement is a first important generalization to multimode bipartitions of
the equivalence between separability and PPT for 1 × N bipartite Gaussian states
[265]. In particular, it implies that no bisymmetric bound entangled Gaussian
states can exist [265, 91] and all the NA × NB multimode block entanglement of
such states is distillable. Moreover, it justifies the use of the negativity and the
logarithmic negativity as measures of entanglement for these multimode Gaussian
states, as will be done in Chapter 5.
In general, the distillability problem for Gaussian states has been also solved
[91]: the entanglement of any non-PPT bipartite Gaussian state is distillable by
LOCC. However, we remind that this entanglement can be distilled only resorting to
non-Gaussian LOCC [76], since distilling Gaussian states with Gaussian operations
is impossible [78, 205, 90].
3.1.1.1. Symplectic representation of PPT criterion. The partially transposed matrix
˜σ of any N-mode Gaussian CM σ is still a positive and symmetric matrix. As
such, it admits a Williamson normal-mode decomposition [267], Eq. (2.29), of the
form
˜σ = S T ˜νS , (3.4)
where S ∈ Sp (2N,R) and ˜ν is the CM
˜ν =
N
˜νk 0
k=1
0 ˜νk
, (3.5)
The N quantities ˜νk’s are the symplectic eigenvalues of the partially transposed
CM ˜σ. While the symplectic spectrum {νk} of σ encodes the structural and informational
properties of a Gaussian state, the partially transposed spectrum {˜νk}
encodes a complete qualitative (and to some extent quantitative, see next Section)
characterization of entanglement in the state. Namely, the PPT condition (3.3),
i.e. the uncertainty relation for ˜σ, can be equivalently recast in terms of the parameters
˜νk’s as
˜νk ≥ 1 . (3.6)