ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 3
Characterizing entanglement of Gaussian
states
In this short Chapter we recall the main results on the qualification and quantification
of bipartite entanglement for Gaussian states of CV systems. We will borrow
some material from [GA22].
3.1. How to qualify bipartite Gaussian entanglement
3.1.1. Separability and distillability: PPT criterion
The positivity of the partially transposed state (Peres-Horodecki PPT criterion
[178, 118], see Sec. 1.3.2.1) is necessary and sufficient for the separability of twomode
Gaussian states [218] and, more generally, of all (1+N)-mode Gaussian states
under 1 × N bipartitions [265] and — as we are going to show — of symmetric
and bisymmetric (M + N)-mode Gaussian states (see Sec. 2.4.3) under M × N
bipartitions [GA5]. In general, the partial transposition ϱ TA of a bipartite quantum
state ϱ is defined as the result of the transposition performed on only one of the
two subsystems (say SA) in some given basis. In phase space, the action of partial
transposition amounts to a mirror reflection of the momentum operators of the
modes comprising one subsystem [218]. The CM σ A|B, where subsystem SA groups
NA modes, and subsystem SB is formed by NB modes, is then transformed into a
new matrix
with
˜σ A|B ≡ θ A|B σ A|B θ A|B , (3.1)
θA|B = diag{1, −1, 1, −1, . . . , 1, −1,
1, 1, 1, 1, . . . , 1, 1}
. (3.2)
2NA
Referring to the notation of Eq. (2.20), the partially transposed matrix ˜σ A|B differs
from σ A|B by a sign flip in the determinants of the intermodal correlation matrices,
Det εij, with modes i ∈ SA and modes j ∈ sB.
The PPT criterion yields that a Gaussian state σ A|B (with NA = 1 and NB
arbitrary) is separable if and only if the partially transposed ˜σ A|B is a bona fide
CM, that is it satisfies the uncertainty principle Eq. (2.19),
2NB
˜σ A|B + iΩ ≥ 0 . (3.3)
This reflects the positivity of the partially transposed density matrix ϱ TA associated
to the state ϱ. For Gaussian states with NA > 1 and not endowed with special
symmetry constraints, PPT condition is only necessary for separability, as bound
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