ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

66 4. Two-mode entanglement
According to the PPT criterion, GLEMS are separable only if
µ ≤ µ1µ2/ µ 2 1 + µ22 − µ21 µ22 .
Therefore, in the range
µ1µ2
µ1µ2
< µ ≤
µ1 + µ2 − µ1µ2 µ 2
1 + µ 2 2 − µ21 µ22 (4.40)
both separable and entangled states two-mode Gaussian states can be found. Instead,
the region
µ >
µ1µ2
µ 2 1 + µ 2 2 − µ2 1 µ2 2
(4.41)
can only accomodate entangled states. The very narrow region defined by inequality
(4.40) is thus the only region of coexistence of both entangled and separable Gaussian
two-mode mixed states, compatible with a given triple of purities. We mention
that the sufficient condition for entanglement (4.41), first obtained in Ref. [GA2],
has been independently rederived in Ref. [87].
Let us also recall that for Gaussian states whose purities saturate the rightmost
inequality in Eq. (4.9), GMEMS and GLEMS coincide and we have a unique class
of entangled states depending only on the marginal purities µ1,2: they are the
Gaussian maximally entangled states for fixed marginals (GMEMMS), introduced
in Sec. 4.3.2.
All the previous necessary and/or sufficient conditions for entanglement —
which constitute the strongest entropic criteria for separability [164] to date in the
case of Gaussian states — are collected in Table 4.I and allow a graphical display
of the behavior of the entanglement of mixed Gaussian states as shown in Fig. 4.2.
These relations classify the properties of separability of all two-mode Gaussian
states according to their degree of global and marginal purities.
Degrees of purity Entanglement properties
µ < µ1µ2
µ1µ2 ≤ µ ≤
µ1µ2 < µ ≤
µ1+µ2−µ1µ2
µ1µ2
µ1+µ2−µ1µ2
µ1µ2 √
µ 2
1 +µ 2 2−µ2 1 µ2 2
µ1µ2 √
µ 2
1 +µ 2 2−µ2 1 µ2 µ1µ2
< µ ≤ µ1µ2+|µ1−µ2|
2
µ >
µ1µ2
µ1µ2+|µ1−µ2|
unphysical region
separable states
coexistence region
entangled states
unphysical region
Table 4.I. Classification of two-mode Gaussian states and of their properties
of separability according to their degrees of global purity µ and of marginal
purities µ1 and µ2.