ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

72 4. Two-mode entanglement
provide the explicit expressions of µ κ p(µ1, µ2), as plotted in Fig. 4.3 for the cases
(a) p = 3, and (b) p = 4,
µ κ 3(µ1, µ2) =
µ κ 4(µ1, µ2) = √ 3 µ1µ2
6
3
µ 2 +
1
3
µ 2 − 2
1
1
2
, (4.52)
µ 2 1 + µ 2 2 − 2µ 2 1µ 2 2 +
(µ 2 1 + µ22 ) (µ21 + µ22 − µ21 µ22 ) + µ41 µ4 1
2
2 .
(4.53)
4.3.4.2. Classifying entangled states with generalized entropic measures. Apart from
the relevant ‘inversion’ feature shown by p−entropies for p > 2, the possibility of
an accurate characterization of CV entanglement based on global and marginal entropic
measures still holds in the general case for any p. In particular, the set of all
Gaussian states can be again divided, in the space of global and marginal Sp’s, into
three main areas: separable, entangled and coexistence region. It can be thus very
interesting to investigate how the different entropic measures chosen to quantify the
degree of global mixedness (all marginal measures are equivalent) behave in classifying
the separability properties of Gaussian states. Fig. 4.4 provides a numerical
comparison of the different characterizations of entanglement obtained by the use
of different p−entropies, with p ranging from 1 to 4, for symmetric Gaussian states
(Sp1 = Sp2 ≡ Spi ). The last restriction has been imposed just for ease of graphical
display. The following considerations, based on the exact numerical solutions of the
transcendental conditions, will take into account nonsymmetric states as well.
The mathematical relations expressing the boundaries between the different
regions in Fig. 4.4 are easily obtained for any p by starting from the relations
holding for p = 2 (see Table 4.I) and by evaluating the corresponding Sp(µ1,2) for
each µ(µ1,2). For any physical symmetric state such a calculation yields
2 − µ 2
i
0 ≤ (p − 1)Sp < 1 − gp
1 − gp
2 − µ 2 i
µi
⇒ entangled,
≤ (p − 1)Sp < 1 − g 2 p
µi
2 − µi
1 − g 2
2 − µi
p
≤ (p − 1)Sp ≤ 1 − g 2
1
p
µi
µi
⇒ coexistence, (4.54)
⇒ separable.
Equations (4.54) were obtained exploiting the multiplicativity of p−norms on product
states and using Eq. (2.44) for the lower boundary of the coexistence region
(which represents GLEMS becoming entangled) and Eq. (2.47) for the upper one
µ 2 i