ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

74 4. Two-mode entanglement
E
E
4
3
2
1
0
0
2
1.5
1
0.5
2
0
0 0.1 0.2
4
SV
S3
0.3
(a)
(c)
6
0.4
1
0
2
3
SVi
0.1 0
0.2
0.1
0.3
0.4
S3i
E
3
2
1
0
0 0.2 0.4
Figure 4.5. Upper and lower bounds on the logarithmic negativity of symmetric
Gaussian states as functions of the global and marginal generalized
p−entropies, for (a) p = 1 (Von Neumann entropies), (b) p = 2 (linear entropies),
(c) p = 3, and (d) p = 4. The blue (yellow) surface represents
GMEMS (GLEMS). Notice that for p > 2 GMEMS and GLEMS surfaces intersect
along the inversion line (meaning they are equally entangled along that
line), and beyond it they interchange their role. The equations of the inversion
lines are obtained from Eqs. (4.51–4.53), with the position Sp1 = Sp2 ≡ Sp i .
directly from Eq. (2.36) and reads
Spi =
E
1.5
1
0.5
0
0
1 − gp(1/µi)
p − 1
0.1
SL
S4
0.6
0.2
(b)
(d)
0.8
0.3
0.2 0
0.4
0.2
0.6
0.8
SLi
0
0.1
0.2
S4i
. (4.55)
We notice prima facie that, with increasing p, the entanglement is more sharply
qualified in terms of the global and marginal p−entropies. In fact the region of
coexistence between separable and entangled states becomes narrower with higher
p. Thus, somehow paradoxically, with increasing p the entropy Sp provides less
information about a quantum state, but at the same time it yields a more accurate
characterization and quantification of its entanglement. In the limit p → ∞ all
the physical states collapse to one point at the origin of the axes in the space of
generalized entropies, due to the fact that the measure S∞ is identically zero.