ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

global generalized entropy S3
global Von Neumann entropy SV
8
6
4
2
0
0.5
0.4
0.3
0.2
0.1
0
unphysical
coexistence
entangled
4.3. Entanglement versus Entropic measures 73
separable
0 1 2 3 4 5 6
marginal Von Neumann entropy SV i
unphysical
coexistence
(a)
entangled
0 0.1 0.2 0.3 0.4 0.5
marginal generalized entropy S3i
(c)
separable
global linear entropy SL
global generalized entropy S4
1
0.8
0.6
0.4
0.2
0
0.3
0.25
0.2
0.15
0.1
0.05
0
unphysical
coexistence
entangled
separable
0 0.2 0.4 0.6 0.8 1
marginal linear entropy SL i
unphysical
coexistence
(b)
entangled
separable
0 0.05 0.1 0.15 0.2 0.25 0.3
marginal generalized entropy S4i
Figure 4.4. Summary of the entanglement properties for symmetric Gaussian
states at fixed 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.
All states in the red region are separable. In the entangled region, the average
logarithmic negativity ĒN (Sp i , Sp) Eq. (4.56) is depicted, growing from green
to magenta. For p > 2 an additional dashed curve is plotted; it represents
the nodal line of inversion. Along it the entanglement is fully determined by
the knowledge of the global and marginal generalized entropies Sp i , Sp, and
GMEMS and GLEMS are equally entangled. On the left side of the nodal line
GMEMS (GLEMS) are maximally (minimally) entangled Gaussian states at
fixed Sp i , Sp. On the right side of the nodal line they are inverted: GMEMS
(GLEMS) are minimally (maximally) entangled states. Also notice how the
yellow region of coexistence (accommodating both separable and entangled
states) becomes narrower with increasing p. The equations of all boundary
curves can be found in Eq. (4.54).
(which expresses GMEMS becoming separable). Let us mention also that the relation
between any local entropic measure Spi and the local purity µi is obtained
(d)