ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

7.2. Distributed entanglement and genuine tripartite quantum correlations 129
states the Gaussian measures of entanglement, including Gτ , have been computed
explicitly in Sec. 4.5.2.2 [GA7], yielding
Q = arcsinh 2
m2 (a, s, d) − 1 + arcsinh 2
m2 (a, s, −d) − 1 , (7.28)
where m = m− if D ≤ 0, and m = m+ otherwise (one has m+ = m− for D = 0).
Here:
m− =
|k−|
(s − d) 2 − 1 ,
2 2a
m+ =
2 (1 + 2s2 + 2d2 ) − (4s2 − 1)(4d2 − 1) − a4 − √
δ
4(s − d)
,
D = 2(s − d) − 2 k2 − + 2k+ + |k−|(k2 − + 8k+) 1/2 /k+ ,
k± = a 2 ± (s + d) 2 , (7.29)
and the quantity δ = (a − 2d − 1)(a − 2d + 1)(a + 2d − 1)(a + 2d + 1)(a − 2s − 1)(a −
2s + 1)(a + 2s − 1)(a + 2s + 1) is the same as in Eq. (7.12). Note (we omitted the
explicit dependence for brevity) that each quantity in Eq. (7.29) is a function of
(a, s, d). Therefore, to evaluate the second term in Eq. (7.28) each d in Eq. (7.29)
must be replaced by −d.
Studying the derivative of m∓ with respect to s, it is analytically proven that,
in the whole range of parameters {a, s, d} defined by Ineq. (7.27), both m− and m+
are monotonically decreasing functions of s. The quantity Q is then maximized
over s for the limiting value
s = s min a + 1
≡ . (7.30)
2
This value of s corresponds to three-mode pure Gaussian states in which the state
of the reduced bipartition 2|3 is always separable, as one should expect because
the bipartite entanglement is maximally concentrated in the states of the 1|2 and
1|3 reduced bipartitions. With the position Eq. (7.30), the quantity D defined in
Eq. (7.29) can be easily shown to be always negative. Therefore, for both reduced
CMs σ12 and σ13, the Gaussian contangle is defined in terms of m−. The latter,
in turn, acquires the simple form
m−(a, s min 1 + 3a + 2d
, d) = .
3 + a − 2d
(7.31)
Consequently, the quantity Q turns out to be an even and convex function of d,
and this fact entails that it is globally maximized at the boundary
|d| = d max a − 1
≡ .
2
(7.32)
We finally have that
Q max ≡ Q a, s = s min , d = ±d max
= arcsinh 2 a 2 − 1 , (7.33)
which implies that in this case the sharing inequality (6.2) is exactly saturated and
the genuine tripartite entanglement is consequently zero. In fact this case yields