ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

6.2. Monogamy of distributed entanglement in N-mode Gaussian states 119
diag{a, a}, βl = diag{b, b}, and γl = diag{c+, c−}, where c+ ≥ |c−| [218, 70]. The
uncertainty condition 2.19 for σS1|Sl is thus equivalent to the following inequalities
[see also Eq. (4.2)]
a ≥ 1 , b ≥ 1 , ab − c 2 ± ≥ 1 ; (6.29)
Det σ S1|Sl + 1 = (ab − c2 +)(ab − c 2 −) + 1 ≥ a 2 + b 2 + 2c+c− . (6.30)
Furthermore, since the state ϱS1|Sl is entangled, we have [218]
(ab − c 2 +)(ab − c 2 −) + 1 < a 2 + b 2 − 2c+c− . (6.31)
From Eqs. (6.30) and (6.31), it follows that c− < 0.
In Eq. (6.28), τG(σ p
S1|Sl ) = f(4Det αp − 4), which is an increasing function of
, see Eq. (6.23).
The infimum of the right-hand side of Eq. (6.28) is achieved by the pure-state CM
σ p
(with σp
S1|Sl S1|Sl ≤ σS1|Sl and σp
S1|Sl + iΩ ≥ 0) that minimizes Det αp . The
minimum value of Det αp is given by
Det α p , where α p is the first 2 × 2 principal submatrix of σ p
S1:Sl
min
0≤θ