ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

124 7. Tripartite entanglement in three-mode Gaussian states
no generality is lost in assuming a standard form CM, because the entanglement
properties of any bipartition of the system are invariant under local (single-mode)
symplectic operations. Now, Eqs. (7.3) and (7.2) may be recast as follows
a 2 3 = a 2 1 + a 2 2 + 2c12d12 − 1 , (7.8)
a 2 3 = (a1a2 − c 2 12)(a1a2 − d 2 12) , (7.9)
showing that we may eliminate one of the two covariances to find the expression
of the remaining one only in terms of the three local mixednesses (inverse purities)
al, Eq. (7.5). Defining the quantity κ as
κ ≡ c12d12 = 1 + a23 − a2 1 − a2 2
,
2
leads to the following condition on the covariance c12,
(7.10)
c 4 12 − 1 2 2
(κ − 1) + a1a 2 2 − a 2 1 − a 2 2
2 c12 + κ 2 = 0 . (7.11)
a1a2
Such a second-order algebraic equation for c 2 12 admits a positive solution if and only
if its discriminant δ is positive,
δ ≥ 0 . (7.12)
After some algebra, one finds
δ = (a1 + a2 + a3 + 1)(a1 + a2 + a3 − 1)
× (a1 + a2 − a3 + 1)(a1 − a2 + a3 + 1)
× (−a1 + a2 + a3 + 1)(a1 + a2 − a3 − 1)
× (a1 − a2 + a3 − 1)(−a1 + a2 + a3 − 1) . (7.13)
Aside from the existence of a real covariance c12, the further condition of positivity
of σ12 has to be fulfilled for a state to be physical. This amounts to impose the
inequality a1a2 − c 2 12 ≥ 0, which can be explicitly written, after solving Eq. (7.11),
as
4 2a 2 1a 2 2 − (κ − 1) 2 + a 2 1a 2 2 − a 2 1 − a 2 √
2 ≥ δ .
This inequality is trivially satisfied when squared on both sides; therefore it reduces
to
2a 2 1a 2 2 − (κ − 1) 2 + a 2 1a 2 2 − a 2 1 − a 2 2 ≥ 0 . (7.14)
Notice that conditions (7.12) and (7.14), although derived by assuming a specific
bipartition of the three modes, are independent on the choice of the modes
that enter in the considered bipartition, because they are invariant under all possible
permutations of the modes. Defining the parameters
a ′ l ≡ al − 1 , (7.15)
the uncertainty principle Eq. (2.35) for single-mode states reduces to
a ′ l ≥ 0 ∀ l = 1, 2, 3 . (7.16)
This fact allows to greatly simplify the existence conditions (7.12) and (7.14), which
can be combined into the following triangular inequality
|a ′ i − a ′ j| ≤ a ′ k ≤ a ′ i + a ′ j . (7.17)
Inequality (7.17) is a condition invariant under all possible permutations of the
mode indexes {i, j, k}, and, together with the positivity of each a ′ l , fully characterizes
the local symplectic eigenvalues of the CM of three-mode pure Gaussian