ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

130 7. Tripartite entanglement in three-mode Gaussian states
states with a2 = a1 and a3 = 1 (if d = dmax ), or a3 = a1 and a2 = 1 (if d = −dmax ),
i.e. tensor products of a two-mode squeezed state and a single-mode uncorrelated
vacuum. Being Qmax from Eq. (7.33) the global maximum of Q, Ineq. (7.23) holds
true and the monogamy inequality (6.2) is thus proven for any pure three-mode
Gaussian state, choosing either the Gaussian contangle Gτ or the true contangle
Eτ as measures of bipartite entanglement [GA10].
The proof immediately extends to all mixed three-mode Gaussian states σ,
but only if the bipartite entanglement is measured by Gτ (σ). 17 Let {π(dσp m), σp m}
be the ensemble of pure Gaussian states minimizing the Gaussian convex roof in
Eq. (6.9); then, we have
G i|(jk)
τ (σ) =
≥
π(dσ p m)G i|(jk)
τ (σ p m)
π(dσ p m)[G i|j
τ (σ p m) + G i|k
τ (σ p m)] (7.34)
≥ G i|j
τ (σ) + G i|k
τ (σ) ,
where we exploited the fact that the Gaussian contangle is convex by construction.
This concludes the proof of the CKW monogamy inequality (6.2) for all three-mode
Gaussian states.
The above proof, as more than once remarked, implies the corresponding
monogamy proof for all three-mode Gaussian states by using the Gaussian tangle
Eq. (6.16) as a bipartite entanglement monotone. Monogamy of the Gaussian
tangle for all N-mode Gaussian states has been established in Sec. 6.2.3 [GA15].
7.2.2. Residual contangle and genuine tripartite entanglement
The sharing constraint leads naturally to the definition of the residual contangle
as a quantifier of genuine tripartite entanglement in three-mode Gaussian states,
much in the same way as in systems of three qubits [59] (see Sec. 1.4.3). However,
at variance with the three-qubit case (where the residual tangle of pure states
is invariant under qubit permutations), here the residual contangle is partitiondependent
according to the choice of the probe mode, with the obvious exception
of the fully symmetric states. A bona fide quantification of tripartite entanglement
is then provided by the minimum residual contangle [GA10]
E i|j|k
τ ≡ min E
(i,j,k)
i|(jk)
τ − E i|j
τ − E i|k
τ , (7.35)
where the symbol (i, j, k) denotes all the permutations of the three mode indexes.
This definition ensures that E i|j|k
τ is invariant under all permutations of the modes
and is thus a genuine three-way property of any three-mode Gaussian state. We
can adopt an analogous definition for the minimum residual Gaussian contangle
G res
τ , sometimes referred to as arravogliament [GA10, GA11, GA16] (see Fig. 7.2 for
a pictorial representation):
G res
τ
≡ G i|j|k
τ
≡ min G
(i,j,k)
i|(jk)
τ − G i|j
τ − G i|k
τ . (7.36)
17 If σ is decomposed into pure non-Gaussian states, it is not known at the present stage
whether the CKW monogamy inequality Eq. (6.2) is satisfied by each of them.