ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

116 6. Gaussian entanglement sharing
How does this entanglement distribution mechanism work? We show here,
based on a simple computation first appeared in [GA10], that it obeys the fundamental
monogamy law.
We will employ the Gaussian contangle Gτ , Eq. (6.13), as a measure of bipartite
entanglement. Due to the convex roof extension involved in the definition of Gτ ,
Eq. (6.9), it will suffice to prove monogamy for pure fully symmetric Gaussian states,
for which the Gaussian contangle Gτ coincides with the true contangle Eτ in every
bipartition thanks to the symmetry of the considered states (see Sec. 6.1.2.1). Such
a proof will also imply the corresponding monogamy property for the (Gaussian)
tangle, Eq. (6.16).
For a pure, fully symmetric Gaussian states of N + 1 modes, we will then prove
the statement
E i|(j1,...,jN )
τ
−
N
l=1
E i|jl
τ ≥ 0 , (6.18)
by induction. Referring to the notation of Eqs. (2.60, 5.13) with L ≡ N, for any N
and for b > 1 (for b = 1 one has a product state), the 1 × N contangle
E i|(j1,...,jN )
τ = log 2 (b − b 2 − 1) (6.19)
is independent of N, while the total two-mode contangle
NE i|jl
τ = N
4 log2
1
N
b 2 (N + 1) − 1
− (b 2 − 1)(b 2 (N + 1) 2 − (N − 1) 2 )
, (6.20)
is a monotonically decreasing function of the integer N at fixed b.
Because the sharing inequality trivially holds for N = 1, it is inductively proven
for any N.
Entanglement — specifically measured by any CV extension of the tangle as
introduced in Sec. 6.1.2.1 — in all (pure and mixed) fully symmetric Gaussian
states, defined in Sec. 2.4.3, is indeed monogamous.
We can now study the difference between left-hand and right-hand sides in
Eq. (6.18), quantifying the residual entanglement not stored in bipartite correlations
between any two modes. As apparent from the above proof, this residual entanglement,
for any squeezing b, strictly grows with N. This proves quantitatively
that (as qualitatively clear from the analysis of Sec. 5.2) with increasing number
of modes N, the entanglement is gradually less encoded in pairwise bipartite form,
being conversely more and more increasingly retrieved in multipartite (specifically,
three-partite, four-partite, . . ., N-partite) quantum correlations among all the single
modes. Crucially, we have just proven that this redistribution is always such
that a general monogamy law on the shared CV entanglement is satisfied.
The multipartite entanglement in (generally mixed) fully symmetric Gaussian
states will be operationally interpreted in terms of optimal success of CV N-party
teleportation networks in Sec. 12.2.
6.2.3. Monogamy inequality for all Gaussian states
Following Ref. [GA15], we state here the crucial result which definitely solves the
qualitative problem of entanglement sharing in Gaussian states. Namely, we prove
that the monogamy inequality does hold for all Gaussian states of multimode CV