ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

118 6. Gaussian entanglement sharing
From Eq. (6.16), the Gaussian tangle for the pure Gaussian state ϱS1|S2...SN is
then written as
N
τG(ϱS1|S2...SN ) = w(Det α) ≡ f Υl , (6.25)
with f(t) = (g −1 (t) − 1/2) 2 , g(t) = √ t + 4 − √ t . (6.26)
We observe that f(t)/t is an increasing function for t > 0 and f(0) = 0 so f is a
star-shaped function: f(ct) ≤ cf(t) for c ∈ [0, 1] and t ≥ 0. 14 Therefore, we have
f(t) ≤ t
s
t+sf(t+s) and f(s) ≤ t+sf(t+s) for t, s ≥ 0 to obtain f(t)+f(s) ≤ f(t+s).
That is, f is superadditive [149]. Hence, 15
N
N
f ≥ f(Υl). (6.27)
l=2
Υl
Each term in the right-hand side is well defined since Υl > 0, Eq. (6.24).
We are now left to compute the right-hand side of Eq. (6.21), i.e. the bipartite
entanglement in the reduced (mixed) two-mode states ϱS1|Sl (l = 2, . . . , N). We
will show that the corresponding Gaussian tangle is bounded from above by f(Υl),
which will therefore prove the monogamy inequality via Eq. (6.27). To this aim,
we recall that any bipartite and multipartite entanglement in a Gaussian state is
fully specified in terms of its CM, as the displacement vector of first moments can
be always set to zero by local unitary operations, which preserve entanglement by
definition. It is thus convenient to express the Gaussian tangle directly in terms of
the CMs. Recalling the framework of Gaussian entanglement measures (Sec. 3.2.2),
the definition (6.15) given in Sec. 6.1.2.1 for the Gaussian tangle of a mixed Gaussian
state with CM σS1|Sl can be rewritten as
τG(σS1|Sl , (6.28)
) = inf
σ p
S 1 |S l
l=2
where the infimum is taken over all CMs σ p
S1|Sl
l=2
τG(σ p
S1|Sl )|σp
S1:Sl ≤ σ S1|Sl
of pure Gaussian states such that
σS1|Sl ≥ σp , see Eq. (3.11).
S1|Sl
The quantities Υl, Eq. (6.24), and τG(σS1:Sl ) for any l, as well as every single-
⊕N
mode reduced determinant, are Sp (2,) -invariants, i.e. they are preserved under
local unitary (symplectic at the CM level) operations, as mentioned in Sec. 2.2.2.
For each two-mode partition described by Eq. (6.23), we can exploit such localunitary
freedom to put the CM σS1:Sl in standard form, Eq. (4.1),16 with α =
14
f(t) is convex for t ≥ 0, which also implies that f is star-shaped.
15If we chose to quantify entanglement in terms of the contangle (rather than of the Gaussian
tangle), defined for pure Gaussian states as the squared logarithmic negativity Eq. (6.5), we would
have, instead of f(t) in Eq. (6.26), the quantity log2 [g(t)/2] which lacks the star-shape property. It
can be confirmed numerically that the function log2 [g(t)/2] (t ≥ 0) is not superadditive. However
this does not imply the failure of the N-mode monogamy inequality for the contangle [GA10],
which might be proven with different techniques than those employed here.
16
The reduced two-mode CMs σS1|S cannot be all brought simultaneously in standard
l
form, as clarified in Appendix A.2.1. However, our argument runs as follows [GA15]. We apply
Sp (2,) ⊕ Sp (2,) operations on subsystems S1 and S2 to bring σS1|S2 in standard form, evaluate
an upper bound on the Gaussian tangle in this representation, and derive an inequality between
local-unitary invariants, Eq. (6.37), that is therefore not relying on the specific standard form in
which explicit calculations are performed. We then repeat such computation for the remaining
matrices σS1|S with l = 3 . . . N. At each step, only a single two-mode CM is in standard
l