ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

114 6. Gaussian entanglement sharing
used the fact that the smallest symplectic eigenvalue ˜ν− of the partial transpose
of a pure 1 × N Gaussian state σ p
i|j is equal to ˜ν− = √ Det σi − √ Det σi − 1, as
follows by recalling that the 1 × N entanglement is equivalent to a 1 × 1 entanglement
by virtue of the phase-space Schmidt decomposition (see Sec. 2.4.2.1) and by
exploiting Eq. (4.13) with ∆ = 2, µ = 1 and µ1 = µ2 ≡ 1/ √ Det σi.
The Gaussian contangle Gτ , like all members of the Gaussian entanglement
measures family (see Sec. 3.2.2) is completely equivalent to the Gaussian entanglement
of formation [270], which quantifies the cost of creating a given mixed
Gaussian state out of an ensemble of pure, entangled Gaussian states.
Gaussian tangle.— Analogously, for a 1×N bipartition associated to a pure Gaussian
state ϱ p
A|B = |ψ〉 A|B〈ψ| with SA = S1 (a subsystem of a single mode) and
SB = S2 . . . SN , we define the following quantity
τG(ϱ p
A|B ) = N 2 (ϱ p
A|B ). (6.14)
Here, N (ϱ) is the negativity, Eq. (1.40), of the Gaussian state ϱ. The functional
τG, like the negativity N , vanishes on separable states and does not increase under
LOCC, i.e. , it is a proper measure of pure-state bipartite entanglement. It can be
naturally extended to mixed Gaussian states ϱA|B via the convex roof construction
τG(ϱ A|B) = inf
{pi,ϱ (p)
i }
i
piτG(ϱ p
i ), (6.15)
where the infimum is taken over all convex decompositions of ϱA|B in terms of
pure Gaussian states ϱ p
i : ϱA|B = p
i piϱi . By virtue of the Gaussian convex roof
construction, the Gaussian entanglement measure τG Eq. (6.15) is an entanglement
monotone under Gaussian LOCC (see Sec. 3.2.2). Henceforth, given an arbitrary
N-mode Gaussian state ϱS1|S2...SN , we refer to τG, Eq. (6.15), as the Gaussian
tangle [GA15]. Obviously, in terms of CMs, the analogous of the definition (6.11) is
valid for the Gaussian tangle as well, yielding it computable like the contangle in
Eq. (6.13). Namely, exploiting Eq. (3.7), one finds
τG(σi|j) ≡ τG(σ opt
i|j ) = w[m2 1 √ √ 2
i|j ], w[x] = x − 1 + x − 1 . (6.16)
4
Refer to the discussion immediately after Eq. (6.13) for the definition of the quantities
involved in Eq. (6.16).
We will now proceed to investigate the entanglement sharing of Gaussian states
and to establish monogamy constraints on its distribution. We remark that, being
the (squared) negativity a monotonic and convex function of the (squared) logarithmic
negativity, see Eq. (6.7), the validity of any monogamy constraint on distributed
Gaussian entanglement using as an entanglement measure the “Gaussian tangle”, is
implied by the proof of the corresponding monogamy inequality obtained using the
“Gaussian contangle”. For this reasons, when possible, we will always employ as a
preferred choice the primitive entanglement monotone, represented by the (Gaussian)
contangle [GA10] (which could be generally referred to as a ‘logarithmic’ tangle
in quantum systems of arbitrary dimension).
6.2. Monogamy of distributed entanglement in N-mode Gaussian states
We are now in the position to prove a collection of results concerning the monogamy
of distributed Gaussian entanglement in multimode Gaussian states.