ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

12.2. Equivalence between entanglement and optimal teleportation fidelity 205
Figure 12.4. Localizable entanglement in the sense of [248]. By optimal local
measurements on N − 2 subsystems in a N-party system, a highly entangled
two-mode state is (probabilistically, in principle) obtained between the two
non-measuring parties. For Gaussian states and measurements the localization
process in indeed deterministic, as the entanglement properties of the resulting
states are independent of the measurement outcomes.
The resulting optimal two-mode state σ loc contains a localized entanglement
which is exactly quantified by the quantity
˜ν loc
− ≡ ˜ν (N)
− .
It is now clear that ˜ν (N)
− of Eq. (12.25) is a proper symplectic eigenvalue, being
the smallest one of the partial transpose ˜σ loc of the optimal two-mode state σloc that can be extracted from a N-party entangled resource by local measurements on
the remaining modes (see Fig. 12.4). Eq. (12.25) thus provides a bright connection
between two operative aspects of multipartite entanglement in CV systems: the
maximal fidelity achievable in a multi-user teleportation network [236], and the CV
localizable entanglement [248].
This results yield quite naturally a direct operative way to quantify multipartite
entanglement in N-mode (mixed) symmetric Gaussian states, in terms of the socalled
Entanglement of Teleportation, defined as the normalized optimal fidelity
[GA9]
E (N)
T
≡ max
0,
F opt
N
− Fcl
1 − Fcl
= max
0,
1 − ˜ν(N) −
1 + ˜ν (N)
, (12.26)
−
and thus ranging from 0 (separable states) to 1 (CV generalized GHZ state). The
localizable entanglement of formation Eloc F of N-mode symmetric Gaussian states
σ of the form Eq. (2.60) is a monotonically increasing function of E (N)
T , namely:
E loc
1 − E
F (σ) = h
(N)
T
1 + E (N)
, (12.27)
T
with h(x) given by Eq. (4.18). For N = 2 the state is already localized and E loc
F ≡
EF , Eq. (12.16)
In the next subsection we will see how the entanglement of teleportation relates,
for three-mode states, to the residual Gaussian contangle introduced in Sec. 7.2.2.