ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

12.2. Equivalence between entanglement and optimal teleportation fidelity 197
12.2. Equivalence between entanglement in symmetric Gaussian resource
states and optimal nonclassical teleportation fidelity
The original CV teleportation protocol [39] has been generalized to a multi-user
teleportation network requiring multiparty entangled Gaussian states in Ref. [236].
The tripartite instance of such a network has been recently experimentally demonstrated
by exploiting three-mode squeezed states, yielding a maximal fidelity of
F = 0.64 ± 0.02 [277].
Here, based on Ref. [GA9], we investigate the relation between the fidelity of
a CV teleportation experiment and the entanglement present in the shared resource
Gaussian states. We find in particular that, while all the states belonging
to the same local-equivalence class (i.e. convertible into each other by local unitary
operations) are undistinguishable from the point of view of their entanglement
properties, they generally behave differently when employed in quantum information
and communication processes, for which the local properties such as the optical
phase reference get relevant. Hence we show that the optimal teleportation
fidelity, maximized over all local single-mode unitary operations (at fixed amounts
of noise and entanglement in the resource), is necessary and sufficient for the presence
of bipartite (multipartite) entanglement in two-mode (multimode) Gaussian
states employed as shared resources. Moreover, the optimal fidelity allows for the
quantitative definition of the entanglement of teleportation, an operative estimator
of bipartite (multipartite) entanglement in CV systems. Remarkably, in the
multi-user instance, the optimal shared entanglement is exactly the “localizable entanglement”,
originally introduced for spin systems [248] (not to be confused with
the unitarily localizable entanglement of bisymmetric Gaussian states, discussed in
Chapter 5), which thus acquires for Gaussian states a suggestive operative meaning
in terms of teleportation processes. Moreover, let us recall that our previous study
on CV entanglement sharing led to the definition of the residual Gaussian contangle,
Eq. (7.36), as a tripartite entanglement monotone under Gaussian LOCC
for three-mode Gaussian states [GA10] (see Sec. 7.2.2). This measure too is here
operationally interpreted via the success of a three-party teleportation network.
Besides these fundamental theoretical results, our findings are of important
practical interest, as they answer the experimental need for the best preparation
recipe for entangled squeezed resources, in order to implement CV teleportation
(in the most general setting) with the highest fidelity. It is indeed crucial in view
of experimental implementations, to provide optimal ways to engineer quantum
correlations, such that they are not wasted but optimally exploited for the specifical
task to be realized. We can see that this was the leitmotiv of the previous Chapter
as well.
We will now detail the results obtained in Ref. [GA9], starting with the twoparty
teleportation instance, and then facing with the general (and more interesting)
N-party teleportation network scenario. Notice that, by the defining structure
itself of the protocols under consideration, the employed resources will be (both in
the two-party and in the general N-party case) fully symmetric, generally mixed
Gaussian states (see Sec. 2.4.3). Therefore the equivalence between optimal nonclassical
fidelity and entanglement strictly holds only for fully symmetric Gaussian
resources. We will discuss this thoroughly in the following, and show how this interesting
connection actually is not valid anymore for nonsymmetric, even two-mode
resources.