ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

212 12. Multiparty quantum communication with Gaussian resources
Depending on the symmetries of the shared resource, the telecloning can be realized
with equal fidelities for all receivers (symmetric telecloning) or with unbalanced
fidelities among the different receivers (asymmetric telecloning). In particular, in
the first case, the needed resource must have complete invariance under mode permutations
in the N-mode block distributed among the receivers: the resource state
has to be thus a 1 × N bisymmetric state [GA4, GA5] (see Sec. 2.4.3 and Chapter
5).
Here, based on Ref. [GA16], we specialize on 1 → 2 telecloning, where Alice,
Bob and Claire share a tripartite entangled three-mode Gaussian state and Alice
wants to teleport arbitrary coherent states to Bob and Claire with certain fidelities.
As the process itself suggests, the crucial resource enabling telecloning is not the
genuine tripartite entanglement (needed instead for a successful ‘multidirectional’
teleportation network, as shown in the previous Section), but the couplewise entanglement
between the pair of modes 1|2 and 1|3. We are assuming that the sender
(Alice) owns mode 1, while the receivers (Bob and Claire) own modes 2 and 3.
12.3.2. Symmetric telecloning
Let us first analyze the case of symmetric telecloning, occurring when Alice aims
at sending two copies of the original state with equal fidelities to Bob and Claire.
In this case it has been proven [51, 50, 238] that Alice can teleport an arbitrary
coherent state to the two distant twins Bob and Claire (employing a Gaussian
cloning machine) with the maximal fidelity
F 1→2
max = 2
. (12.31)
3
Very recently, unconditional symmetric 1 → 2 telecloning of unknown coherent
states has been demonstrated experimentally [134], with a fidelity for each clone of
F = 0.58 ± 0.01, surpassing the classical threshold of 0.5, Eq. (12.1).
The argument accompanying Eq. (12.31) inspired the introduction of the ‘nocloning
threshold’ for two-party teleportation [104], basically stating that only a
fidelity exceeding 2/3 — thus greater than the previously introduced “classical”
threshold of 1/2, Eq. (12.1), which implies the presence of entanglement — ensures
the realization of actual two-party quantum teleportation of a coherent state. In
fact, if the fidelity falls in the range 1/2 < F < 2/3, then Alice could have kept
a better copy of the input state for herself, or sent it to a ‘malicious’ Claire. In
this latter case, the whole process would result into an asymmetric telecloning,
with a fidelity F > 2/3 for the copy received by Claire. It is worth remarking
that two-party CV teleportation beyond the no-cloning threshold has been recently
demonstrated experimentally, with a fidelity F = 0.70±0.02 [226]. Another important
and surprising remark is that the fidelity of 1 → 2 cloning of coherent states,
given by Eq. (12.31), is not the optimal one. As recently shown in Ref. [52], using
non-Gaussian operations as well, two identical copies of an arbitrary coherent state
can be obtained with optimal single-clone fidelity F ≈ 0.6826.
In our setting, dealing with Gaussian states and Gaussian operations only,
Eq. (12.31) represents the maximum achievable success for symmetric 1 → 2 telecloning
of coherent states. As previously anticipated, the basset hound states σ p
B of
Sec. 7.4.3 are the best suited resource states for symmetric telecloning. Such states
belong to the family of multiuser quantum channels introduced in Ref. [238], and
are 1 × 2 bisymmetric pure states (see Fig. 5.1), parametrized by the single-mode