ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

12.3. 1 ➝ 2 telecloning with bisymmetric and nonsymmetric three-mode resources 217
In this case the fidelity of Claire’s clone saturates the classical threshold, F opt:1→2
asym:3 =
1/2, while the fidelity of Bob’s clone reaches F opt:1→2
asym:3 = 4/5, which is the maximum
allowed value for this setting [204]. Also, choosing t = 1/5, Bob’s fidelity gets
classical and Claire’s fidelity is maximal.
In general, a telecloning with F opt:1→2
asym:2
only in the window
1.26 ≈ 2 √ 2
2 −
≥ 2/3 and F opt:1→2
asym:3
≥ 1/2 is possible
1 + √
2 ≤ a ≤ 2 √
2 2 + 1 + √
2 ≈ 10.05 (12.38)
and, for each a falling in the region defined by Ineq. (12.38), in the specific range
a − 2 √ a + 1 + 2
≤ t ≤
a − 1
2 √ 2 √ a + 1 − 2
. (12.39)
a − 1
For instance, for a = 3, the optimal asymmetric telecloning (with Bob’s fidelity
above no-cloning and Claire’s fidelity above classical bound) is possible in the whole
range 1/2 ≤ t ≤ 2 √ 2 − 1, where the boundary t = 1/2 denotes the basset hound
state realizing optimal symmetric telecloning (see Fig. 12.7). The sum
S opt:1→2 = F opt:1→2
asym:2
+ F opt:1→2
asym:3
can be maximized as well, and the optimization is realized by values of a falling
in the range 2.36 a ≤ 3, depending on t. The absolute maximum of Sopt:1→2 is
reached, as expected, in the fully symmetric instance t = 1/2, a = 3, and yields
Sopt:1→2 max = 4/3.
We finally recall that optimal three-mode Gaussian resources, can be produced
by implementing the allotment operator (see Sec. 10.1.1) [GA16], and employed to
perform all-optical symmetric and asymmetric telecloning machines [238, 204].