ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

12.3. 1 ➝ 2 telecloning with bisymmetric and nonsymmetric three-mode resources 215
a3
1
2
3
4
5
1 2 3 4 5
a2
0.5
0.6
0.7
0.8
1
0.9
Figure 12.8. Fidelities for asymmetric telecloning with three-mode pure
Gaussian resources, at a fixed a1 = 2, as functions of a2 and a3, varying in the
allowed range of parameters constrained by Ineq. (7.17) (see also Fig. 7.1). The
darker surface on the right-hand side of the diagonal a2 = a3 (along which the
two surfaces intersect) is the fidelity of Bob’s clone, F 1→2
asym:2 , while the lighter,
‘mirror-reflected’ surface on the left-hand side of the diagonal is the fidelity of
Claire’s clone, F 1→2
asym:3 . Only nonclassical fidelities (i.e. F > 1/2) are shown.
12.3.3. Asymmetric telecloning
We focus now on the asymmetric telecloning of coherent states, through generic pure
three-mode Gaussian states shared as resources among the three parties. Consid-
ering states in standard form, Eq. (7.19) (see Sec. 7.1.2), parametrized by the local
single-mode mixednesses ai of modes i = 1, 2, 3, the fidelity F 1→2
asym:2 of Bob’s clone
(employing the 1|2 two-mode reduced resource) can be computed from Eq. (12.4)
and reads
F 1→2
asym:2 = 2
− 2a 2 3 + 2a1a2 + 4 (a1 + a2) + 3 a 2 1 + a 2 2
− (a1 + a2 + 2)
[(a1 + a2 − a3) 2 − 1][(a1 + a2 + a3) 2 − 1]
+ 2
a1a2
(12.33)
− 1
2
Similarly, the fidelity F 1→2
asym:3 of Claire’s clone can be obtained from Eq. (12.33) by
exchanging the roles of “2” and “3”.
It is of great interest to explore the space of parameters {a1, a2, a3} in order
to find out which three-mode states allow for an asymmetric telecloning with the
fidelity of one clone above the symmetric threshold of 2/3, while keeping the fidelity
of the other clone above the classical threshold of 1/2. Let us keep a1 fixed. With
increasing difference between a2 and a3, one of the two telecloning fidelities increases
at the detriment of the other, while with increasing sum a2 + a3 both fidelities
decrease to fall eventually below the classical threshold, as shown in Fig. 12.8. The
asymmetric telecloning is thus optimal when the sum of the two local mixednesses of
,