Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
a
in
OPA
ψ
b
b 1 2
Figure 3.3:
Optical scheme of a displacement-covariant cloner. The input mode ain is injected
on the signal mode of an optical parametric amplifier (opa) of gain 2, the idler
mode being denoted as b1. After amplification, the signal mode is divided at a
balanced beam splitter (bs), resulting in two clones in modes a1 and a2. The
second input mode of the beam splitter is noted b2. If both b1 and b2 are initially in
the vacuum state, the corresponding cloner is the Gaussian cloner of [53,54,55]. In
contrast, if we inject a specific two-mode state |ψ〉 into b1 and b2, we can generate
the whole set of displacement-covariant cloners, in particular the non-Gaussian
optimal one. Picture and caption are taken from [a].
If the input state is the vacuum state |0〉〈0|, the single-copy fidelities are the expectation
values of the operators
F1 = e −(Q1+Q2)2 /4−(P1−P2) 2 /4 ,
F2 = e −(Q1−Q2)2 /4−(P1−P2) 2 /4
in the state |ψ〉〈ψ|. These operators differ from those in Eq.(3.29a) only by the
symplectic transformation which describes the action of a beam splitter, i.e. by the
mapping a1 ↦→ (a1 + a2)/ √ 2 and a2 ↦→ (a1 − a2)/ √ 2, resulting in
a
1
BS
Q1 ↦→ (Q1 + Q2)/ √ 2, P1 ↦→ (P1 − P2)/ √ 2 ,
Q2 ↦→ (Q1 − Q2)/ √ 2, P2 ↦→ (P1 + P2)/ √ 2 .
Cerf and Navez [a] argue that it is not necessary to implement the exact state
ρT = |ψ〉〈ψ| to get substantial improvements over the fidelities of a Gaussian cloner.
Already an approximation of the optimal state by a linear combination of a small
number of few-photon states yields fidelities which clearly exceed the Gaussian limit.
For example, the exact state for the symmetric cloner,
62
|ψ〉 =
∞
cn |2n〉|2n〉, (3.40)
n=0
a
2