Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
Replacing t(ξ) in (3.27) and letting ξi = (q, p), the i-th single-copy fidelity for the
fixed input state |0〉〈0| is fi = tr[ρT Fi], where
dq dp
Fi = χin(q, p)
2π
2
exp i (p Pi − q
i=j Qj)
= exp −P 2
i /2 −
i=j Q2 j /2 . (3.28)
In the following, we study the weighted single-copy fidelities
i λi fi by numerical
computation and analytical arguments. For simplicity, we restrict this discussion
to the case of 1-to-2 cloning. While in principle the method can be generalized,
numerical computations of the fidelities might get more involved. The operator F
for the weighted single-copy fidelity λf1(T) + (1 − λ)f2(T) = tr[ρT F] is composed
of the weighted sum of the respective Fi:
F = λ1 e −(Q2
2 +P2
1 )/2 + λ2 e −(Q2
1 +P2
2 )/2
(3.29a)
≃ λ1 e −(Q2
1 +Q2
2 )/2 + λ2 e −(P2
1 +P2
2 )/2 , (3.29b)
where the second expression is obtained by applying an orthogonal, symplectic transformation
such that Q1 ↦→ −P1 and P1 ↦→ Q1. Both forms are equivalent for the purpose
of computing eigenvalues. The largest eigenvalue of F gives the maximal singlecopy
fidelity, the corresponding eigenvector describes the optimal cloner. Before we
detail their approximate computation, we discuss the results depicted in Fig. 3.2.
Since a linear combination of Gaussian operators as in (3.29b) does in general
not have Gaussian eigenfunctions, the optimal cloners are not Gaussian. In fact,
comparing the optimal symmetric cloner yielding f1 = f2 ≈ 0.6826 with the best
Gaussian cloner (see [53,54,55] and below), limited to f1 = f2 = 2
3 , already indicates
the enhancement in fidelity by non-Gaussian cloners. A more detailed study of the
best Gaussian 1-to-2 cloners (see below) results in the dotted curve of fidelity pairs
in Fig. 3.2. Clearly, the non-Gaussian cloners perform better for every region of the
diagram. The two symmetric cloners can be found at the intersection of the dashdotted
diagonal with the dotted curve of best Gaussian cloners and the solid curve
of optimal cloners. At the points of the singular cloners with f1, f2 = 1, the solid
curve of optimal cloners has a nonfinite slope s = ∞ and s = 0, respectively, while
the dotted curve of the best Gaussian cloners has a finite slope (see in particular (b)
in Fig. 3.2). By the arguments of Section 3.2, this implies that the optimal cloners
for f1 = 1, f2 = 1 do not coincide with the singular cloners. In contrast, the best
Gaussian cloners for f1 ≈ 1 and f2 ≈ 1 are determined by the respective singular
cloners; see also below. The trivialcopy-throughcloners, which yield fidelity f1 = 1
or f2 = 1, are singular in any case by Corollary 3.3.
The following subsection gives details on the approximate, numerical computation
of the largest eigenvalue of F and the corresponding eigenfunctions. To complement
the results on optimal cloners, the last two subsections briefly investigate the best
Gaussian cloner for 1-to-2 cloning with arbitrary weights and for symmetric 1-to-n
cloning. Before this, we summarize the results in
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