Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
Transformation Ω
In (Q, P)-block representation and with a square matrix (n)i,j = 1 for i, j =
1, 2, . . ., n we have
Σ(ξ, η) = ξ T
·
· η = σ(Ω ξ, Ω η) choosing 3
Ω =
0 n −n
n − n 0
0n − n
n
0
. (3.19)
For later use, we compute detΩ = (−1) n (det n)det(n − n). Since we will also
need the eigenvalues ofn, we more generally compute its characteristic polynomial
det(n − λ n). By inspection, we find the recursion relation
det(n − λ n) = (2 − λ − n) det(n−1 − λ n−1) + (n − 1)λ det(n−2 − λ n−2)
and prove by induction that
Letting λ = 1, this yields
The inverse of Ω is
3.4 Optimization
det(n − λ n) = (−1) n λ n (λ − n). (3.20)
Ω −1
=
detΩ = 1 − n . (3.21)
0 n
n/(n − 1) − n 0
. (3.22)
A key ingredient of our optimization method is the linearity of the fidelities in T and
hence in ρT. Using again the abbreviation f = fjoint or f =
i λi fi, we can thus
write the fidelity as the expectation value of a linear operator F in the state ρT:
f(T, ρ) = tr[ρT F] . (3.23)
The applicable operators F = Fjoint and F =
i λi Fi are obtained by expressing
the fidelity in terms of characteristic functions by noncommutative Fourier transform
and the Parseval relation (2.11), regrouping the factors and transforming back to
new operators4 ρT and F. The latter depends only on the symplectic geometry via
the transformation Ω mediating between symplectic forms, but not on the cloner T.
In principle, F also depends on the input state ρ. However, since we can restrict the
3 This choice is not unique, but can involve arbitrary symplectic transformations, i.e. S −1 Ω S for
S ∈ Sp(2n,Ê) is permissible, too.
4 A similar method has been used independently by Wódkiewicz et al. to obtain results on the
teleportation of continuous-variable systems [60] and the fidelity of Gaussian channels [61].
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