Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
Proposition 3.6:
The weighted single-copy fidelities for 1-to-n cloning of coherent states are optimized
by non-Gaussian cloners. For 1-to-2 cloning, the optimal symmetric cloner
yields fidelities f1,2 ≈ 0.6826. The optimal cloners are nonsingular except for the
cases f1,2 = 1. The best Gaussian 1-to-2 cloners are described by rotation invariant,
squeezed Gaussian wave functions. They are nonsingular for weight 1
5
and correspond to the singular cloners beyond this regime. In the symmetric case
λ = 1
. The best symmetric Gaussian 1-to-n cloners
2 , the fidelities are f1,2 = 2
3
yield fidelities fi = (2 − 1/n) −1 .
Numerical optimization
< λ < 4
5
In order to approximately calculate the largest eigenvalue of F in (3.29b), we numerically
compute the expectation value 〈φn| F |φn〉 of F in a state obtained from
the iteration φn+1 = F φn/F φn. This power iteration effectively suppresses the
parts of φ0 outside the eigenspace to the largest eigenvalue, so 〈φn| F |φn〉 approximates
the largest eigenvalue of F. From the resulting function φn, the singlecopy
fidelities can be computed as the expectation values of the constituents of F,
f1 = 〈φn| e −(Q2 1 +Q2 2 )/2 |φn〉 and f2 = 〈φn| e −(P2 1 +P2 2 )/2 |φn〉. Varying the weight λ
yields the points on the solid curve in Fig. 3.2.
The starting point for the power iteration is a rotation invariant Gaussian function
φc(x, y) ∝ exp(−c (x 2 + y 2 )). The squeezing value c is taken from the optimal
Gaussian cloner where available, i.e. for 0.2 < λ < 0.8 (see the discussion of optimal
Gaussian 1-to-2 cloners below). Samples for the solid curve in Fig. 3.2 from
this regime are taken in the interval 0.25 ≤ λ ≤ 0.75 with increment 0.05 and an
iteration depth of eight steps. Alternatively, we start from the state φc resulting
from the iteration for λ = 0.79 with nine steps and scale the squeezing parameter
c by a heuristically determined factor of (− log l) 9 . Sampling the parameter l for
0.24 ≤ l ≤ 0.36 and 0.64 ≤ l ≤ 0.76 with increment 0.02 yields further points on the
outskirts of the curve. The fidelity pairs obtained by this method are well separated
from the points representing the singular cloners and the iteration does not tend towards
a singular state. Moreover, the eigenstate to the largest eigenvector is a pure
state with wave function φ(x). It is unique by the following argument: Both operators
exp −(Q 2 1 + Q 2 2)/2 and exp −(P 2 1 + P 2 2 )/2 correspond to positive integral
kernels, hence replacing any wave function ψ(x) by |ψ(x)| yields larger expectation
values while preserving the norm. Assume two states ψ1(x) ≥ 0 and ψ2(x) ≥ 0
were both eigenstates to the largest eigenvalue. Then so is any linear combination
p1 ψ1(x) − p2 ψ2(x). However, since |p1 ψ1(x) − p2 ψ2(x)| yields larger expectation
values, the conclusion is ψ1(x) ≡ ψ2(x) and the eigenstate to the largest eigenvector
is unique.
Addendum: On suggestion of a referee, we complement this discussion with more
details. Note that the following paragraphs have been added after acceptance of the
thesis.
50
All expressions arising in the iteration φn+1 = F φn/F φn have been ob-