Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
area of achievable fidelity pairs. The optimal cloners yield fidelities corresponding to
points on thehigh fidelityrim of this set, schematically indicated by the arcs in
Fig.3.1. Any 1-to-2 cloning fidelity pair allowed by quantum physics can be reached
by classically mixing an optimal cloner with a fixed output state (depicted by the
dotted line). The fidelities beyond the curve of optimal cloners are not accessible
(indicated in Fig.3.1(a) by the shaded region).
An additional aspect of the interpretation of the diagrams is provided by the
tangents, depicted by the thin solid lines in Fig.3.1(b). Following from the total
weighted single-copy fidelity for 1-to-2 cloning, f = λf1 + (1 − λ)f2, all cloners
on the line f2 = f/(1 − λ) − f1 λ/(1 − λ) yield the total fidelity f for weight λ.
Conversely, a line with slope s = −λ/(1 − λ) and abscissa t = f/(1 − λ) comprises
all cloners yielding f for weight λ. Moving a line with slope s parallel to itself until
it touches the set fsc results in the optimal cloner for the corresponding weight (the
dot in Fig. 3.1(b) for λ = 1
2 ). Moreover, the slope of the tangent in (0, 1) and (1, 0)
conveys important information about the optimality of the trivial cloners, which
solely map the input state into one of the two output subsystems. If the line with
slope corresponding to some λ0 > 0 touches the curve of optimal cloners in (0, 1),
the optimal cloner for weight λ0 is the trivial cloner with (f1, f2) = (0, 1), degraded
by a fixed output state (f1, f2) = (0, 0) with weight λ0 and total fidelity f = (1−λ0).
This is illustrated in Fig.3.1(b). In contrast, Fig. 3.1(a) corresponds to a case where
the trivial cloners are optimal only for λ = 0 and λ = 1, since the tangent in the end
points of the arc is horizontal or vertical.
Since we show below that the optimal worst-case fidelities can be reached by
cloners which are covariant with respect to phase space translations of the input
state, we simultaneously optimize the average fidelities.
3.3 Covariance
In this section, we will show that for every cloner we can define a cloner which is
covariant with respect to translations of the input state in phase space and which
yields at least the same worst-case fidelity for coherent input states. For 1-to-n
cloning, such cloners are necessarily quasi-free, i.e. they map Weyl operators to
multiples of Weyl operators, and are essentially determined by a state on the output
ccr algebra.
A map on states is phase space covariant in the above sense if displacing the input
state in phase space gives the same result as displacing the output by the same
amount. If we define the shifted cloner Tξ by
T∗ξ(ρ) = W ⊗n∗
ξ T∗(Wξ ρ W ∗ ξ ) W ⊗n
ξ , (3.3)
translational covariance means T∗ξ(ρ) = T∗(ρ). Note that the same phase space
translation ξ is used for the input system as well as for all output subsystems. This
is justified from the intention to replicate the input state as closely as possible. Given
covariance of T∗ in the Schrödinger picture, the covariance of T in the Heisenberg picture
follows immediately: If T∗ is covariant with respect to phase space translations,
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