Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
If one of the states is pure, as in our case, this expression reduces to f(ρ1, ρ2) =
tr[ρ1 ρ2]. We employ this functional to quantify the quality of the clones with respect
to the input state.
For example, we could require that the overlap between the joint output T∗(ρ) of
the cloner and a tensor product of n perfect copies of the input state ρ becomes as
large as possible. This is accomplished by maximizing the joint fidelity
fjoint(T, ρ) = tr T∗(ρ)ρ ⊗n . (3.1)
Since this criterion compares the complete output state, including correlations between
subsystems, with a tensor product state, which is noncorrelated, it might be
too strong. Instead, one could measure the quality of individual clones by comparing
a single output subsystem, e.g. the i-th, to the input state with an appropriate
fidelity expression:
fi(T, ρ) = tr T∗(ρ)( ⊗ · · · ⊗ ⊗ ρ (i) ⊗ · · · ⊗ ) , (3.2)
where the upper index (i) indicates the position in the tensor product. However, a
single suchone-clone-onlyfidelity could be trivially put to one by a cloner which
does essentially nothing, but merely returns the input state in the i-th subsystem of
the output and yields a suitable fixed stated for the other subsystems. So, optimizing
the fidelities for all i in sequence would result in different cloners for each fidelity. To
avoid this, we optimize over a weighted sum of such fidelities, n i=1 λi fi(T, ρ) with
positive weights λi. The relative weights determine which clones should resemble the
input state more closely and thus allow to describe nonsymmetric cloners.
For a similar reason it is not useful to optimize the cloner for each input state
separately, because that would yield a source which perfectly produces the respective
quantum state. Instead, we can either consider the average or the worst-case
quality with respect to an ensemble of input states. However, both approaches face
conceptual difficulties. In the first case, the process of averaging over the pure Gaussian
states is not well defined, because this amounts to averaging over the group
Sp(2n,Ê) of symplectic transformations, which is noncompact. In the latter case,
for every given cloner squeezed states exist which for sufficiently large squeezing
bring the fidelity arbitrarily close to zero (see end of Section 3.4.1). While this can
in principle be compensated for a fixed and known squeezing by a modified cloner
(desqueeze, clone unsqueezed state and resqueeze output), it is not possible to circumvent
this behavior for arbitrary, unknown squeezing. We address the problem by
optimizing the cloner only for coherent states, which constitute a subset of all pure
Gaussian states.
As the figure of merit, we choose the worst-case fidelities fjoint(T) and fi(T),
defined as the infima of (3.1) and (3.2) over the set coh = |ξ〉〈ξ|
ξ ∈ Ξin of all
coherent states,
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fjoint(T) = inf
ρ∈coh fjoint(T, ρ) and fi(T) = inf
ρ∈coh fi(T, ρ).