Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
By virtue of this lemma, the fidelity bound for classical cloning also applies to
cloners which are not purely classical but make use of supplemental ppt-bound
entangled states to link measurement and preparation. However, assisting the process
with non-ppt entanglement can result in substantially higher fidelities, as this
operation describes the teleportation of coherent states [58, 59]. Our derivation of
the limit (3.34) thus proves and extends a success criterion for continuous-variable
teleportation [63,64], cf. Section 3.6. As the result of this section, we obtain
Proposition 3.9:
Classical cloning of coherent states realized by measuring the input state and
repreparing output states depending on the results is limited to fidelities f ≤ 1
2 .
Supplemental ppt-bound entangled states do not improve this limit. The optimal
cloner is Gaussian and covariant.
For the case of an unassisted measure-and-prepare scheme, an independent proof
has been given in [46]. In Fig.3.2, the achievable fidelities for classical cloners lie in
the lower left quadrant with f1 ≤ 1
2 and f2 ≤ 1
2 .
3.4.4 Bosonic output
Symmetric cloners yield the same single-copy fidelity for each clone. It is an obvious
question if this implies further symmetries for the output state of the cloner. In
particular, the output might lie in the bosonic sector, i.e. be invariant under the
interchange of two clones. Note that this is not necessarily true since different states
for individual clones could lead to the same single-copy fidelity. We show below
that the output of symmetric covariant cloners belongs to the bosonic sector if the
cloner is described by a bosonic state. Moreover, this condition is met by all optimal
symmetric cloners considered in this chapter (cf. Proposition 3.11 below).
To formalize the statement, we introduce the flip operator(i,j) which acts on
vectors |ψ〉 ∈ H ⊗n by interchanging tensor factors i and j:
(i,j) |ψ1〉 ⊗ · · · ⊗ |ψi〉 ⊗ · · · ⊗ |ψj〉 ⊗ · · · ⊗ |ψn〉
= |ψ1〉 ⊗ · · · ⊗ |ψj〉 ⊗ · · · ⊗ |ψi〉 ⊗ · · · ⊗ |ψn〉,
where i, j ∈ {1, 2, . . ., n}. For i = j we define(i,i) = . Since (i,j) 2 = , the
eigenvalues of(i,j) are +1 and −1. A vector |ψ+〉 which belongs to the eigenspace
of +1 for all(i,j) describes a state which is invariant under interchange of subsystems,
i.e. a bosonic state. Similarly, the intersection of all eigenspaces of −1 for the
(i,j) with i = j contains the fermionic states. 10 With this, we state the claim as
10 The intersections of all eigenspaces to eigenvalue +1 or to −1 of the flip operators are called
the bosonic or fermionicsectors, respectively.
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