Quantum Information Theory with Gaussian Systems

3.3 Covariance
In the case of joint fidelity, the respective expression contains only ωΛn(Fjoint) since
Fjoint = |α〉〈α| ⊗n is compact on all tensor factors.
To investigate the fidelities of a possibly singular, covariant cloner T∗, consider
the restriction ω = T∗(ρ)| D ⊗n of its output to D ⊗n . Denote by T∗Λ the map which
takes the density operator ρ on H to T∗Λ(ρ) = ωΛ, the unique density operator
on the tensor factors Λ from the decomposition of ω. Since T∗ is covariant, so is
T∗Λ. However, it lacks normalization, as only the overall T∗(ρ) is normalized. To
renormalize T∗Λ, we introduce the normalization operator NΛ which implements the
bounded linear map ρ ↦→ tr T∗Λ(ρ) = tr[ρ NΛ] ≤ 1. As T∗Λ is covariant, NΛ has to
commute with all Weyl operators and is thus a multiple of the identity, NΛ = pΛ
with 0 < pΛ ≤ 1. We define by
T∗Λ(ρ) = T∗Λ(ρ)/pΛ = ωΛ/pΛ
(3.14)
a family of normalized, covariant 1-to-|Λ| cloning transformations, where |Λ| denotes
the number of elements in the set Λ. Note that the normalization constant pΛ does
not depend on the input state. T∗Λ(ρ) is normal, since the output ωΛ/pΛ is a density
operator. With the help of T∗Λ, the fidelity of possibly singular cloners can be
expressed in terms of nonsingular cloners. For joint fidelity, we get:
fjoint T∗ = fjoint T∗, |0〉〈0|
by covariance of T∗
= ω(Fjoint) for ω = T∗ |0〉〈0|
= tr
ωΛn Fjoint
by (3.11), Fjoint is compact on Λn
= pΛn tr |0〉〈0| Fjoint by (3.14)
T∗Λn
= pΛn fjoint(TΛn) by (3.1).
Since 0 < pΛ ≤ 1, this fidelity is enlarged if pΛn = 1 and hence pΛ = 0 for Λ = Λn,
i.e. if T∗ = T∗Λn . But this better cloner is covariant and normal, which proves (ii)
and (i) for joint fidelity, where the linear combination consists of a single covariant
cloner which yields normal output for all clones.
For a proof of (iii), we discuss the role of zero and nonzero coefficients λi in the
weighted single-copy fidelity n i=1 λi fi. If one of the weights is zero, e.g. λn = 0,
the figure of merit does not care for the respective clone n. A 1-to-n cloner can thus
be optimized by using the optimal, covariant 1-to-(n − 1) cloner and amending the
output with an arbitrary state for the n-th output system. However, if this additional
state is a normal state, the resulting cloner is not covariant (see above). If this cloner
is subjected to the averaging procedure from Lemma 3.1, the averaged cloner will
be covariant and hence the state of the n-th clone in its output will be singular.
Consequentially, if a clone is not contained in the figure of merit, the optimal cloner
is either not covariant with respect to all clones or it is covariant but singular. This
proves (iii).
Consider now the single-copy fidelity with nonzero weights λi > 0. We denote the
respective fidelity operator by F(λ) =
i λi Fi, where λ = (λ1, λ2, . . .,λn). With
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