Quantum Information Theory with Gaussian Systems

3.3 Covariance
clone). A more general type of cloner yields normal output on some of the clones
and a singular state on the rest, i.e.
T∗(ρ) = (T∗Λ ⊗ T∗Λc)(ρ), (3.6)
where T∗Λ is a normal cloner on the clones indicated by Λ ⊂ {1, 2, . . ., n} and T∗Λc is
a purely singular cloner on the rest. The following lemma shows that cloners which
contain purely singular parts in the output are not optimal:
Lemma 3.2:
For covariant 1-to-n cloners optimized with respect to worst-case single-copy or
joint fidelity, the following holds:
(i) A cloner with a purely singular contribution to the output state cannot be
optimal. The optimal cloner is a linear combination of covariant cloners which
yield a normal state for some of the clones and purely singular output for the
rest, i.e. a linear combination of the cloners in (3.6).
(ii) For joint fidelity, the optimal cloner is normal.
(iii) If the cloner is to be covariant with respect to more clones than enter the
fidelity criterion, then the optimal cloner is singular.
Remark: If the weighted single-copy fidelity f = n
i=1 λi fi contains terms with
λi = 0, these clones do not enter the fidelity criterion but formally require a 1-to-n
cloner which is covariant with respect to all n clones. In this case, the proof shows
that the optimal cloner is either not covariant for all clones or singular. We cope with
this issue by disregarding clones with λi = 0. Instead, we consider a cloner which
is restricted to the clones with λi = 0. For the only exception, see the following
Corollary 3.3.
Proof: The proof follows [43]. In general, a state ω is a positive linear functional on
the algebra of observables, i.e. ω ∈ B(H ⊗f ), cf. Section 2.2.4. However, since we are
only interested in expectation values for fidelities, we can restrict states to a specially
tailored subalgebra. For a single system, we define D ⊂ B(H) as the algebra of all
operators of the form
D = C + d , yielding expectation values ω(D) = ω(C) + ω( )d, (3.7)
where C is a compact operator on H and d ∈. This definition separates contributions
to the expectation value from normal and purely singular parts of a state ω:
Since a purely singular state ω ′ yields an expectation value of zero on compact operators,
the parameter d can be obtained as d = ω ′ (C + d ) from any such state. The
decomposition (3.7) is thus unique. Hence any state ω on D consists of two parts:
a linear functional on the compact operators, which necessarily corresponds to a
(nonnormalized) density operator ω1 by ω(C) = tr[ω1 C], and a term proportional
to d, which introduces another parameter ω0 ∈Ê. Expectation values of ω are thus
given by
ω(D) = ω(C) + ω( )d = tr[ω1 C] + d
tr[ω1] + ω0 . (3.8)
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