Quantum Information Theory with Gaussian Systems

the expectation value of an arbitrary observable A in a state ρ obeys
tr ρ T(A) = tr T∗(ρ)A = tr W ⊗n∗
ξ T∗(Wξ ρ W ∗ ξ )W ⊗n
ξ A
= tr ρ W ∗ ξ T(W⊗n
∗
)Wξ .
ξ AW⊗n
ξ
3.3 Covariance
(3.4)
Thus, covariance of T follows from covariance of T∗ and we will in the following use
the one or the other interchangeably. Using T, the fidelities can be written in a unified
form as f(T, ρ) = tr[ρ T(A)], where A = ρ⊗n and A =
i λi ⊗· · ·⊗ ⊗ρ (i) ⊗ · · ·⊗
for f = fjoint and f =
i λi fi, respectively. Furthermore, for coherent states we get
f T, |ξ〉〈ξ| = tr |0〉〈0| Tξ(A) = f Tξ, |0〉〈0| , where
Tξ(A) = W ∗ ξ T W ⊗n
ξ AW⊗n
∗
ξ
Wξ
in strict analogy with (3.3).
By applying an average Mξ over the symmetry group of phase space translations,
we can define for every map T a covariant map which we denote by Tξ. However,
since the group of translations is noncompact, Mξ has to be aninvariant mean[42]
which does exist only by virtue of the Axiom of Choice. The cloner Tξ yields worstcase
fidelities which are not lower than those achieved by T [43]. For a discussion,
see the proof of
Lemma 3.1:
For every 1-to-n cloner T there exists a covariant cloner Tξ such that for f = fjoint
or f =
i λi fi
f(T) ≤ f( Tξ).
Remark: The cloner Tξ might besingular, i.e. its output for normal states described
by a density operator ρ could be a purely singular state, which cannot be
connected to any density operator. This issue is addressed in the next Section 3.3.1,
where it is shown that such cloners are not optimal.
Proof: The invariant mean Mξ will not be applied to T directly but to bounded
phase space functions g(ξ), where Mξ g(ξ) is linear in g, positive
if
g is positive,
normalized as Mξ[1] = 1 and indifferent to translations, Mξ g(ξ + η) = Mξ g(ξ) .
For expectation functionals of a bounded operator A on the cloner output T∗ξ(ρ), the
invariant mean Mξ tr T∗ξ(ρ)A is well-defined as the argument is a function bounded
by A. Moreover, by the properties of Mξ it is a covariant, bounded, normalized,
positive linear functional on A, which describes a state on the output ccr algebra.
Since it is also linear in ρ, we can introduce a linear operator T∗ such that T∗(ρ) is
the respective state. However, this state might be singular (see Section 3.3.1 below),
hence T∗ has to map density operators of the input system onto the linear functionals
on the output ccr algebra:
T∗(ρ): B∗(H) → B ∗ (H ⊗n ), such that
T∗(ρ)[A] = Mξ tr T∗ξ(ρ)A .
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