Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
Normalization of ω imposes ω( ) = tr[ω1]+ω0 = 1 and positivity requires that ω0 ≥ 0
and ω1 ≥ 0. An intuitive interpretation would suggest that ω0 is the probability for
a system in state ω to beat infinity, 2 while ω1/(1 − ω0) is the normalized density
operator describing the nonsingular part of ω. Note that the fidelity of ω with respect
to a coherent state α is determined by the compact operator |α〉〈α| as
f ω, |α〉〈α| = ω |α〉〈α| = tr ω1|α〉〈α| , (3.9)
which is independent of ω0. Hence one can expect that the optimization of the cloner
output state reduces the weight at infinity ω0.
In order to obtain a more rigorous argument, we introduce the tensor product
subalgebra D⊗n ⊂ B H⊗n generated by the identity as well as all operators of the
form ⊗ · · · ⊗ C ⊗ · · · ⊗ which have a compact operator C in a single tensor
factor. A product of such operators is characterized by the set Λ ⊂ Λn = {1, 2, . . ., n}
of tensor factors with compact entry; the complement Λc = Λn \ Λ contains a factor
⊗Λ c
. A general element D ∈ D⊗n is thus decomposed according to
D =
Λ
DΛ ⊗ ⊗Λc
, (3.10)
where DΛ is the respective compact part on the tensor factors indexed by Λ and
the sum runs over all subsets of Λn. Similarly, a state ω is decomposed into parts
which are labeled by a set of tensor factors Λ on which the state is normal and
thus described by a density operator ωΛ; on the complement Λc , the part describes
systemsat infinity. The purely singular contribution is denoted by ω∅. A part ωΛ
⊗Λ′c
yields nonzero expectation value for D from (3.10) only on terms DΛ ′ ⊗ for
which Λ ′ ⊂ Λ because else a singular part would meet a compact operator. Hence
ω(D) = ω Λ ′DΛ ′ ⊗
⊗Λ′c
=
tr ωΛ DΛ ′ ⊗
⊗Λ\Λ′
. (3.11)
Λ
Λ ′ ⊂Λ
Positivity of ω is assured if all ωΛ ≥ 0 and normalization requires
Λ tr[ωΛ] = 1.
This relation implies that the fidelity of the i-th clone with ω is obtained as
ω(Fi) =
tr
ωΛ Fi| Λ , (3.12)
Λ∋i
where the sum runs over all Λ containing the index i and Fi| Λ is the restriction
of Fi to the tensor factors Λ. For a weighted sum of such fidelities, determined by
F(λ) =
i λi Fi with λi ≥ 0, the expectation value is given by a sum over the above
expression,
ω F(λ) =
tr ωΛ F(λ)
. (3.13)
Λ
Λ
2 This interpretation can be made rigorous by a correspondence between spaces of functions and
spaces of operators, which allows aone-point compactificationof the phase space, i.e. the
process of adjoining a point at infinity to the real vector space. By using only a single point at
infinity we identify all purely singular states, which is justified since they do not contribute to
fidelities as explained above. This is a main motivation in the definition of D.
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