Quantum Information Theory with Gaussian Systems

3.4 Optimization
Lemma 3.10:
The output states ρout of symmetric covariant 1-to-n cloners for continuousvariable
states lie in the bosonic sector, i.e. the output states have expectation
value +1 with every flip operator, if and only if the cloner is described by a bosonic
state ρT in (3.18). In particular,
tr ρout(i,j) = tr ρT(i,j) for i, j ∈ {1, 2, . . ., n} .
Proof: To simplify notation, we understand ρ ≡ ρout as the output state of a 1-to-n
cloner described by a state ρT. Since we only consider deterministic cloners, the
output states ρ are normalized anyway, tr[ρ] = 1, and the proof can be restricted
to flip operators(i,j) with i = j. To shorten expressions, we drop the phase space
arguments of modes which are not considered and indicate the remaining modes by
upper indices, e.g. for a characteristic function χ(ξ):
χ (i,j) (ξ, η) = χ(0, . . .,0, ξ,
0, . . .,0, η,
0, . . .,0).
i j−i
The same convention is used for other functions as well as Weyl operators and in a
similar way for a single mode.
We start by discussing properties of=(1,2) for two modes and generalize later.
In order to transport the action ofto phase space, note that
W(ξ, η) = W(η, ξ). (3.36)
We introduce the parity operatorÈ(j) , which acts on the field operators of mode j
byÈ(j) RkÈ(j) = −Rk for k = 2j − 1 and k = 2j in standard ordering of R. On
Weyl operators,È(j) induces a change of sign for the respective argument,
È(2) W(ξ, η) = W(ξ, −η)È(2) .
Under a symplectic transformation S which maps two modes to symmetric and
antisymmetric combinations according to
ξ+η
S : (ξ, η) ↦→ √2 , ξ−η
,
U ∗ SU S acts as (1) ⊗È(2) :
U ∗ SW(ξ, η)US = U ∗ S W(η, ξ)U S
= U ∗ SU S W ξ+η
√2 , ξ−η
η+ξ
= W √2 , η−ξ
∗
USU S .
√ 2
Hence the expectation value of(i,j) can be written as an expectation value of
(i) ⊗È(j) ,
tr ρ(i,j) = tr ρ ′ (i) ⊗È(j) , where ρ ′ = U ∗
S (i,j) ρ U S (i,j)
√ 2
√ 2
(3.37)
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