Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
Best symmetric Gaussian 1-to-n cloners
By a symmetric Gaussian cloner we understand a cloning map which is invariant
under interchanging the output modes and which is described by a Gaussian state ρT.
To investigate these cloners, we use the characteristic function t with respect to the
twisted symplectic form Σ = ( n −n) ⊗ σin, confer Eq. (3.16) and its discussion in
Section 3.3.2.
For the cloner to be symmetric and Gaussian, t has to have the form
t(ξ) = exp −ξ T
· (a n ⊗ 2 + bn ⊗ 2) · ξ/4 . (3.31)
The map is completely positive if an only if (a n ⊗ 2 + bn ⊗ 2) − iΣ ≥ 0.
Introducing the abbreviations
A = a 2 − iσin , B = b 2 + iσin and X = n ⊗ A +n ⊗ B , (3.32)
this condition is equivalent to X ≥ 0, which in turn is true if and only if 〈φ| X |φ〉 ≥ 0
for all φ = n j=1 φj, φj ∈2 . The evaluation of this condition is simplified by
rewriting φj = ψj + ψ0 where ψ0 =
j φj/n and hence
j ψj = 0:
〈φ| X |φ〉 =
=
n
〈φj| A |φj〉 +
j=1
n
〈φj| B |φi〉
i,j=1
n
〈ψj| A |ψj〉 + n 〈ψ0| A |ψ0〉 + n 2 〈ψ0| B |ψ0〉.
j=1
By evaluating this expression for particular ψj it is easily seen that A ≥ 0 and
n B + A ≥ 0 are necessary and sufficient conditions for X ≥ 0:
ψ1 = −ψ2 = 0 , ψi=1,2 = 0 ⇒ 〈φ| X |φ〉 = 2 〈ψ1| A |ψ1〉,
ψ0 = 0 , ψi=0 = 0 ⇒ 〈φ| X |φ〉 = n 〈ψ0| A |ψ0〉 + n 2 〈ψ0| B |ψ0〉.
The definitions in (3.32) imply that the above conditions on A and B are variants
of the state conditions on covariance matrices (2.22) which are fulfilled if and only
if a ≥ 1 and a + n b ≥ n − 1.
Since the cloner is symmetric with respect to interchanging the output modes, all
single-copy fidelities are identical. They are calculated as the overlap between one
output subsystem, e.g. the first, and the fixed input state |0〉〈0| with characteristic
function χin(ξ) = exp(−ξ2 /4):
T, |0〉〈0| = tr T |0〉〈0| ⊗ ⊗ . . . ⊗ |0〉〈0|
54
fsymmetric(T) = f1
dξ
=
2π t(ξ, 0, . . . , 0)χin(ξ) 2
dξ
=
2π e−(a+b+2) ξ2 /4 2
=
a + b + 2
(3.33a)
≤
n 1
→
2n − 1 2
for n → ∞ , (3.33b)