Quantum Information Theory with Gaussian Systems

4.2 Reversible Gaussian qca
case, which corresponds to sin α(k) = 0. The imaginary part is readily obtained
from (4.23) as ImPk = cosα(k) −Γ(k) / 2 sinα(k) , which proves (4.22). For the
remaining projector, we get Re Pk = RePk and ImPk = − Im Pk from the beginning
of the proof. Hence Pk is indeed the complex conjugate of Pk.
Convergence
The decomposition (4.21) of ˆ Γ(k) is particularly useful for a compact description of
the iterated transformation ˆ Γt(k). By (4.15),
ˆΓt(k) = ˆ Γ(k) t = e itα(k) Pk + e −itα(k) Pk , (4.24)
since as projectors on disjoint eigenspaces Pk and Pk obey P 2 k = Pk , P 2
k = Pk and
Pk · Pk = 0. With this relation, the time-dependent correlation function γt(x) is
obtained by inverse Fourier transform from (4.17) as
γt(x) = 1
2π
= 1
2π
π
−π
π
−π
+ 1
2π
dk e ikx ˆ Γ T
t (k) · γ0(k) · ˆ Γ t(k)
ikx
dk e e 2itα(k) P T
k · γ0(k) · Pk + e −2itα(k)
T
Pk · γ0(k) · Pk
π
−π
ikx
dk e P T
T
k · γ0(k) · Pk + Pk · γ0(k) · Pk .
(4.25)
In (4.25), the transformation is separated into a time-dependent, oscillating part
in the first term and a stationary part in the second. In the limit of large time
t, the rapidly oscillating term vanishes and the correlation function converges by
an argument similar to the method of stationary phase: Starting from a product
state (or any clustering state), γ0(k) is continuous; since we excluded the degenerate
case, ˆΓ(k), Pk and Pk are continuous, too, and the whole integrand is well-behaved.
Note that α(k) is differentiable and has only finitely many extrema (cf. Fig. 4.3
and caption). The main contribution to the integral stems from intervals where
α ′ (k) ≈ 0, i.e. from around the extrema of α(k) at kn, ordered such that kn ≤ kn+1
for n = 1, 2, . . .N < ∞. We explain the argument for the first term in the first
integrand of (4.25). Splitting the integral at the extrema of α(k) at kn and writing
ĉ(k) = P T
k · γ0(k) · P k , we obtain:
π
dk ĉ(k)exp ikx + 2 itα(k) =
−π
N
n=1
kn−ǫ
kn+ǫ
dk ĉ(k)exp ikx + 2 itα(k)
N
+
≡A
n=1
dk ĉ(k)exp ikx + 2 itα(k)
+R
kn+1−ǫ
kn+ǫ
≡B
85