Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
where
R =
k1−ǫ
dk ĉ(k)exp ikx + 2 itα(k) π
+ dk ĉ(k)exp ikx + 2 itα(k) .
−π
kN+ǫ
Integrals of type A cover intervals around the extrema of α(k), integrals B the
intervals in between extrema. The other two integrals in R cover remainders at the
ends of the whole integration interval [−π, π]; they are effectively of type B. If the
derivative of α(k) is nonvanishing, α ′ (k) = 0, we can substitute u = 2α(k) and
k = α −1 (u) to obtain for integrals of type B:
dk ĉ(k)exp ikx + 2 itα(k) = 2α ′ (k) u
−1
′ n
du ĉ α −1 (u) exp ixα −1 (u) + 2 itu ,
kn+1−ǫ
kn+ǫ
where u n = 2α(k n + ǫ) and u ′ n = 2α(k n+1 − ǫ). Since this integrand is absolutely
integrable, the Riemann-Lebesgue lemma [7, Ch. IX] assures that the integral vanishes
for t → ∞. For integrals A, this substitution is not possible since α ′ (kn) = 0.
However, we can expand α(k) to second order around kn, yielding
kn+ǫ
dk ĉ(k)exp ikx + 2 itα(k) = e 2itα(kn)
kn+ǫ
dk ĉ(k)exp ikx + 2 it(k − kn) 2 α ′′ (kn) .
kn−ǫ
kn−ǫ
Again, the integral vanishes for t → ∞. These arguments would be spoiled by any
ˆΓ(k0) with real eigenvalues, which would turn the phase factor e2itα(k) into a realvalued
exponential and thus result in continued squeezing of the respective mode.
Hence we restrict the dynamics to small coupling parameter f.
While the correlation function γ(x) converges, the amplitude part
un
x∈ξ T
x · d of
a translationally invariant state in (4.4) does not unless d = 0: Under time evolution
for t steps, the initial sum is mapped to
↦−→
x∈ξx
ξ)x =
x∈(Γt ˆ Γt(0) · ˆ ξ(0)
= e itα(0) P0 + e −itα(0)
P0 · ξ(0) ˆ itα(0)
= Re(e P0) · ˆ ξ(0).
This expression clearly depends on t since α(k) = 0 was excluded as the degenerate
case. Hence the convergence of an initial state under the dynamics of the qca is
restricted to states with vanishing first moments.
It is remarkable that while the initial state is a pure, uncorrelated state and the
dynamics is reversible for the whole system as well as for every mode, the system
exhibits convergence under interplay of the plane-wave modes. However, we only
consider observables with finite support on the chain; hence this behavior suggests
that correlations areradiated to infinityduring time evolution. Since the whole
range of intermediate states is mapped to the same limit state, the system exhibits
the signs of irreversibility we are interested in:
86