Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
respectively, which have to hold true independently of each other. Hence require
g(x) = 0 for x /∈ M. Recall that for causality, g(x) = 0 for x /∈ N − N is necessary
in any case, see above. If N − N ⊆ M, even σ(Γ T ξ, Γ η) ≡ 0 and the condition is
satisfied independently of Γ. Otherwise, however, (4.37) imposes restrictions on Γ
also, which are obtained in the same way as for (4.10):
∀∆ ∈ (N − N) \ M:
x∈N
Γ +
x Γ ∆+x = 0 . (4.38)
While the combinationI+IIof cases I and II with M = {0} thus assures that
the global rule can be inferred from the local rule, a concatenation of two such
systems is in general not of this type. Hence they comply with property (i), but not
with (ii). The concatenation of two channels T1 and T2 from (4.32) determined by
transformations Γi(x) with support on Ni and noise forms gi(x) with support on
Mi for i = 1, 2 results in a combined dynamics T according to
T W(ξ)
= T2 T1(W(ξ)) = W(Γ2 Γ1 ξ) exp −g2(Γ1 ξ, Γ1 ξ)/4 − g1(ξ, ξ)/4
= W(Γ ξ) exp −g(ξ, ξ)/4 , where Γ = Γ 2 Γ 1 and g = g 1 + Γ T
1 g 2 Γ 1 .
The support of Γ and g can be found from the respective translationally invariant
functions:
(Γ2 Γ1)(x) =
− z) · Γ1(z) =⇒ N = N1 + N2 ,
z∈Γ2(x
g(x) = g1(x) +
1(y − x) · g2(y − z) · Γ1(z)
y,z∈Γ T
=⇒ M = M1 ∪ (M2 + N1 − N1).
This implies that two systems of type I+II with Mi = Ni−Ni can be concatenated
to yield a system with the same characteristics since M = (N1 + N2) − (N1 + N2).
However, for Mi = {0} an additional condition on g2 is necessary. In this case,
gi(x) = δ(x)gi(0) due to the restricted support of gi. Hence
g(x) = δ(x)g 1 (0) +
= δ(x)g 1 (0) +
To get g(x) = δ(x)g(0), we need
y,z∈Γ T
1 (y − x) · δ(y − z)g2 (0) · Γ1 (z)
y∈Γ T
1 (y − x) · g2 (0) · Γ1 (y).
y∈Γ T
1 (y − x) · g 2 (0) · Γ 1 (y) = δ(x)g′ 2
(0) (4.39)
with a suitable g ′ 2 (0) such that g and Γ obey the condition (4.34). However, there
exist systems of type I+II with M = {0} which cannot meet this condition. As a
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