Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
While in this way the local rule can be directly inferred from the global rule, the
definition of a particular qca is not constructive. One would possibly rather start
with a prescribed neighborhood scheme together with a local interaction and obtain
a global rule to match. We join the authors of [70] on that a satisfactory theory
of qcas should connect the global transition rule T and the local rule such that
either can be uniquely inferred from the other. They argue that the class of global
rules should have an axiomatic specification, with locality and the existence of a
finite neighborhood scheme as the most important aspect. In contrast, the local rule
should be characterized constructively. This is easily possible for reversible qcas.
For later reference, we provide the relevant Lemma 2 from [70] and its proof: 2
Lemma 4.2:
For a reversible qca, global and local rule are equivalent, i.e.
(i) The global automorphism T is uniquely determined by the local transition
rule T0.
(ii) An automorphism T0: A0 → A(N) is the transition rule of a reversible qca
if and only if for all x ∈s
such that N ∩ (N + x) = ∅ the algebras T0(A0)
and τx T0(A0) commute elementwise.
Remark: Note that for all x ∈s
not affected by (ii), i.e. those with N ∩(N+x) = ∅,
the algebras T0(A0) and τx T0(A0) commute anyway, because T0(A0) ⊂ A(N).
Proof: By translational invariance of T it suffices to consider T0, since Tx: Ax →
A(x + N) is recovered as Tx(Ax) = τx T0 τ−x(Ax). Because T is an automorphism,
it can be expressed in terms of Tx: any finite tensor product
x∈Λ Ax of one-site
operators Ax gives rise to
T
x∈ΛAx
= T x∈ΛAx
= Tx(Ax). (4.2)
x∈Λ
For the first equality sign we have identified Ax with a subalgebra of A(Λ) by tensoring
with unit operators (see above) and the second identity is due to T being an
automorphism. Since the operators on the right hand side have overlapping localization
regions x + N, their product cannot be replaced by a tensor product. However,
the argument of T is a product of commuting operators, hence is the right hand side.
The commutativity condition of (ii) is thus necessary.
It is also sufficient because if the factors Tx(Ax) commute, their product is unambiguously
defined. Moreover, every local observable can be expressed as a linear
combination of finite tensor products. Consequently, Eq.(4.2) defines an automorphism
on the quasi-local algebra, proving (i).
The commutation relation in (ii) above is in fact a key to the notion of a qca
from [70]. While it is automatically satisfied for reversible qcas, it becomes an
issue for the irreversible case (cf. Section 4.3). We therefor illustrate it in Fig. 4.1
2 The lemma is slightly restated to match the modified definition of a qca.
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