Quantum Information Theory with Gaussian Systems

4.3 Irreversible Gaussian qca
holds. Note that the trace norm bounds also imply bounds on the respective traces:
A − B 1 < ε 2 =⇒ tr[A − B] ≤ tr |A − B| = A − B 1 < ε 2 .
This allows to establish a bound on (4.31) for all t ≥ T:
tr[ρt] − tr[PρtP] ≤ tr[ρt] − tr[ρ∞] + tr[ρ∞] − tr[Pρ∞P]
+ tr[Pρ∞P] − tr[PρtP]
≤ 0 + ρ∞ − Pρ∞P 1 + Pρ∞P − PρtP 1
< 2ε 2 ,
since tr[ρt] = tr[ρ∞] = 1. Hence, by (4.31), ρt − PρtP 1 < 2 √ 2ε and finally
from (4.30)
ρ∞ − ρt 1 < ε 2 + ε 2 + 2 √ 2 ε < 6ε.
This proves convergence of ρt and thus of ρ0 under the dynamics to ρ∞ in trace
norm with respect to finitely localized observables, i.e. finite lattice regions.
4.3 Irreversible Gaussian QCA
By an irreversible qca, we understand a qca with a global rule T which has, however,
no completely positive inverse. The dynamics thus cannot be inverted by physical
operations. In contrast to the reversible case, irreversible qcas still resist a detailed
characterization. So far, investigations have been restricted to special classes of such
systems, e.g. in [88]. In this chapter, we highlight a few problems in the characterization
of irreversible qcas for Gaussian systems.
As mentioned above, several desirable features which come built in for reversible
qcas pose difficulties in the irreversible case. While Definition 4.1 covers the essential
properties of a qca, it does, however, not consider two important principles:
(i) the local rule should determine the global rule (Lemma 4.2) and
(ii) the concatenation of qcas should again be a qca (Corollary 4.3).
The first principle allows to explicitly obtain the global rule of a qca for every valid
local dynamics. This complements the axiomatic definition of the class of qcas with
a constructive approach for individual automata. The second property allows to
build a qca out of set ofmoduleqcas. In particular, two steps of any given qca
would result in a combined dynamics which again is a (different) qca. We will in
the following investigate how these properties influence the definition of irreversible
Gaussian qcas.
As above, an irreversible Gaussian qca has a quasi-free dynamics, which maps
Weyl operators to multiples of Weyl operators. This is accomplished by a linear
transformation Γ of the phase space argument and additional noise to assure complete
positivity (cf. Section 2.3). In the reversible case, a symplectic Γ renders
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