Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
Since our notion of a quantum cellular automaton is based on an infinite lattice,
any attempt to define a qca has to deal with the infinite number of quantum systems
at the lattice sites. As discussed in [70], several previous definitions found in the
literature suffer from conceptual shortcomings which prevent a successful application
to infinite lattice systems. In particular, the notion of localization as implemented
by states on the infinite lattice is problematic. For example, the basic operation of
applying the same unitary transformation to each cell separately would require the
multiplication of an infinite number of phase factors, which does not allow for a
well-defined unitary operator describing the global state change.
In order to circumvent these problems, we work in the Heisenberg picture and
define the dynamics of observables. This approach was motivated by methods used
in statistical mechanics of quantum spin systems, where infinite arrays of simple
quantum systems play a prominent role [70]. In contrast to a notion of localized
states, localized observables are clearly defined: they require a measurement of a
finite collection of cells only. If the lattice sites are labeled by s-tuples of integers,
where s is the lattice dimension, we denote by Ax the algebra of observables which
are localized on the single lattice site x ∈s . This algebra could be an algebra of
d × d matrices for a spin system or a ccr algebra for a continuous-variable system.
The set of all observables which are localized on a finite region Λ ⊂s of the lattice
constitutes the algebra A(Λ) =
x∈Λ Ax associated with this region. For two regions
Λ1 ⊂ Λ2, we take A(Λ1) as a subalgebra of A(Λ2) by tensoring with unit operators
as necessary, i.e. on Λ2 \ Λ1. This allows us to properly define the product of two
operators A1A2 from different local algebras A(Λ1) and A(Λ2), respectively, as the
corresponding element from A(Λ1 ∪ Λ2). Since this procedure does not affect the
norm, all local algebras are normed and their completion is the quasi-local algebra
[84], denoted by A(s ).
This inclusion of algebras is especially instructive in connection with the neighborhood.
If N ⊂s is defined as the finite neighborhood of the cell x = 0, we can
install it as the uniform neighborhood scheme and obtain the neighborhood of any
cell x as the set x + N ≡ {x + n | n ∈ N }. Accordingly, the neighborhood of a
finite region Λ ⊂s of the lattice is the set Λ + N ≡ {x + n | x ∈ Λ, n ∈ N }.
The observables on any finite region Λ are contained in the algebra on the region
enlarged by its neighborhood, A(Λ) ⊂ A(Λ + N), if and only if Λ ⊂ Λ + N. This
is only true if the neighborhood scheme explicitly contains the origin. While this
need not necessarily be the case, we can formally enlarge the neighborhood without
actually considering the additional elements in the interaction. Hence we can always
assume 0 ∈ N. By the same argument, we can w.o.l.g. assume the neighborhood N
to be simply connected. Note that by the above definition thepointwise difference
of two sets is in general not empty, e.g. N − N = {x − y | x, y ∈ N }.
The dynamics of the system is implemented as linear transformations on the observable
algebras. In particular, one time step in the global evolution of the qca
is a transformation T on the observable algebra A(s ) of the infinite system. To
describe a proper time evolution, T has to be completely positive. Since we only
consider deterministic dynamics, it has also to be unital, T( ) = , i.e. it has to
be a quantum channel. In addition, uniformity of the whole system requires that T
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