Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
as well by (4.3b). In addition, T0 inherits the necessary and sufficient commutation
properties for application of Lemma 4.2(ii) from T2
. 0
Again, this property becomes an issue for the concept of irreversible qcas, see
Section 4.3.
4.2 Reversible Gaussian QCA
A Gaussian quantum cellular automaton is a continuous-variable system which conforms
to the Definition 4.1 of a qca and evolves under a quasi-free dynamics, i.e.
a dynamics that maps Gaussian states to Gaussian states in the Schrödinger picture.
For the sake of clarity, we discuss our methods by means of a simple example
system: an infinite one-dimensional chain of one-mode harmonic oscillators with
nearest-neighbor coupling and translational invariance. The single-site algebras are
thus isomorphic to the ccr algebra of one mode. Setting the lattice dimension s = 1,
the quasi-local algebra becomes A(s ) = A(). While this restricts the generality
of some of the results, the presented ideas are valid for arbitrary lattices with translational
symmetry and a suitable elementary cell. 3 However, even for this restricted
case there is an instance which exhibits the characteristics of irreversibility we are
looking for (see Section 4.2.4).
4.2.1 Phase space and basics
While the rest of this thesis is concerned with Gaussian systems of finitely many
modes, in this chapter the lattice structure requires a concept for infinitely many
degrees of freedom. The phase space of such systems is an infinite-dimensional linear
space of functions. Since we are interested in localized observables only and the
ccr algebra is spanned by the Weyl operators, we can restrict ourselves to localized
functions. Hence the phase space of the systems under consideration is the set Ξ =
ξ:→Ê2 ξx ≡ ξ(x) = 0 almost everywhere , whereÊ2 is the phase space of a
single oscillator and the functions ξ vanish everywhere except for a finite number of
sites. The global phase spacevectorsξ relate every site x with a proper local phase
space vector ξx ∈Ê2 for a single mode. This generalizes the concept of a direct sum
of one-site phase spaces to an infinite set of such systems. The symplectic form on
this phase space is defined in terms of the symplectic form on the one-mode phase
space, σs, as
σ(ξ, η) =
x∈σs(ξx, ηx).
3 In [70], the authors argue that any qca can be converted into a qca with nearest-neighbor
interaction at the expense of loosing full translational symmetry. We will not pursue this further,
though.
4 To avoid too many indices, we write the arguments of Weyl operators in parentheses in this
chapter, W(ξ) ≡ Wξ .
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