Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
In the case of one mode per site, the requirements of (4.11) simplify readily: The
conditions for u = ±2, meaning σ(Γ − ξ, Γ + η) = σ(Γ + ξ, Γ − η) = 0 for all ξ, η ∈Ê2 ,
imply that Γ − and Γ + project onto the same, one-dimensional subspace ofÊ2 . Hence
both are multiples of a common matrix Γ ± with rank one. For any real 2×2 matrix
M we get M +
· M = (det M) and thus immediately have Γ +
= 0 as
− · Γ = Γ+
− + · Γ +
well as Γ +
0 ·Γ0 = from the condition for u = 0. But for one mode, this is equivalent
to Γ0 being a symplectic matrix. If we choose the one-dimensional subspace of Γ− and Γ + as the direction of the position variable, 6 we get
Γ + = Γ − = f Γ ±
with Γ ± =
(Γ0)2,1 (Γ 0)2,2
0 0
, (4.12)
where f is a common, arbitrary, real-valued coupling parameter and (Γ 0)i,j denotes
the respective matrix entries of Γ 0. The shape of Γ ± is a consequence of the conditions
for u = ±1. We summarize these results in
Proposition 4.4:
The quasi-free quantum channel
T W(ξ) = W(Γ ξ),
where Γ is translationally invariant by (4.8) and symplectic by the conditions in
(4.11), results in a reversible qca on an infinite linear chain of harmonic oscillators
with nearest-neighbor interaction. For the case of one mode per site, the on-site
transformation Γ 0 is symplectic and determines the interaction Γ ± , except for the
coupling constant f, according to (4.12).
Remark: The fact that in this case the coupling is identical in both directions implies
that theleft-andright-shiftermentioned as examples in the introduction
cannot be realized with one mode per site. Instead, they require a spareswap
system and an alternating partitioning scheme in order to avoid collision problems.
For details, see [70].
Proof: These definitions result in a qca in the sense of Definition 4.1. The local observable
algebra Ax is spanned by the Weyl operators on single lattice sites, wx(ξx)
with ξx ∈Ê2 . The global Weyl operators W(ξ) with ξ ∈ Ξ span the quasi-local
algebra A(). Since Γ is a symplectic transformation and translationally invariant,
T as defined above is a translationally invariant automorphism of A(). The requirement
of locality and finite propagation speed is met by the nearest-neighbor
coupling inherent in Γ. The local rule is the restriction of T to the algebra of singlesite
observables.
A single time step of the system is implemented by applying T to the observable in
question. For Weyl operators, this is by the definition in (4.7) the same as applying
6 This choice can be interpreted either as a specification of the interaction Γ± or as a choice of
the symplectic basis in the phase spaceÊ2 of a single site.
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