Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
this out in components of µ and using the definitions of γ and σ above results in the
detailed condition
x,y∈µ T
x ·
γ(x − y) + iδ(x − y)σs · µy ≥ 0 (4.6)
for all µ as above. Here δ denotes the Kronecker delta with δ(x) = 1 for x = 0 and
δ(x) = 0 otherwise.
4.2.2 Transition rule
To implement a Gaussian system, the global transition rule T has to be quasi-free,
i.e. it has to map Gaussian states into Gaussian states in the Schrödinger picture.
In the Heisenberg picture, this is accomplished by mapping the Weyl operators to
Weyl operators subject to a symplectic transformation Γ:
Clearly, the so-defined T is a homomorphism, since
T W(ξ) = W(Γ ξ). (4.7)
T W(ξ) T W(η) = e −iσ(Γ ξ,Γ η) W(Γ ξ + Γ η)
= e −iσ(ξ,η) W(Γ ξ + Γ η) = T W(ξ) W(η) .
It is also an automorphism, since as a symplectic transformation Γ is invertible.
Hence together with a suitable locality condition T could indeed be the global rule of
a Gaussian qca. In fact, this is the only possible configuration: any transformation
Γ resulting in a homomorphism would have to be linear in the arguments of the
Weyl operators and fulfill σ(Γ ξ, Γ η) = σ(ξ, η) for all ξ and η, which is exactly the
definition of a symplectic transformation.
As with the generalization of matrices above, Γ acts on phase space functions by
sitewise applying suitable real 2×2 matrices Γx,z,
(Γ ξ)x =
z∈Γx,z · ξz .
For T to be translationally invariant, i.e. invariant under lattice translations 5 τ∆,
where (τ∆ ξ)x = ξx+∆ with ∆ ∈, the transformation Γ has to be invariant, too. It
has thus to commute with τ∆ for all ξ ∈ Ξ and all x, ∆ ∈:
(Γ τ∆ ξ)x = (τ∆ Γ ξ)x ⇐⇒ Γx,z = Γx−z .
We assume nearest-neighbor coupling for the example, which imposes Γx−z = 0
unless |x − z| ≤ 1. Consequently, Γ is completely determined by three real-valued
5 We denote both the isomorphism of local algebras on different sites, τx Ay = Ay+x, and the
shifting of phase space functions, (τ∆ ξ)x = ξx+∆, by the same symbol τ. This is justified
because both transformations represent the same change of origin of the lattice.
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