Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
some k0, this property will be spread out by inverse Fourier transform to the whole
of γ(x). In this case, γ determines the characteristic function of a pure Gaussian
state plus additional Gaussian noise and therefore corresponds to a mixed state.
Conversely, γ(k) describes a pure Gaussian state if σs γ(k) 2 = − (cf. Section 2.2).
The state condition on the bilinear form γ over the infinite chain is thus transformed
into a condition of the same form on finite matrices under Fourier transform. This
is summarized in the following
Lemma 4.5:
A function γ which maps x ∈to real 2×2 matrices, is absolutely summable,
x∈γ < ∞, and symmetric, γ(−x) = γ(x) T
,
(i) defines a translationally invariant Gaussian state on the linear chain labeled
byif and only if the Fourier transform γ(k) fulfills γ(k) + iσs ≥ 0 for all
k ∈ [−π, π] and
(ii) corresponds to a pure Gaussian state if and only if γ(k) + iσs is not strictly
positive for any k ∈ [−π, π], i.e. if σs γ(k) 2 = − .
During time evolution of the system, the correlation function γ changes according
to the symplectic transformation Γ of the phase space argument in (4.7) as
γt(x) =
Γt(y)
y,z∈ T
· γ0(x + y − z) · Γt(z) or (4.17a)
γt(k) = ˆ Γ T
t (k) · γ0(k) · ˆ Γ t(k), (4.17b)
where γ0 denotes the correlation function of the initial state.
4.2.4 Example system
To gain more specific results, we consider a more concrete instance of the above
system: The initial state is a coherent product state described by the correlation
function γ0(0) = and γ0(x) = 0 otherwise, resulting in the Fourier transform
γ0(k) = . Clearly, γ conforms to the requirements of Lemma 4.5 and thus describes
a translationally invariant, pure Gaussian state. For the on-site part Γ0 of the dynamics,
we choose a rotation, Γ0 = cos φ −sin φ
sin φ cos φ , where −π ≤ φ ≤ π, which by
(4.12) determines Γ up to the coupling parameter f. Repeating (4.15) and (4.12),
the Fourier transform is
ˆΓ(k) = Γ 0 + 2f cos(k)Γ ±
with Γ ± =
(Γ0)2,1 (Γ0)2,2 . (4.18)
0 0
Since Γ ± contains a row of Γ 0, the determinant is det ˆ Γ(k) = detΓ 0 = 1. Hence
ˆΓ(k) induces a symplectic transformation on every single mode k. The value of the
coupling parameter f determines whether the two eigenvalues of ˆ Γ(k) are real and
82