Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
For clarity, the proof is formulated for the case 0 < φ < π. However, it holds for
−π < φ < 0 by the same arguments. An exception is made for the state condition
of ˆε(k), where positivity requires consideration of both cases.
Firstly, the Fourier transformed correlation function ε(k) is indeed symmetric
under interchange of k and −k since ˆΓ(−k) = ˆΓ(k) by (4.18) as well as α(−k) = α(k)
by (4.19) and thus P−k = Pk, P−k = Pk from (4.22). Moreover, ε(k) is invariant
under the dynamics. To see this, note from (4.21) that ˆ Γ(k) commutes with Pk and Pk
since Pk Pk = Pk Pk = 0. For a single mode per site, as in our example, det ˆ Γ(k) = 1
by the definition in (4.18) implies that ˆ Γ(k) is a symplectic transformation and thus
leaves σs invariant. The following equalities then show that ε(k) does not change
under the action of Γ:
ˆΓ T
(k) · ε(k) · ˆ Γ(k) = i ˆ Γ T
(k)σs ˆ Γ(k) · (Pk − Pk) = i σs (Pk − Pk) = ε(k).
In order to see that ε(k) also fulfills the state condition ε(k) + iσs ≥ 0 from
Lemma 4.5 consider the identity
ε(k) + iσs = iσs · (Pk − Pk + ) = 2iσs Pk .
Since Pk has rank one, the only nonzero eigenvalue of iσs Pk is given by the trace,
tr[iσs Pk] = (sinφ − f cos(k)cosφ)/ sin α(k). As α(k) is restricted w.l.o.g. to the
interval (0, π), cf. Lemma 4.6, the denominator is always positive, sinα(k) > 0. By
the condition on f from (4.20), the numerator and thus the nonzero eigenvalue is
positive for 0 < φ < π and negative for −π < φ < 0. (Note that we have excluded the
degenerate cases with φ ∈ {0, ±π} for which the numerator is zero.) Hence ε(k) obeys
the state condition with the appropriate differentiation of cases from the statement
of the theorem. In addition, ε(k) corresponds to a pure state since σs · ε(k) 2 = − .
Moreover, ε(k) can be modified modewise by a factor g(k) = g(−k) ≥ 1 without
affecting the above relations, except for the pure state condition. Hence g(k) plays
the role of atemperaturefor the plane-wave modes.
It remains to connect the stationary states g(k) ε(k) to the limit state of Eq.(4.26).
This is accomplished by the choice g(k) = c(k) φk. Note that c(k) is real-valued
and obeys c(k) = c(−k) since ˆγ0(k) as well as Pk and Pk are reflection symmetric
(see beginning of proof). Hence g(k) = g(−k) ∈Ê. The first task is to connect
iσs P k with P ∗ k P k . Since ˆ Γ(k) is a symplectic transformation, expanding the identity
P T
k · σs ·P k
= P T
k · ˆ Γ T (k)σs ˆ Γ(k)·P k
implies P T
k σs P k = 0 and in turn the relation
iσs P k = i(P T
k + P ∗ k ) · σs P k = P ∗ k · iσs · P k . (4.27)
The nonorthogonal projector Pk can be written as Pk = |φk〉〈ψk|, where we assume
w.l.o.g. that ψk = 1 while in general φk = 1. However, the condition P 2 k = P k
requires 〈ψk|φk〉 = 1. With this, we have iσs P k = P ∗ k · iσs · P k = r |ψk〉〈ψk|, where
r = 〈φk|iσs|φk〉. Indeed, r is real-valued since
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r 2 = 〈φk|iσs|φk〉 2 = 〈φk|iσs · P k P ∗ k · iσs|φk〉 = 〈φk|ψk〉 2 φk 2 = φk 2 .