Quantum Information Theory with Gaussian Systems

4.2 Reversible Gaussian qca
Proposition 4.7:
A translationally invariant linear chain of single harmonic oscillators which evolves
⊲ from a pure Gaussian state with finite correlation length (clustering state)
and vanishing first moments
⊲ under a quasi-free dynamics governed by a nonsqueezing symplectic transformation
reaches a stationary state in the limit of large time.
The limit state of the time evolution is determined by the second, stationary term
in (4.25). For all reflection symmetric states, i.e. states with γ(x) = γ(−x) and thus
γ0(k) = γ0(−k) as in our example system, the projection character of Pk and Pk
effectively reduces γ0(k) to a single matrix element c(k). The limit state is thus
described by a single parameter for each mode:
γ∞(x) = 1
2π
= 1
2π
π
dk e ikx
T
T
T
Pk · γ0(−k) · Pk + Pk · γ0(k) · Pk −π
π
dk e ikx
c(k) P T
T
k · Pk + Pk · Pk −π
(4.26)
Reversing the argument, we can describe any stationary, reflection symmetric state
by a unique pure such state and a modewisetemperatureparameter. Casting the
expression into a different form gives rise to
Theorem 4.8:
All stationary, translationally invariant and reflection symmetric Gaussian states
of the linear chain of single harmonic oscillators with nondegenerate, nearestneighbor
dynamics ˆ Γ(k) from (4.18) are thermal equilibrium states, described by
their Fourier transformed correlation function γstat(k) = g(k) ε(k) comprising
⊲ the correlation function of a pure state with Fourier transform
ε(k) = iσs(Pk − Pk) for 0 < φ < π
ε(k) = iσs(Pk − Pk) for −π < φ < 0 where Γ 0 =
cosφ
− sinφ
sin φ cosφ
⊲ and a continuous function g(k) of modewisetemperatureparameters with
g(k) = g(−k) ≥ 1 .
Proof: The proof is divided into several parts. First, ˆε(k) is shown to possess the
properties claimed in the prelude. Second, ˆε(k) has to obey the state condition.
Third, it corresponds to a pure state and is modified to a mixed state by g(k).
And finally, there exists a g(k) such that g(k) ˆε(k) describes the limit state (4.26).
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