Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
convergence in trace norm, i.e.
ρt − ρ∞ 1 = tr |ρt − ρ∞| → 0 as t → ∞ .
Note that the knowledge about the limit state simplifies the proof. Since the
characteristic function χ∞ is continuous, the limit state is indeed described by a
density operator ρ∞. Moreover, χ∞(0) = 1 implies tr[ρ∞] = 1, cf. Section 2.1.1.
In contrast to the general case considered in [85], we restrict the discussion to
expectation values of the ρt with operators from the quasi-local algebra A(), which
is generated by the Weyl operators with finite support. Therefore, it suffices to assure
convergence with respect to these operators. But expectation values with such Weyl
operators are exactly the pointwise values of the characteristic function:
tr ρ∞ W(ξ) = χ∞(ξ) = lim
t→∞ χt(ξ) = lim
t→∞ tr ρt W(ξ) .
This is the statement of weak convergence ρt
w
−→ ρ∞ on A().
To establish convergence in trace norm, we closely follow the proof of Lemma 4.3
in [86], which we provide for completeness: Given 0 < ε < 1, let P be a spectral
projector for ρ∞ with finite rank and ρ∞ −Pρ∞P 1 < ε. By the triangle inequality,
we can bound the trace norm distance of any ρt and ρ∞ as
ρ∞ − ρt 1 ≤ ρ∞ − Pρ∞P 1 + Pρ∞P − PρtP 1 + PρtP − ρt 1 . (4.30)
Assuming the spectral decomposition ρt = ∞
m=1 rm|em〉〈em|, where {|em〉} ∞ m=1 is
an orthonormal basis of the Hilbert space, the authors of [86] derive an upper bound
for the last term:
As ρt
ρt − PρtP 1 ≤ ρt − Pρt 1 + Pρt − PρtP 1
≤
= 2
∞
m=1
∞
m=1
rm
|em〉〈em| − P |em〉〈em| + 1
∞
m=1
rm
∞
≤ 2
≤ 2
m=1
rmP
|em〉〈em| − P |em〉〈em|P
1
1 − P |em〉 2 1
rm
tr[ρt] −
1/2 ∞
∞
m=1
m=1
rm
rm
1/2
P |em〉 2
1 − P |em〉 2
1
1/2 1
1/2
= 2 tr[ρt] − tr[PρtP] 1/2 . (4.31)
w
−→ ρ∞, PρtP converges weakly to Pρ∞P. Since P is of finite rank, there exists
a number T ∈Æsuch that for all time steps t ≥ T the bound PρtP −Pρ∞P 1 < ε 2
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