Quantum Information Theory with Gaussian Systems

Im
e iα(k)
e−iα(k)
Re
4.2 Reversible Gaussian qca
Im
Re
ea(k) e−a(k)
Figure 4.2:
Depicting the eigenvalues of ˆ Γ(k): for small coupling parameter |f| < fcrit the
two eigenvalues are complex phases and conjugated to each other (left figure); for
large coupling |f| > fcrit, the eigenvalues are real and inverse to each other (right
figure). All eigenvalues meet at a value of 1 for f → fcrit.
inverse or complex and conjugate to each other. 7 To obtain a quantitative statement,
we write the eigenvalues as e ±iα(k) , where α(k) is either real- or purely imaginaryvalued,
and consider the trace as their sum:
tr ˆ Γ(k) = e iα(k) + e −iα(k) = 2 cosα(k) = 2 cosφ + 2f cos(k)sin φ
⇒ α(k) = arccos cosφ +f cos(k)sin φ .
(4.19)
If |tr ˆ Γ(k)| ≤ 2, then α(k) is real-valued, |e ±iα(k) | = 1 and ˆ Γ(k) is a rotation on
mode k. Otherwise, α(k) is purely imaginary-valued, the eigenvalues are real and
ˆΓ(k) corresponds to a squeezing. For |tr ˆ Γ(k)| = 2 the eigenvalues meet at a value of
1. The relevance of the eigenvalues lies in their direct consequence for the dynamics:
if some ˆ Γ(k0) had real eigenvalues larger than 1, the respective mode would be
constantly squeezed, which would transform any input state over time into aninfinitely
squeezed state[44]. The limit state of such dynamics is highly singular; for
example, the probability for any oscillator in the chain to be finitely excited is zero.
The nonsqueezing regime with real eigenvalues for all ˆ Γ(k) is given by the inequality
cosα(k) = cosφ +f cos(k)sin φ ≤ 1 ,
which has to hold for all k ∈ [−π, π]. Except for cases where sin φ = 0 or cos(k) = 0
and the above inequality is trivially true, the respective condition on f is
1 − |cosφ|
|f| ≤ fcrit =
|sinφ|
⇐⇒
|f| ≤ tan(φ/2) for |φ| ≤ π/2 ,
|f| ≤ (4.20)
cot(φ/2) for π/2 < |φ| ≤ π .
(Note that either none or both conditions hold, since tan(φ/2) ≤ cot(φ/2) for
|φ| ≤ π/2 and vice versa.) In order to retain the possibility of finding (normal) limit
7 The general case of eigenvalues which are complex and inverse to each other is excluded since
ˆΓ(k) is real-valued. Hence the characteristic polynomial of ˆ Γ(k) has real coefficients and complex
solutions are conjugated to each other.
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