Quantum Information Theory with Gaussian Systems

characteristic function
χrand(ξ) = tr
1
T(ρ)Wξ = dη e
c
−ηT ·G·η/4
tr Wη ρ W ∗ η Wξ
= 1
dη e
c
−ηT ·G·η/4 iσ(η,ξ)
e χin(ξ)
= exp −ξ T
· (σ T
G −1 σ) · ξ/4 χin(ξ).
5.1 Setup
In the Heisenberg picture, T maps Weyl operators to multiples of themselves, so it is
a completely positive map: Wξ ↦→ Wξ exp −ξ T ·(σ T G −1 σ)·ξ/4 , where σ T G −1 σ ≥ 0
(cf. Section 2.3). Since the factor c assures normalization, it is even a channel. A
Gaussian input state with covariance matrix γ and displacement α is transformed
into a Gaussian state with characteristic function
χrand(ξ) = exp −ξ T
· (γ + σ T
G −1 σ) · ξ/4 − iσ(ξ, α) , (5.2)
i.e. the covariance matrix is changed according to γ ↦→ γ + σ T G −1 σ, but the (average)
displacement is not affected. This is visualized in Fig. 5.1: an initial coherent
state |α〉〈α| is represented by alollipop stick, where amplitude and phase are depicted
by the vector α and the uncertainty is indicated by the circle corresponding
to the covariance matrix γ = (cf. Section 2.2); adding classical, uncorrelated Gaussian
noise with isotropic variance g, i.e. with covariance matrix G = /g, enlarges
the uncertainty by σ T G −1 σ = g and hence the radius of the circle by g (the dotted
and dashed circles for medium and larger g). Since the displacement is not affected,
these circles are centered around the endpoint of α.
In view of the discretization we define two variants of the above channel, a cutoff
version T [ ] where the integration is restricted to phase space translations with absolute
value |ξ| ≤ a and its discretized counterpart T Σ, which replaces the integration
by a summation over a finite set of phase space displacements {ξk}k=1,...,K suitable
to approximate the integral:
T [](ρ) = 1
c [ ]
T Σ(ρ) = 1
c Σ
|ξ|≤a
K
k=1
dξ e −ξT ·G·ξ/4 Wξ ρ W ∗ ξ , (5.3)
e −ξT k ·G·ξk/4
Wξk ρ W∗ , (5.4)
ξk
where c [ ] and c Σ provide normalization. The set {ξk} and the cutoff radius a remain
to be determined below. For convenience, we introduce a short-hand notation for
randomized coherent input states |α〉〈α| of Nf modes,
T(α) = T |α〉〈α|
and likewise for T [ ](α) and T Σ(α). Furthermore, we can write T(α) = W α T(0)W ∗ α
for all three flavors of T.
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