Quantum Information Theory with Gaussian Systems

5 Gaussian private quantum channels
This integral is estimated for isotropic, uncorrelated Gaussian noise with uniform
covariance g as follows:
1
c
dξ e −ξT
·G·ξ/4 2Nf−1 Nf −1
= 2 g (Nf − 1)!
|ξ|≥a
a
∞
dr r 2Nf−1 e −r2 /(4g)
by introducing polar coordinates and
integrating over angular coordinates
= 2 2Nf g Nf (Nf − 1)! ∞
−1
a 2
dt t Nf−1 e −t/(4g)
substituting t = r 2
≤ 2 2Nf g Nf (Nf − 1)! ∞
−1
dt e −t/(8g)
if a 2 is large enough to ensure that
t Nf−1 e −t/(4g) ≤ e −t/(8g) for t ≥ a 2
a 2
(5.10)
= 2 2Nf−3 g Nf−1 (Nf − 1)! −1 e −a 2 /(8g) . (5.11)
Note that for the single-mode case Nf = 1 the inequality in the second to last
line becomes an equality and there is no additional condition on a. Otherwise, the
condition reads a 2 ≥ t0 , where t0 is the larger, real solution of t = 8 g (Nf −1)logt.
This solution exists, if 8 g (Nf − 1) ≥ e, which we assume to be true in the case
Nf ≥ 2 due to g ≫ 1. Combining Eqs.(5.8), (5.9) and (5.11), we arrive at the
bound
T[ ](α) − T(α) 1 ≤ 2 2Nf−4 g Nf−1 (Nf − 1)! −1 e −a 2 /(8g) . (5.12)
In the next step, the cutoff integral (5.3) over a hypersphere of the phase space is
replaced by a summation (5.4) over a discrete, regular grid of hypercubes (cf. Fig.5.2).
Each cell is labeled by a positive integer k and described by a corner point ξk and
the characteristic function of a set, χk(ξ) = 1 if ξ belongs to the k-th hypercube and
zero otherwise. The length δ of the diagonal of the hypercubes yields the maximal
distance |ξk − ξ| ≤ δ between a point in phase space and the corner of the cell in
which it is situated. The vectors ξk will constitute the set of encryption operations.
The error introduced is estimated as follows:
108
T Σ(α) − T [](α) 1 =
1
cΣ
dξ
|ξ|≤a
K
k=1
χk(ξ)e −ξT
k ·G·ξ k /4 W ξk |α〉〈α|W∗ ξk −
1
c []
|ξ|≤a
dξ e −ξT ·G·ξ/4 Wξ |α〉〈α|W ∗ ξ
,
1