Quantum Information Theory with Gaussian Systems

5 Gaussian private quantum channels
Repeating our task in this notation, we want to ensure that any two discretely
randomized tensor products T Σ(α) and T Σ(β) of N coherent input states with f
modes each are nearly indistinguishable, T Σ(α) − T Σ(β) 1 < ǫ, if they obey the
energy constraint |α| 2 , |β| 2 ≤ 2NfEmax, i.e. if each single mode contributes at most
energy Emax.
5.2 Security estimation
Since the relevant distinguishability T Σ(α) − T Σ(β) 1 is not easily accessible, we
use the triangle inequality and derive a bound in terms of the trace norm distances
T Σ(α) −T [](α) 1 and T [](α) −T(α) 1 , which determine the quality of the involved
approximations and can thus be bounded, and T(α)−T(β) 1, which can be bounded
by the relative entropy distance. These quantities are introduced by applying the
triangle inequality for the trace norm:
TΣ(α) − T Σ(β) 1 ≤ T Σ(α) − T [](α) 1 + T Σ(β) − T [ ](β) 1 + T [](α) − T [ ](β) 1
≤ T Σ(α) − T [](α) 1 + T Σ(β) − T [ ](β) 1 + T [](α) − T(α) 1
+ T [](β) − T(β) 1 + T(α) − T(β) 1 .
(5.5)
We proceed by deriving bounds for each term. The trace norm distance of two
density operators ρ, ρ ′ can be estimated by the relative entropy distance S(ρ ρ ′ ) =
tr[ρ (log ρ − log ρ ′ )] between the operators [97, Thm. 5.5]. This is used to establish
2
T(α) − T(β)1 ≤ 2 S T(α) T(β) . (5.6)
The exponential form (2.34a) for the density operator of a Gaussian state allows to
express the relative entropy in terms of the symplectic eigenvalues γn of its covariance
matrix:
106
S T(α) T(β) = tr T(α) log T(α) − log T(β)
= tr T(0) − T(α − β) log T(0)
= 1
2
= 1
2
= 1
2
2Nf
i,j=1
2Nf
i,j=1
2Nf
i,j=1
since T(α) = W α T(0)W ∗ α
M ′ i,j tr T(0) − T(α − β) R ′ i R ′
j by (2.35)
M ′ i,j
M ′ i,j
tr T(0)R ′ i R ′ ′
j − tr T(α − β)R i R ′
j
tr T(0)R ′ i R ′
j −
tr T(0) R ′ i − (α′ − β ′ ′
)i R j − (α ′ − β ′
)j