7.9 Lower Bound on the Conditional Entropy: Analytical Proof
145
Note that in the parameter regimes of x and y it holds that
0 ≤
1 − x − y
1
2
− y
≤ 1.
(7.124)
We finally analyse the properties of g(
1
2
, y), which is convex in y since its second
derivative is always positive:
∂
2 g(
1
2
, y)
∂ y 2
=
1
y ln(2) + y 2 ln(4)
> 0.
(7.125)
A convex function has a unique minimum if it exists in the parameter regime. In our
case this is given by:
∂ g(
1
2
, y)
∂ y
= log(2y) − log(
1
2
+ y)
!
= 0 ⇔ y =
1
2
(7.126)
for which g(
1
2
,
1
2
) = 0 holds. Thus in general it holds:
g
1
2
, y
≥ 0.
(7.127)
By combining these considerations we prove the inequality in (7.116):
D
(7.120)
≥ g(λ
α
01 , λ
α
00 )
(7.123)
≥
1 − λ
α
01 − λ
α
00
1
2
− λ
α
00
g
1
2
, λ
α
00
≥ 0,
(7.128)
where in the last inequality we used the fact that the pre-factor is positive (7.124) and
that g(
1
2
, λ
α
00 ) is lower bounded by zero (7.127). This concludes the proof of inequality (7.113), thus completing the analytical solution of the optimization problem in
(7.102).
References
1. Yao, A., Mayers, D., & (1998). Quantum cryptography with imperfect apparatus. In 2013 IEEE
54th Annual Symposium on Foundations of Computer Science, , Los Alamitos, CA (p. 503).
USA: IEEE Computer Society.
2. Acín, A., Gisin, N., & Masanes, L. (2006). From Bell’s theorem to secure quantum key distribution. Physical Review Letters, 97, 120405.