7.9 Lower Bound on the Conditional Entropy: Analytical Proof
143
By proving the following result (with the assumption λ
α
01 ≥ λ
α
00 and (7.106)):
H (R A |E) ρ α ≥ H (R A |E) τ (λ
α
01 ) ∀ ρ α
(7.113)
we obtain the solution of the optimization problem. In order to realize this, let us
use the last expression on the state ρ
∗
α , which is the solution of the minimization in
(7.102):
F(S α ) = H (R A |E) ρ ∗
α
≥ H (R A |E) τ (λ
α,∗
01 )
≥ H (R A |E) τ (ν α ).
(7.114)
The last inequality in (7.114) is motivated by the following two observations:
• By applying (7.109) to the state ρ
∗
α , we obtain S τ (λ
α,∗
01 ) ≥ S α,∗ ≥ S α , which combined with (7.112) implies that λ
α,∗
01 ≥ ν α since S τ (ν) in (7.108) is monotonically
increasing in the interval ν ∈ [
1
2
, 1].
• The entropy H (R A |E) τ (ν) in (7.110) is monotonically increasing in the interval
ν ∈ [
1
2
, 1].
The above observations lead to the second inequality in (7.114).
By combining (7.114) with (7.111), we obtain the desired lower bound:
F(S α ) = H (R A |E) τ (ν α ) = 1 − h(ν α )
= 1 − h
⎛
⎝ 1
2
+
1
2
S α
2
2
− 1
⎞
⎠ ,
(7.115)
where the last equality is obtained by reverting Eq. (7.112).
The bound (7.115) is a tight solution of the optimization problem in (7.102).
Indeed, for every violation S α , there exists a state τ (ν α ) such that its entropy coincides
with the bound and such that it can produce a violation equal to S α , thanks to S τ (ν α ) =
S α (7.112) and to the fact that the maximal CHSH violation (7.103) is achievable.
We are left to prove the inequality in (7.113), which can be made explicit by using
(7.101) and (7.110):
D := h(λ
α
01 ) − H ({λ
α
i j }) + h(λ
α
00 + λ
α
01 ) ≥ 0.
(7.116)
We simplify the first two terms in D:
h(λ
α
01 ) − H ({λ
α
i j }) = −(1 − λ
α
01 ) log(1 − λ
α
01 ) +
(i, j) =(0,1)
λ
α
i j log λ
α
i j . (7.117)