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7 Device-Independent Quantum Cryptography
By noticing that the second term in (7.103) is larger than the first if and only if
λ
α
01 ≥ λ
α
00 , we can simplify the maximal CHSH violation to:
S α = 2
√
2
(λ
α
00 − λ
α
10 ) 2 + (λ
α
01 − λ
α
11 ) 2 .
(7.104)
Then, a necessary condition for S α ≥ 2 can be derived by upper bounding (7.104)
as follows:
2 ≤ S α ≤ S α ≤ 2
√
2
(λ
α
00 ) 2 + (λ
α
01 ) 2 ≤ 2
√
2
λ
α
01 (λ
α
01 + λ
α
00 ) ≤ 2
√
2
λ
α
01 ,
(7.105)
which implies the following necessary condition on λ
α
01 :
λ
α
01 ≥
1
2
.
(7.106)
Consider the following class of states parametrized by ν ∈ [
1
2
, 1]:
τ (ν) = (1 − ν)|ψ 00 ψ 00 | + ν|ψ 01 ψ 01 |,
(7.107)
whose maximal CHSH violation (7.103) reads:
S τ (ν) = 2
√
2
ν 2 + (1 − ν) 2 .
(7.108)
It is straightforward to verify, by using the last expression, that
S τ (λ
α
01 ) ≥ S α ∀ ρ α ,
(7.109)
where S α is given in (7.104). Moreover, the entropy (7.101) of the states (7.107)
reads:
H (X |E) τ (ν) = 1 − h(ν),
(7.110)
where we used the binary entropy h(x) = −x log x − (1 − x) log(1 − x).
By definition of the optimization problem (7.102), the solution of the optimization
for a given S α is upper bounded by the entropy of any particular state with S α = S α .
Thus for the states (7.107) we have:
F(S α ) ≤ H (R A |E) τ (ν α )
(7.111)
where ν α is fixed such that the maximal violation S τ (ν α ) of the state τ (ν α ) is exactly
given by S α :
S τ (ν α ) = 2
√
2
ν 2
α + (1 − ν α ) 2 = S α ,
(7.112)
where we choose the solution ν α ≥ 1/2.