7.5 Conditional Entropy Bound
119
ρ α =
1
i, j=0
λ
α
i j |ψ i j ψ i j | with λ
α
0 j ≥ λ
α
1 j ∀ j ∈ {0, 1}.
(7.31)
The proof of Theorem 7.1 is given in the Appendix of this chapter (Sect. 7.8) and is
rearranged in order to coherently fit with the more general result proved in [42].
The second crucial ingredient to derive the bound on H (R A |E) is the analytical
expression of the maximal CHSH violation S ρ that can achieved on a given two-qubit
state ρ. In other words, there exist measurements performed by Alice and Bob on
ρ such that the observed CHSH violation is S = S ρ , and any other measurement
setting leads to violations S ≤ S ρ . This is a well-known result derived in [43].
Theorem 7.2 ([43]) The maximum violation S ρ of the CHSH inequality (7.27),
attained by a two-qubit state ρ, is given by:
S ρ = 2
√
t 0 + t 1
(7.32)
where t 0 and t 1 are the largest and second-to-the-largest eigenvalues of the matrix
T ρ T
T
ρ , where T ρ is the correlation matrix of ρ, with elements: [T ρ ] i j = Tr[ρ(σ i ⊗
σ j )] for i, j = 1, 2, 3 (σ i are the Pauli matrices).
For the state ρ α in (7.31), the maximal CHSH violation reads:
S α = 2
√
2 max
(λ
α
00 − λ
α
11 ) 2 + (λ
α
01 − λ
α
10 ) 2 ,
(λ
α
00 − λ
α
10 ) 2 + (λ
α
01 − λ
α
11 ) 2
.
(7.33)
We are now ready to derive the lower bound on the conditional entropy H (R A |E)
in terms of the observed CHSH violation S. The derivation provided here, although
based on the same concepts used in [3], presents further details in order to better
guide the reader in the various steps. Moreover, the last step of the proof leading
to the final result substantially differs from [3] as it employs a completely different
approach.
To start with, Theorem 7.1 says that we can restrict the computation of the conditional entropy of interest over a mixture of states ρ α of the form (7.31). We emphasize that the total information available to Eve includes the knowledge of the value
α stored in a random variable , therefore we must compute the conditional entropy
H (R A |E tot ), where E tot = E.
More specifically, the aim is to derive a lower bound F(S) of H (R A |E tot ) where
F is a function of the observed CHSH violation S. The bound is tight if for every
violation S there exist a quantum state and a set of measurements that achieve that
violation and whose conditional entropy is exactly given by F(S).
119
ρ α =
1
i, j=0
λ
α
i j |ψ i j ψ i j | with λ
α
0 j ≥ λ
α
1 j ∀ j ∈ {0, 1}.
(7.31)
The proof of Theorem 7.1 is given in the Appendix of this chapter (Sect. 7.8) and is
rearranged in order to coherently fit with the more general result proved in [42].
The second crucial ingredient to derive the bound on H (R A |E) is the analytical
expression of the maximal CHSH violation S ρ that can achieved on a given two-qubit
state ρ. In other words, there exist measurements performed by Alice and Bob on
ρ such that the observed CHSH violation is S = S ρ , and any other measurement
setting leads to violations S ≤ S ρ . This is a well-known result derived in [43].
Theorem 7.2 ([43]) The maximum violation S ρ of the CHSH inequality (7.27),
attained by a two-qubit state ρ, is given by:
S ρ = 2
√
t 0 + t 1
(7.32)
where t 0 and t 1 are the largest and second-to-the-largest eigenvalues of the matrix
T ρ T
T
ρ , where T ρ is the correlation matrix of ρ, with elements: [T ρ ] i j = Tr[ρ(σ i ⊗
σ j )] for i, j = 1, 2, 3 (σ i are the Pauli matrices).
For the state ρ α in (7.31), the maximal CHSH violation reads:
S α = 2
√
2 max
(λ
α
00 − λ
α
11 ) 2 + (λ
α
01 − λ
α
10 ) 2 ,
(λ
α
00 − λ
α
10 ) 2 + (λ
α
01 − λ
α
11 ) 2
.
(7.33)
We are now ready to derive the lower bound on the conditional entropy H (R A |E)
in terms of the observed CHSH violation S. The derivation provided here, although
based on the same concepts used in [3], presents further details in order to better
guide the reader in the various steps. Moreover, the last step of the proof leading
to the final result substantially differs from [3] as it employs a completely different
approach.
To start with, Theorem 7.1 says that we can restrict the computation of the conditional entropy of interest over a mixture of states ρ α of the form (7.31). We emphasize that the total information available to Eve includes the knowledge of the value
α stored in a random variable , therefore we must compute the conditional entropy
H (R A |E tot ), where E tot = E.
More specifically, the aim is to derive a lower bound F(S) of H (R A |E tot ) where
F is a function of the observed CHSH violation S. The bound is tight if for every
violation S there exist a quantum state and a set of measurements that achieve that
violation and whose conditional entropy is exactly given by F(S).