7.5 Conditional Entropy Bound
123
We can now formulate the optimization problem (stated in Remark 7.1) whose
solution is the lower bound (7.36) on H (R A |E) ρ α . The optimization problem reads
as follows:
F(S α ) := min
{λ
α
i j }
1 − H ({λ
α
i j }) + h(λ
α
00 + λ
α
01 )
sub. to S α ≥ S α ; λ
α
0 j ≥ λ
α
1 j ;
i, j=0,1
λ
α
i j = 1,
(7.50)
and its detailed solution is given in the Appendix of this chapter (Sect. 7.9). We
remark that the solution presented in this book is based on a completely different
approach with respect to the original derivation in [3]. In particular, the proposed
solution is inspired by similar proofs contained in [42] where the results of this
section are extended to multiparty scenarios.
The solution of the above optimization is given by:
F(S α ) = 1 − h
⎛
⎝ 1
2
+
1
2
S α
2
2
− 1
⎞
⎠ ,
(7.51)
which is a convex function as required by (7.37). Hence, by employing (7.51) in
(7.37), we finally obtain the lower bound on the conditional von Neumann entropy
of Alice’s raw key bit as a function of the observed CHSH violation S:
H (R A |E tot ) ≥ 1 − h
⎛
⎝ 1
2
+
1
2
S
2
2
− 1
⎞
⎠ .
(7.52)
By employing the derived bound e.g., in the asymptotic secret key rate (7.29) of
the described DIQKD protocol, one can obtain a lower bound on the achievable key
rate in terms of the observed CHSH violation S. We stress the fact that the bound
in (7.52) plays a crucial role in obtaining an analytical expression for the secret key
rate of any DIQKD protocol based on the CHSH inequality.
7.5.1 Privacy Certification in Standard- and DI-QKD
The conditional entropy bound in (7.52) can be seen as a quantitative certification
of the privacy of Alice’s key bit, in the context of a DIQKD protocol based on the
CHSH inequality. It is interesting to compare this result with the analogous privacy
certification (conditional entropy bound) used in a standard QKD protocol, namely
the BB84 protocol studied in Sect. 3.2.
In order to carry out a fair comparison, we set equal grounds for the DIQKD
protocol and the BB84 protocol. In particular, we have seen that the ideal resource
123
We can now formulate the optimization problem (stated in Remark 7.1) whose
solution is the lower bound (7.36) on H (R A |E) ρ α . The optimization problem reads
as follows:
F(S α ) := min
{λ
α
i j }
1 − H ({λ
α
i j }) + h(λ
α
00 + λ
α
01 )
sub. to S α ≥ S α ; λ
α
0 j ≥ λ
α
1 j ;
i, j=0,1
λ
α
i j = 1,
(7.50)
and its detailed solution is given in the Appendix of this chapter (Sect. 7.9). We
remark that the solution presented in this book is based on a completely different
approach with respect to the original derivation in [3]. In particular, the proposed
solution is inspired by similar proofs contained in [42] where the results of this
section are extended to multiparty scenarios.
The solution of the above optimization is given by:
F(S α ) = 1 − h
⎛
⎝ 1
2
+
1
2
S α
2
2
− 1
⎞
⎠ ,
(7.51)
which is a convex function as required by (7.37). Hence, by employing (7.51) in
(7.37), we finally obtain the lower bound on the conditional von Neumann entropy
of Alice’s raw key bit as a function of the observed CHSH violation S:
H (R A |E tot ) ≥ 1 − h
⎛
⎝ 1
2
+
1
2
S
2
2
− 1
⎞
⎠ .
(7.52)
By employing the derived bound e.g., in the asymptotic secret key rate (7.29) of
the described DIQKD protocol, one can obtain a lower bound on the achievable key
rate in terms of the observed CHSH violation S. We stress the fact that the bound
in (7.52) plays a crucial role in obtaining an analytical expression for the secret key
rate of any DIQKD protocol based on the CHSH inequality.
7.5.1 Privacy Certification in Standard- and DI-QKD
The conditional entropy bound in (7.52) can be seen as a quantitative certification
of the privacy of Alice’s key bit, in the context of a DIQKD protocol based on the
CHSH inequality. It is interesting to compare this result with the analogous privacy
certification (conditional entropy bound) used in a standard QKD protocol, namely
the BB84 protocol studied in Sect. 3.2.
In order to carry out a fair comparison, we set equal grounds for the DIQKD
protocol and the BB84 protocol. In particular, we have seen that the ideal resource