7.7 Device-Independent Conference Key Agreement
135
similar techniques. The bound would guarantee a more accurate security analysis
and hence a better performance.
Finally, we mention that the only other DICKA protocol proposed so far [9] is
based on a Bell inequality that can be seen as a particular case of the one introduced
in [10] and analysed here in the tripartite case. In particular, the inequality used in
[9] is recovered when one imposes that Alice i for i ≥ 3 has only one measurement
setting at her disposal used for testing, instead of two.
The DICKA protocol proposed in [9] also lacks a tight security proof, and would
benefit from tighter analytical bounds on H (R A |E) like the one derived in [42].
Appendix
In this Appendix we prove some statements made in the main text, whose articulated
proof would have altered the cohesion and flow of the text.
7.8 State Reduction in the CHSH Scenario
Here we present the proof of Theorem 7.1, whose statement is reported for clarity.
Theorem 7.5 ([3]) Let Alice and Bob perform the DIQKD protocol described in
Subsect. 7.4.2. It is not restrictive to assume that, in each round, Eve distributes a
mixture
α p α ρ α of two-qubit states ρ α , together with a flag |α (known to her) which
determines the measurements performed on ρ α given the parties’ inputs. Without loss
of generality, the measurements performed by Alice’s and Bob’s devices on ρ α are
rank-one binary projective measurements in the (x, y)-plane of the Bloch sphere.
Moreover, each state ρ α is diagonal in the Bell basis (3.15) and reads:
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}.
(7.74)
Proof The proof follows the same principles of the original proof in [3], however
we apply some modifications and add details in a way which is coherent with its
generalization valid for N parties that we prove in [42].
Reduction to qubits Firstly we reduce the state shared by Alice and Bob in one
round to a convex combination of two-qubit states.
Recall that the statistics of a general quantum measurement (POVM, c.f. Sect. 2.2)
is reproduced by a projective measurement in a larger Hilbert space, due to the
Naimark theorem [55, 56]. Since in a DI scenario the Hilbert space dimensions are
135
similar techniques. The bound would guarantee a more accurate security analysis
and hence a better performance.
Finally, we mention that the only other DICKA protocol proposed so far [9] is
based on a Bell inequality that can be seen as a particular case of the one introduced
in [10] and analysed here in the tripartite case. In particular, the inequality used in
[9] is recovered when one imposes that Alice i for i ≥ 3 has only one measurement
setting at her disposal used for testing, instead of two.
The DICKA protocol proposed in [9] also lacks a tight security proof, and would
benefit from tighter analytical bounds on H (R A |E) like the one derived in [42].
Appendix
In this Appendix we prove some statements made in the main text, whose articulated
proof would have altered the cohesion and flow of the text.
7.8 State Reduction in the CHSH Scenario
Here we present the proof of Theorem 7.1, whose statement is reported for clarity.
Theorem 7.5 ([3]) Let Alice and Bob perform the DIQKD protocol described in
Subsect. 7.4.2. It is not restrictive to assume that, in each round, Eve distributes a
mixture
α p α ρ α of two-qubit states ρ α , together with a flag |α (known to her) which
determines the measurements performed on ρ α given the parties’ inputs. Without loss
of generality, the measurements performed by Alice’s and Bob’s devices on ρ α are
rank-one binary projective measurements in the (x, y)-plane of the Bloch sphere.
Moreover, each state ρ α is diagonal in the Bell basis (3.15) and reads:
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}.
(7.74)
Proof The proof follows the same principles of the original proof in [3], however
we apply some modifications and add details in a way which is coherent with its
generalization valid for N parties that we prove in [42].
Reduction to qubits Firstly we reduce the state shared by Alice and Bob in one
round to a convex combination of two-qubit states.
Recall that the statistics of a general quantum measurement (POVM, c.f. Sect. 2.2)
is reproduced by a projective measurement in a larger Hilbert space, due to the
Naimark theorem [55, 56]. Since in a DI scenario the Hilbert space dimensions are