140
7 Device-Independent Quantum Cryptography
We observe that the two states ¯
ρ + and ¯
ρ − yield the same probability distribution of
the outcomes:
p(a, b) ¯
ρ + = Tr
a b ¯
ρ +
= Tr
a b ¯
ρ −
= p(a, b) ¯
ρ − ,
(7.94)
where the projectors a and b represent Alice and Bob’s projective measurements in the (x, y)-plane relative to some non-specified inputs. The projectors can
be parametrized by writing the corresponding observables A and B as convex combinations of the Pauli operators X and Y :
A = cos(ϕ A )X + sin(ϕ A )Y
B = cos(ϕ B )X + sin(ϕ B )Y ,
(7.95)
for some unknown angles ϕ A , ϕ B . The eigenstates of the observables in (7.95) read:
|a A =
1
√
2
(|0 + (−1)
a e
iϕ A |1)
|b B =
1
√
2
(|0 + (−1)
b e
iϕ B |1)
(7.96)
where the measurement outcomes are defined as a, b ∈ {0, 1} (a = 0 corresponds
to eigenvalue +1 and a = 1 to eigenvalue −1). Then the projectors a and b are
simply given by a = |aa | A and b = |bb| B .
Furthermore, the states ¯
ρ + and ¯
ρ − provide Eve with the same information, i.e.
their conditional entropies coincide:
H (R A |E) ¯
ρ + = H (R A |E) ¯
ρ − .
(7.97)
Additionally, it is not disadvantageous for Eve to prepare the balanced mixture:
ρ α :=
¯
ρ + + ¯
ρ −
2
,
(7.98)
rather than preparing one of the two states with certainty, if she knows which of the
two states she prepared:
H (R A |E) ρ α ≤ H (R A |E) ¯
ρ + .
(7.99)
The proofs of the observations (7.94), (7.97) and (7.99) follow by direct computation
and are omitted. Nevertheless, the interested reader can find analogous proofs in the
Supplementary Information in [42], valid for the general N -party scenario.
We conclude that it is not restrictive to assume that Eve distributes to the parties
the mixture (7.76) of two-qubit states ρ α together with ancillae that fix the parties’
possible measurements. Each state ρ α is given by (7.98) and is diagonal in the Bell
basis:
7 Device-Independent Quantum Cryptography
We observe that the two states ¯
ρ + and ¯
ρ − yield the same probability distribution of
the outcomes:
p(a, b) ¯
ρ + = Tr
a b ¯
ρ +
= Tr
a b ¯
ρ −
= p(a, b) ¯
ρ − ,
(7.94)
where the projectors a and b represent Alice and Bob’s projective measurements in the (x, y)-plane relative to some non-specified inputs. The projectors can
be parametrized by writing the corresponding observables A and B as convex combinations of the Pauli operators X and Y :
A = cos(ϕ A )X + sin(ϕ A )Y
B = cos(ϕ B )X + sin(ϕ B )Y ,
(7.95)
for some unknown angles ϕ A , ϕ B . The eigenstates of the observables in (7.95) read:
|a A =
1
√
2
(|0 + (−1)
a e
iϕ A |1)
|b B =
1
√
2
(|0 + (−1)
b e
iϕ B |1)
(7.96)
where the measurement outcomes are defined as a, b ∈ {0, 1} (a = 0 corresponds
to eigenvalue +1 and a = 1 to eigenvalue −1). Then the projectors a and b are
simply given by a = |aa | A and b = |bb| B .
Furthermore, the states ¯
ρ + and ¯
ρ − provide Eve with the same information, i.e.
their conditional entropies coincide:
H (R A |E) ¯
ρ + = H (R A |E) ¯
ρ − .
(7.97)
Additionally, it is not disadvantageous for Eve to prepare the balanced mixture:
ρ α :=
¯
ρ + + ¯
ρ −
2
,
(7.98)
rather than preparing one of the two states with certainty, if she knows which of the
two states she prepared:
H (R A |E) ρ α ≤ H (R A |E) ¯
ρ + .
(7.99)
The proofs of the observations (7.94), (7.97) and (7.99) follow by direct computation
and are omitted. Nevertheless, the interested reader can find analogous proofs in the
Supplementary Information in [42], valid for the general N -party scenario.
We conclude that it is not restrictive to assume that Eve distributes to the parties
the mixture (7.76) of two-qubit states ρ α together with ancillae that fix the parties’
possible measurements. Each state ρ α is given by (7.98) and is diagonal in the Bell
basis: