138
7 Device-Independent Quantum Cryptography
and by substituting it into (7.79), we observe that all the coherences relative to Bell
states such that j = l are set to zero:
¯
ρ =
1
i, j,k=0
ρ (i j),(k j) |ψ i j ψ k j |.
(7.81)
The matrix representation of the state in (7.81) in the Bell basis is thus block-diagonal
and reads as follows, upon relabelling the coefficients
13 :
¯
ρ =
⎡
⎢
⎢
⎣
λ 00 r 0 + is 0
0
0
r 0 − is 0 λ 10
0
0
0
0
λ 01 r 1 + is 1
0
0
r 1 − is 1 λ 11
⎤
⎥
⎥
⎦ ,
(7.82)
where λ i j , r j and s j are real numbers.
Exploiting the orientation of the local reference frames The state in (7.82) can be
further reduced by carefully choosing the orientation of the parties’ local reference
frames. Indeed, although we already fixed the measurement directions of Alice and
Bob to lie in the (x, y)-plane, we can still choose the orientation of the axes with
respect to the measurement directions by applying rotations R(θ ) along the z direction
on the qubit spaces. In particular, the state distributed by Eve can be rotated w.l.o.g.
as follows:
¯
ρ + = R A (θ A ) ⊗ R B (θ B ) ¯
ρ R
†
A (θ A ) ⊗ R
†
B (θ B ),
(7.83)
where the rotation R A (θ A ) acts on Alice’s Hilbert space and is given by:
R A (θ A ) = cos
θ A
2
1 A + i sin
θ A
2
Z A ,
(7.84)
and similarly for Bob. The resulting rotated state ¯
ρ + is still block-diagonal and reads:
¯
ρ + =
⎡
⎢
⎢
⎣
λ
00
r 0 + is
0
0
0
r 0 − is
0
λ
10
0
0
0
0
λ
01
r 1 + is
1
0
0
r 1 − is
1
λ
11
⎤
⎥
⎥
⎦ ,
(7.85)
where the new matrix coefficients are given by:
13 Recall that ¯
ρ is a Hermitian operator, hence the matrix representing it must be Hermitian.
7 Device-Independent Quantum Cryptography
and by substituting it into (7.79), we observe that all the coherences relative to Bell
states such that j = l are set to zero:
¯
ρ =
1
i, j,k=0
ρ (i j),(k j) |ψ i j ψ k j |.
(7.81)
The matrix representation of the state in (7.81) in the Bell basis is thus block-diagonal
and reads as follows, upon relabelling the coefficients
13 :
¯
ρ =
⎡
⎢
⎢
⎣
λ 00 r 0 + is 0
0
0
r 0 − is 0 λ 10
0
0
0
0
λ 01 r 1 + is 1
0
0
r 1 − is 1 λ 11
⎤
⎥
⎥
⎦ ,
(7.82)
where λ i j , r j and s j are real numbers.
Exploiting the orientation of the local reference frames The state in (7.82) can be
further reduced by carefully choosing the orientation of the parties’ local reference
frames. Indeed, although we already fixed the measurement directions of Alice and
Bob to lie in the (x, y)-plane, we can still choose the orientation of the axes with
respect to the measurement directions by applying rotations R(θ ) along the z direction
on the qubit spaces. In particular, the state distributed by Eve can be rotated w.l.o.g.
as follows:
¯
ρ + = R A (θ A ) ⊗ R B (θ B ) ¯
ρ R
†
A (θ A ) ⊗ R
†
B (θ B ),
(7.83)
where the rotation R A (θ A ) acts on Alice’s Hilbert space and is given by:
R A (θ A ) = cos
θ A
2
1 A + i sin
θ A
2
Z A ,
(7.84)
and similarly for Bob. The resulting rotated state ¯
ρ + is still block-diagonal and reads:
¯
ρ + =
⎡
⎢
⎢
⎣
λ
00
r 0 + is
0
0
0
r 0 − is
0
λ
10
0
0
0
0
λ
01
r 1 + is
1
0
0
r 1 − is
1
λ
11
⎤
⎥
⎥
⎦ ,
(7.85)
where the new matrix coefficients are given by:
13 Recall that ¯
ρ is a Hermitian operator, hence the matrix representing it must be Hermitian.