7.8 State Reduction in the CHSH Scenario
137
where
x,α
±1 are the projectors on the eigenvalues ±1 of the qubit projective measurement defined by Alice’s choice of input x and by the particular state ρ α she is
measuring.
We now fix α and proceed in specifying the form of the two-qubit state ρ α (we
omit the symbol α in the following). At the moment, we can only say that ρ α is a
normalized positive Hermitian operator acting on the four-dimensional Hilbert space
H A ⊗ H B .
Symmetrization of the marginal distributions We define the planes individuated
by the two measurements of Alice and of Bob to be the (x, y)-plane of the Bloch
sphere. We can now assume w.l.o.g. that the marginal distributions of the outcomes,
p(a|x) and p(b|y), are symmetrized. In other words, the expectation values of the
corresponding observables are null:
A x = =B y = 0 ∀ x, y ∈ {0, 1},
(7.78)
where A x and B y are the observables of Alice and Bob, respectively, defining rankone binary projective measurements in the (x, y)-plane. If (7.78) were not true, Alice
and Bob could enforce it by agreeing on flipping their outcomes with probability 1 /2
in every round. This classical procedure would not change the observed CHSH value
(7.27) nor the QBER, since either both parties flip or none of them does. Additionally,
it would require classical communication between Alice and Bob, known to Eve.
Given that the experiment statistics satisfies (7.78), we can assume that it is Eve
herself who flips the outcomes in place of the parties. However, instead of doing this
classically on the outcomes of the devices, she could provide Alice and Bob with a
suitable state which already embodies the symmetry of the outcomes distributions.
By calling ρ the generic state she initially prepares, the state she distributes that
satisfies (7.78) is given by:
¯
ρ =
1
2
ρ + Z A ⊗ Z B ρ Z
†
A ⊗ Z
†
B
,
(7.79)
where Z A , Z B are Pauli operators on Alice’s and Bob’s qubits. As a matter of fact,
the outcome of a measurement in the (x, y)-plane is flipped if one first applies the
Z operator.
We remark that it is safe to assume that Eve distributes the state (7.79) since this
is not disadvantageous to her. Indeed, her uncertainty on Alice’s raw key bit R A ,
quantified by the conditional von Neumann entropy H (R A |E), does not increase
when she sends the state ¯
ρ instead of ρ. The proof of this fact follows the same lines
of the proof given in Sect. 3.5, hence we omit it.
By expressing the initial generic state ρ in the Bell basis (3.15):
ρ =
1
i, j,k,l=0
ρ (i j),(kl) |ψ i j ψ kl | ρ (i j),(kl) ∈ C,
(7.80)
137
where
x,α
±1 are the projectors on the eigenvalues ±1 of the qubit projective measurement defined by Alice’s choice of input x and by the particular state ρ α she is
measuring.
We now fix α and proceed in specifying the form of the two-qubit state ρ α (we
omit the symbol α in the following). At the moment, we can only say that ρ α is a
normalized positive Hermitian operator acting on the four-dimensional Hilbert space
H A ⊗ H B .
Symmetrization of the marginal distributions We define the planes individuated
by the two measurements of Alice and of Bob to be the (x, y)-plane of the Bloch
sphere. We can now assume w.l.o.g. that the marginal distributions of the outcomes,
p(a|x) and p(b|y), are symmetrized. In other words, the expectation values of the
corresponding observables are null:
A x = =B y = 0 ∀ x, y ∈ {0, 1},
(7.78)
where A x and B y are the observables of Alice and Bob, respectively, defining rankone binary projective measurements in the (x, y)-plane. If (7.78) were not true, Alice
and Bob could enforce it by agreeing on flipping their outcomes with probability 1 /2
in every round. This classical procedure would not change the observed CHSH value
(7.27) nor the QBER, since either both parties flip or none of them does. Additionally,
it would require classical communication between Alice and Bob, known to Eve.
Given that the experiment statistics satisfies (7.78), we can assume that it is Eve
herself who flips the outcomes in place of the parties. However, instead of doing this
classically on the outcomes of the devices, she could provide Alice and Bob with a
suitable state which already embodies the symmetry of the outcomes distributions.
By calling ρ the generic state she initially prepares, the state she distributes that
satisfies (7.78) is given by:
¯
ρ =
1
2
ρ + Z A ⊗ Z B ρ Z
†
A ⊗ Z
†
B
,
(7.79)
where Z A , Z B are Pauli operators on Alice’s and Bob’s qubits. As a matter of fact,
the outcome of a measurement in the (x, y)-plane is flipped if one first applies the
Z operator.
We remark that it is safe to assume that Eve distributes the state (7.79) since this
is not disadvantageous to her. Indeed, her uncertainty on Alice’s raw key bit R A ,
quantified by the conditional von Neumann entropy H (R A |E), does not increase
when she sends the state ¯
ρ instead of ρ. The proof of this fact follows the same lines
of the proof given in Sect. 3.5, hence we omit it.
By expressing the initial generic state ρ in the Bell basis (3.15):
ρ =
1
i, j,k,l=0
ρ (i j),(kl) |ψ i j ψ kl | ρ (i j),(kl) ∈ C,
(7.80)