108
7 Device-Independent Quantum Cryptography
p(a, b|x, y, λ) = p(a|b, x, y, λ)p(b|x, y, λ)
= p(a|x, y, λ)p(b|x, y, λ),
(7.3)
where the second equality represents the fact that, conditioned on the knowledge
of λ, the residual indeterminacy of a is local and not due to a lack of knowledge
of b.
2. Parameter independence: For each microstate λ, the probability of Alice (Bob)
obtaining outcome a (b) is independent of the input y (x) selected by Bob (Alice):
p(a|x, y, λ) = p(a|x, λ)
p(b|x, y, λ) = p(b|y, λ).
(7.4)
This assumption is justified by special relativity, according to which spacelike
separated measurements do not influence each other’s outcome probability distribution.
3. Free will: The measurements inputs are uncorrelated from the underlying state
of the systems described by λ:
p(x, y, λ) = p(x, y) p(λ).
(7.5)
In other words, Alice and Bob are free to choose their inputs independently
of the value of the hidden variables λ. By combining this assumption with the
previous one, we are basically assuming that spacelike separated parties cannot
communicate superluminally, which is the causality constraint of relativity.
By employing the assumptions (7.3), (7.4) and (7.5) in (7.2), we obtain the Bell
experiment description of a local hidden variable (LHV) model:
p(a, b|x, y) =
dλ p(λ) p(a|x, λ)p(b|y, λ).
(7.6)
We will now show that the correlations predicted by quantum mechanics on certain
implementations of the Bell experiment cannot be expressed in the form (7.6). To
this aim, we define the correlator:
a x b y =
a,b=±1
ab p(a, b|x, y),
(7.7)
as the expectation value of the product of Alice and Bob’s outcomes, given that they
selected inputs x and y. We then define the CHSH value [23]:
S CHSH = =a 0 b 0 + +a 0 b 1 + +a 1 b 0 − −a 1 b 1
(7.8)
and prove that if the probabilities p(a, b|x, y) are explainable in terms of a LHV
model (7.6), then the CHSH inequality holds:
7 Device-Independent Quantum Cryptography
p(a, b|x, y, λ) = p(a|b, x, y, λ)p(b|x, y, λ)
= p(a|x, y, λ)p(b|x, y, λ),
(7.3)
where the second equality represents the fact that, conditioned on the knowledge
of λ, the residual indeterminacy of a is local and not due to a lack of knowledge
of b.
2. Parameter independence: For each microstate λ, the probability of Alice (Bob)
obtaining outcome a (b) is independent of the input y (x) selected by Bob (Alice):
p(a|x, y, λ) = p(a|x, λ)
p(b|x, y, λ) = p(b|y, λ).
(7.4)
This assumption is justified by special relativity, according to which spacelike
separated measurements do not influence each other’s outcome probability distribution.
3. Free will: The measurements inputs are uncorrelated from the underlying state
of the systems described by λ:
p(x, y, λ) = p(x, y) p(λ).
(7.5)
In other words, Alice and Bob are free to choose their inputs independently
of the value of the hidden variables λ. By combining this assumption with the
previous one, we are basically assuming that spacelike separated parties cannot
communicate superluminally, which is the causality constraint of relativity.
By employing the assumptions (7.3), (7.4) and (7.5) in (7.2), we obtain the Bell
experiment description of a local hidden variable (LHV) model:
p(a, b|x, y) =
dλ p(λ) p(a|x, λ)p(b|y, λ).
(7.6)
We will now show that the correlations predicted by quantum mechanics on certain
implementations of the Bell experiment cannot be expressed in the form (7.6). To
this aim, we define the correlator:
a x b y =
a,b=±1
ab p(a, b|x, y),
(7.7)
as the expectation value of the product of Alice and Bob’s outcomes, given that they
selected inputs x and y. We then define the CHSH value [23]:
S CHSH = =a 0 b 0 + +a 0 b 1 + +a 1 b 0 − −a 1 b 1
(7.8)
and prove that if the probabilities p(a, b|x, y) are explainable in terms of a LHV
model (7.6), then the CHSH inequality holds: