FOUNDATIONS OF QUANTUM MECHANICS

154 CHAPTER VII. BELL’S INEQUALITIES
1. Outcome independence
The probability to find a value a for ⃗a · ⃗σ is ‘completely’ determined by the settings of the spin
meters and by λ, particularly, it is not necessary to also give outcome b, likewise for finding a
value b,
p ⃗a, ⃗ b
(a | b ∧ λ) = p ⃗a, ⃗ b
(a | λ) and p ⃗a, ⃗ b
(b | a ∧ λ) = p ⃗a, ⃗ b
(b | λ). (VII. 61)
2. Parameter independence
The probability to find the outcome of measurement a or b is independent of the settings of the
remote spin meter,
p ⃗a, ⃗ b
(a | λ) = p ⃗a (a | λ) and p ⃗a, ⃗ b
(b | λ) = p ⃗b (b | λ). (VII. 62)
3. Source independence
The distribution of λ in the source does not depend on the settings of the spin meters,
ρ ⃗a, ⃗ b
(λ) = ρ(λ). (VII. 63)
In principle we can adjust the spin meters ‘at the last moment’, long after the particles have left
the source. It is reasonable to assume that the source is not influenced by what happens to the
measuring devices in the future.
Now we will prove the next theorem.
BELL’S THIRD THEOREM:
A stochastic HVT which is in agreement with outcome, parameter and source independence
is empirically inconsistent with quantum mechanics.
Proof
As a consequence of the aforementioned properties, in every local stochastic HVT, (VII. 60) becomes
p ⃗a, ⃗ b
(a, b, λ) = p ⃗a (a | λ) p ⃗b (b | λ) ρ(λ), (VII. 64)
or
p ⃗a, ⃗ b
(a, b | λ) = p ⃗a (a | λ) p ⃗b (b | λ), (VII. 65)
which means that the quantities A and B are statistically independent of each other for given λ.
This statement is often called factorizability or conditional independence.