FOUNDATIONS OF QUANTUM MECHANICS

VII. 5. STOCHASTIC HIDDEN VARIABLES 157
3. Source independence. We already discussed the possibility of violation of the Bell inequalities
by a super-deterministic theory without source independence. It is a philosophical question whether
we can somehow establish if we have free will or not, therefore, it is a possibility, but not an inevitability,
leaving outcome and parameter independence.
2. Parameter independence. Describing the pairs of particles in the singlet state |Ψ 0 ⟩, (III. 165),
by a pure hidden - variables state, the probability distribution is a delta - distribution,
ρ Ψ0 (λ) = δ λ0 (λ) := δ(λ − λ 0 ), (VII. 71)
which leads to
∫
p ⃗a, ⃗ b,λ0
(a, b, λ) ρ Ψ0 (λ) dλ = p ⃗a, ⃗ b,λ0
(a, b). (VII. 72)
Λ
The probabilities for the outcomes of measurement are given by (III. 176),
p ⃗a, ⃗ b,λ0
(a = 1 ∧ b = 1) = 1 2 sin2 1 2 θ ⃗a, ⃗ b ,
p ⃗a, ⃗ b,λ0
(a = 1 ∧ b = −1) = 1 2 cos2 1 2 θ ⃗a, ⃗ . (VII. 73)
b
EXERCISE 33. Also calculate the other two joint probabilities, that is, for a = 1 ∧ b = 1
and a = −1 ∧ b = 1.
The marginal probabilities are, using (VII. 73),
p ⃗a, ⃗ b
(a | λ 0 ) = p ⃗a, ⃗ b,λ0
(a = 1 ∧ b = 1) + p ⃗a, ⃗ b,λ0
(a = 1 ∧ b = −1) = 1 2 ,
p ⃗a, ⃗ b
(b | λ 0 ) = p ⃗a, ⃗ b,λ0
(a = 1 ∧ b = 1) + p ⃗a, ⃗ b,λ0
(a = −1 ∧ b = 1) = 1 2
, (VII. 74)
which means that, both being equal to 1 2
, they are not dependent of the settings of a remote measuring
device. Consequently, even the quantum mechanical correlations in the singlet cannot be used for
signaling, there is no actio in distans, leading to the following theorem.
NO - SIGNALING THEOREM:
Quantum mechanics satisfies parameter independence, i.e., if subsystems of a composite
physical system no longer interact, the probability of finding certain outcomes of measurement
for an arbitrary quantity of subsystem 1 is independent of which quantity of
subsystem 2 is measured, and vice versa.
EXERCISE 34. Prove that the EPRB experiment is an example of the no - signaling theorem.
Optional: prove, in general, the no - signaling theorem using state operators. Whoever cannot
solve this problem, is advised to consult Ghirardi, Rimini and Weber (1980).