FOUNDATIONS OF QUANTUM MECHANICS

140 CHAPTER VII. BELL’S INEQUALITIES
from each other and do not interact, it follows, according to EPR, that the result of a measurement
of any spin component is determined in advance, i.e., that it is an element of physical reality. This
suggests that there there should be a more complete description of the state of the particles, including
hidden variables.
Specify this description of the pair of particles with variables λ ∈ Λ as we did in chapter V. We
write the quantities corresponding to (⃗σ 1·⃗a)⊗(⃗σ 2·⃗b) as the pair (A, B), having values a,b = ±1. In a
contextual HVT, these values are dependent on the hidden variable λ and the total measuring context,
which can be specified here by means of the measurement directions ⃗a and ⃗ b, leading to
A = A(⃗a, ⃗ b, λ) and B = B(⃗a, ⃗ b, λ). (VII. 1)
Now the essential assumption is the requirement of locality that the quantity A does not depend
on the reading ⃗ b of a remote spin meter, and vice versa for B and ⃗a. These quantities therefore only
depend upon the local context,
A(⃗a, ⃗ b, λ) = A(⃗a, λ), a = ±1,
B(⃗a, ⃗ b, λ) = B( ⃗ b, λ), b = ±1. (VII. 2)
b = +1
b = −1
B( ⃗ b, λ)
A(⃗a, λ)
a = +1
a = −1
b ′ = +1
b ′ = −1
B( ⃗ b ′ , λ)
A(⃗a ′ , λ)
a ′ = +1
a ′ = −1
Spin meter B
ρ(λ)
Spin meter A
Source
Figure VII. 1: Thought experiment of Einstein, Podolsky and Rosen on the singlet
The source emitting the particle pairs probably does not prepare the pairs in the same state λ each
time. We assume that the source can be characterized by a probability density ρ,
∫
ρ(λ) dλ = 1, (VII. 3)
Λ
where we also assume that this probability density does not depend on the measuring directions ⃗a
and ⃗ b, which, after all, can be established long after the particles have left the source. The expectation
value of the product of A and B in this HVT is therefore
∫
E(⃗a, ⃗ b) = A(⃗a, λ) B( ⃗ b, λ) ρ(λ) dλ. (VII. 4)
Λ