FOUNDATIONS OF QUANTUM MECHANICS

VII. 5. STOCHASTIC HIDDEN VARIABLES 155
Using (VII. 64), another Bell inequality can be derived for E(⃗a, ⃗ b) by means of the relation
∫
E(⃗a, ⃗ (
b) = p⃗a, ⃗ b
(1, 1, λ) − p ⃗a, ⃗ b
(1, −1, λ) (VII. 66)
Defining
Λ
− p ⃗a, ⃗ b
(−1, 1, λ) + p ⃗a, ⃗ b
(−1, −1, λ) dλ )
∫
(
= p⃗a (1 | λ) − p ⃗a (−1 | λ) ) ( p ⃗b (1 | λ) − p ⃗b (−1 | λ) ) ρ(λ) dλ.
Λ
f (⃗a, λ) := p ⃗a (1 | λ) − p ⃗a (−1 | λ) (VII. 67)
and
g( ⃗ b, λ) := p ⃗b (1 | λ) − p ⃗b (−1 | λ), (VII. 68)
we see that
|f (⃗a, λ)| 1 and |g( ⃗ b, λ)| 1, (VII. 69)
which brings us back to (VII. 25) and the subsequent equations so that again we obtain the Bell
inequality (VII. 13). Violation of this Bell inequality means that (VII. 64) can not apply and
therefore no HVT can guarantee both outcome independence (VII. 61) and parameter independence
(VII. 62). □
VII. 5. 1
OUTCOME, PARAMETER AND SOURCE INDEPENDENCE
The importance of the distinction between outcome and parameter independence was first brought
to attention by J. Jarrett (1984).
1. Outcome independence, (VII. 61), means that the probability of outcome b, for given λ, does
not depend on the outcome a. This is motivated by the idea that λ gives a complete description of
the state of the pair of particles; the variable λ contains an exhaustive specification of all factors
which are relevant for the outcomes of measurement. Therefore, specifying the extra information that
outcome a has occurred can, if λ is already known, not lead to new information on b.
The purpose of the requirement can be illustrated by giving the next example, in which it is not
satisfied. Suppose that two people, without looking, each draw a little ball out of a box containing two
little balls, one black and one white. Hereafter they separate, one travels to New York, the other to
Tokyo. Now consider a ‘stochastic hidden variable’ with probability 1 2
for the little balls to be black
or white. On arrival at Tokyo the traveler opens his hand and sees that his little ball is black, which
instantaneously enables him to predict the color of the little ball in New York, it has to be white. Here
the outcome of measurement of the one little ball does provide relevant information on the outcome
of a measurement of the other little ball.