FOUNDATIONS OF QUANTUM MECHANICS

158 CHAPTER VII. BELL’S INEQUALITIES
1. Outcome independence. In quantum mechanics it is indeed the requirement of outcome independence
that is not satisfied. The conditional probabilities, i.e., the probabilities for the spin of
particle 1 to be found in the direction ⃗a, given that the spin of particle 2 was found in the direction ⃗ b
and vice versa, which were defined in (III. 172), p. 74, are clearly not independent,
p ⃗a, ⃗ b
(a = 1 | λ 0 ∧ b = 1) = sin 2 1 2 θ ⃗a, ⃗ , (VII. 75)
b
p ⃗a, ⃗ b
(a = −1 | λ 0 ∧ b = 1) = cos 2 1 2 θ ⃗a, ⃗ . (VII. 76)
b
According to quantum mechanics, physical systems are inseparable. However, this interdependence
of outcomes cannot be used to exchange signals since we do not control the outcomes of spin measurements
and therefore we are unable to actively influence the probability distribution over the outcomes
of measurements from a distance. Shimony (1984, p. 227) called it passion at a distance. The experimenter
at B can, on the basis of his observation, indeed do a better prediction concerning an outcome
at A than that which is possible on just the knowledge of the singlet state, but he cannot warn the
observer at A, he only can watch passively.
The singlet |Ψ 0 ⟩ ∈ C 4 does violate the Bell inequalities for suitably chosen spin quantities.
The singlet is not factorizable, i.e., it cannot be written as a direct product of two states in C 2 , it is
entangled. We can raise the question if types of quantum mechanical states exist which do not violate
a Bell inequality for any choice of four spin quantities.
Capasso, Fortunato and Selleri (1973) proved that the CHSH inequality, (VII. 13), is upheld for
every choice of four spin quantities by all factorizable states and by all mixtures thereof. Violations
are therefore only possible for entangled states. Vice versa, Home and Selleri (1991, pp. 22 - 26)
proved that for every entangled pure state, that is, a state which cannot be written as a direct product,
it is always possible to choose spin quantities in such a way that the CHSH inequality is violated.
These results can be summarized in the statement that entanglement and violation of Bell inequalities
are equivalent. It confirms Schrödinger’s insight from 1935 (Schrödinger 1935a) that the
existence of entangled states marks the cardinal difference between classical and quantum mechanics.
VII. 6
AN ALGEBRAIC PROOF WITHOUT INEQUALITIES
The contradiction between a local deterministic or a local stochastic HVT, both either autonomous
or contextual, on the one hand, and quantum mechanics on the other hand, is statistical in nature, because
it concerns inequalities in terms of expectation values or probabilities, like all Bell’s theorems.
But Kochen and Specker’s theorem, which we discussed in V. 3, does not contain any inequalities. In
this case it is customary to speak of algebraic proof.
This raises the question whether an algebraic proof of Bell’s theorems is also possible, that is,
without appealing to the measurement postulate. The answer is affirmative. Using a spin state of a
composite system of four particles, D.M. Greenburger, M.A. Horn en A. Zeilinger (1989) showed
that it is mathematically impossible to locally and separably assign values to all spin quantities. Here
we will show a simplified version given by N.D. Mermin (1993), where, in using |GHZ⟩, we refer to
the aforementioned authors.