22 Noncommutative Mathematics for Quantum Systems
In the GNS representation of this state (see the lecture
‘Quantum dynamical systems from the point of view of
noncommutative mathematics’ by Adam Skalski in this volume
for an introduction to the GNS representation), the projections on
which the valuation takes the value 1 would leave the cyclic vector
invariant, whereas the projections on which the valuation takes
the value 0 would map the cyclic vector to zero, since
‖PΩ‖^2 =〈PΩ,PΩ〉=〈Ω,PΩ〉=φ(P)for any projectionP. As the linear span ofP(H)is weakly dense
inB(H), it would follow that the GNS representation ofφis one-
dimensional, which is absurd.
The classical proofs of the Kochen–Specker Theorem proceed by
finding a family of orthonormal bases ofHsuch that it is impossible
to assign to the basis vectors the values 0 and 1 in such a way that
any basis contains exactly one vector with the value 1. See [KS67,
GK13].
This theorem proves that there exist no ‘realistic non-contextual’
models of quantum probability. If quantum mechanics is correct,
as all experiments carried out so far seem to indicate, then it is
impossible to describe quantum systems in a realistic
non-contextual way by classical probability spaces.
It is possible to describe, e.g., the EPR experiment by acontextual
commutative probability space,cf.[Gil98], but in anon-contextual
model the measurement of one observable should not depend on
the choice that other independent measurements are carried out in
parallel.
The standard choice in conventional quantum physics is to give
up realism. The outcome of an experiment does not exist until we
have chosen which experiment we will carry out. In this sense the
randomness of an experiment described by a quantum probability
space has a different, more fundamental flavor than the way in
which classical probability spaces describe randomness. The
Wikipedia page [wiki Bell] gives more information on various
versions of Bell’s inequalities, for example, the CHSH inequality,
their practical experimental tests, their interpretation, and a few
references for further reading.