Independence and L ́evy Processes in Quantum Probability 211.3.3 The Kochen–Specker theorem
We will now review the Kochen–Specker theorem, which again
states that it is impossible to assign values to quantum mechanical
observables independently of a choice of measurements.
Definition 1.3.3 A valuation (or ‘dispersion-free’ additive
probability measure) is an additive probability measure
μ:P(H)→{0, 1}.
A valuation is a function that assigns to any event either the
value ‘1’ (=‘true’) or the value ‘0’ (=‘false’) in a consistent way, that
is, in any collection of mutually orthogonal events there is at most
one that is true. In a realisticmodel of a quantum system the
outcome of any possible experiment should exist no matter which
experiment we will carry out. In anon-contextualrealistic model
these outcomes should not depend on the ‘context’, that is, on the
choice of other independent experiments. E.g., in the EPR
experiment the spin measured for one particle should not depend
on the choice of the direction of the spin measurement carried out
on the other distant particle. In this case one speaks also of alocal
model.
Non-contextual realistic models with ‘hidden variables’ are
similar to classical probabilistic models. If we knowω ∈ Ωis
realized, then we can predict the outcome of all possible
experiments; however, there might be obstacles that make it
impossible to know the values of the hidden variables.
The Kochen–Specker Theorem shows that it is actually
impossible to assign consistently truth values to the events on a
Hilbert space with dimension bigger than or equal to three, that is,
that such hidden variables not only cannot be known, but that
they cannot exist. In our context it says that there cannot exist any
classical probability space with events that correspond to the
events of a quantum mechanical probability space(B(H),φ)with
dimH≥3.
Theorem 1.3.4 (Kochen–Specker theorem)[nlab KS] Letdim(H)≥ 3.
Then there exists no valuation onP(H).
If such a valuationμ : P(H) → {0, 1}existed, then it would
extend to a stateφonB(H), by Gleason’s theorem. Let us assume
that either dim(H)<∞or thatμisσ-additive, so that the stateφis
normal.