# Kochen-Specker theorem and measurement map in the formalism of QM

There is (or rather was) a famous debate between Born and Einstein about interpretational issues of Quantum Mechanics. In this debate Einstein was advocating the so called theory of hidden variables. There are various no-go theorems ruling out the possibility of the existence of such hidden variables. One possible formulation of existence of hidden variables is as follows: there should exist a measurable space $$(\Omega,\mathcal{F})$$ and two maps $$F: \mathcal{O} \to L^{\infty}(\Omega), A \mapsto F_A$$ (where $$L^{\infty}(\Omega)$$ is the space of all bounded measurable functions on $$\Omega$$ and $$\mathcal{O}$$ is the space of all self-adjoint operators on a Hilbert space $$\mathcal{H}$$) and $$\rho: \mathcal{S} \to \mathcal{M}(\Omega), \psi \mapsto \rho^{\psi}$$ (from the space of unit vectors in $$\mathcal{H}$$ (where $$dim(\mathcal{H})>2$$) to the space of probability measures on $$\Omega$$). These maps should satisfy two conditions:
a) $$F_{u(A)}=u \circ F_A$$ for every real valued measurable bounded function $$u$$ and $$A \in \mathcal{O}$$
b) $$\mu^{\psi}_A(\Delta)=\rho^{\psi}(F_A^{-1}(\Delta))$$. One possible formulation of the Kochen-Specker theorem states that such two maps do not exist: in fact Kochen-Specker theorem states that there is no valuation from $$\mathcal{O}$$ to $$\mathbb{R}$$ (where valuation is by definition a scalar valued map which satisfies property a) as above).

Now let me mention another (whether is it really another problem will be the content of my question) problem, namely: there is no honest mapping $$\mathcal{O} \times \mathcal{S} \to \mathbb{R}$$ within the formalism of QM which would correspond to the outcome of a measurement of an observable in the given state. Suppose that actually there are hidden variables and after all QM can be viewed clasically (which is not the case)

What is then the correct definition of the ,,measurement map'' (involving states and hidden variables)?

So to be clear at this point: I know that there are no hidden variables and also there is no reasonable measurement map. However what am I aiming at is as follows: I would like to see a natural definition of the measurement map (involving hidden variables) and then prove a theorem: ,,such measurement map exists $$\iff$$ there exists a valuation on $$\mathcal{O}$$ (or any equivalent statement of Kochen-Specker theorem)-therefore as Kochen-Specker forbids the existence of valuation thus such a measurement map cannot exist''.

• "I know that there are no hidden variables [...]" Isn't Bohmian mechanics an example of a hidden variable theory, albeit a nonlocal and contextual one? Feb 8, 2022 at 17:52