Kochen–Specker Theorem for von Neumann Algebras

Abstract

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the nonexistence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I n factor as algebra of observables, including I ∞ . Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von Neumann algebra R without summands of types I 1 and I 2 , using a known result on two-valued measures on the projection lattice P(R). Some connections with presheaf formulations as proposed by Isham and Butterfield are made.

A topos foundation for theories of physics: I. Formal languages for physics

“…However, in the present paper, only one system at a time is being considered, and so the truncated notation is fine. 27 We recall that the objects in V(H) are the unital, commutative von Neumann subalgebras of the algebra, B(H), of all bounded operators on H. object, respectively.…”

“…Papers II and III in the series are concerned with quantum theory [1,2] which serves as a paradigmatic example for the general theory. These ideas are motivated by earlier work by one of us (CJI) and Butterfield on interpreting quantum theory in a topos [21,22,23,24,26,25]; see also [20,27].…”

Categorical Quantum Models and Logics

“…The previous theorem holds for more general C*-algebras than just Hilb(H, H) (for large enough Hilbert spaces H); see [74] for results on von Neumann algebras. A C*-algebra A is called simple when its closed two-sided ideals are trivial, and infinite when there is an a ∈ A with a * a = 1 but aa * = 1 [61].…”

On the Interpretation of Probabilities in Generalized Probabilistic Models

“…This means that, for a vast family of non-Kolmogorovian models, we will not be able to think about the elements of the event structure as representing actual properties of an individual system. And with regard to the algebraic formulation of physical probabilistic theories, a generalized version of the KS theorem exists for von Neumann algebras [40] (see also [41]). Due to the existence of these results, an interpretation based on bundles of actual properties for generalized probabilistic models seems to be problematic.…”