A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations

TL;DR

This paper introduces a topos-theoretic framework for quantum states, providing a new type of valuation that assigns multi-valued, contextual truth-values to quantum propositions, addressing the Kochen-Specker theorem's limitations.

Contribution

It develops a novel topos-based approach to quantum valuations, allowing all operators to be assigned generalized, contextual truth-values consistent with quantum theory.

Findings

Kochen-Specker theorem is equivalent to a presheaf having no global sections.

Quantum states induce generalized valuations as sieves of operators.

The approach offers a multi-valued, contextual logic for quantum propositions.

Abstract

The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.