Complexity of Satisfiability in Kochen-Specker Partial Boolean Algebras

TL;DR

This paper investigates the computational complexity of propositional satisfiability in partial Boolean algebras related to quantum mechanics, revealing NP-completeness, existential theory of reals completeness, and undecidability results.

Contribution

It establishes the complexity classifications of satisfiability problems in various classes of partial Boolean algebras derived from quantum structures, using Kochen-Specker sets as key tools.

Findings

NP-complete for non-trivial partial Boolean algebras

Complete for the existential theory of the reals in finite-dimensional cases

Undecidable for all Hilbert spaces

Abstract

The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiablity for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiablity problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.