Kochen-Specker Sets and Generalized Orthoarguesian Equations

TL;DR

This paper investigates Kochen-Specker sets using lattice representations satisfying generalized orthoarguesian equations, clarifies misconceptions about their diagrammatic representations, and demonstrates the strict inclusion of nOA classes, advancing the understanding of quantum logic structures.

Contribution

It shows how Kochen-Specker sets can be represented by lattices satisfying nOA, corrects previous diagrammatic misconceptions, and proves the strict inclusion of 6OA and 7OA classes.

Findings

Peres' KS set is a lattice where 6OA passes and 7OA fails.

The 7OA class properly contains the 6OA class.

nOA form an infinite hierarchy of stronger equations.

Abstract

Every set (finite or infinite) of quantum vectors (states) satisfies generalized orthoarguesian equations ($n$OA). We consider two 3-dim Kochen-Specker (KS) sets of vectors and show how each of them should be represented by means of a Hasse diagram---a lattice, an algebra of subspaces of a Hilbert space--that contains rays and planes determined by the vectors so as to satisfy $n$OA. That also shows why they cannot be represented by a special kind of Hasse diagram called a Greechie diagram, as has been erroneously done in the literature. One of the KS sets (Peres') is an example of a lattice in which 6OA pass and 7OA fails, and that closes an open question of whether the 7oa class of lattices properly contains the 6oa class. This result is important because it provides additional evidence that our previously given proof of noa =< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form an infinite sequence of successively stronger equations.