Kochen-Specker Sets and Generalized Orthoarguesian 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.

16 pages and 5 figures

References in corpus (3)

• Quantum Scholasticism: On Quantum Contexts, Counterfactuals, and the Absurdities of Quantum Omniscience
• Graph Approach to Quantum Systems
• Hilbert Lattice Equations

Cited by in corpus (9)

• Quantum Contextuality
• Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces
• Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets
• Automated generation of Kochen-Specker sets
• Hypergraph Contextuality
• Non-Kochen-Specker Contextuality
• A natural deduction system for orthomodular logic
• Vector Generation of Contextual Sets
• Quantum Contextual Hypergraphs, Operators, Inequalities, and Applications in Higher Dimensions