Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1109.1299 · Journal of Physics A Mathematical and Theoretical

Parity proofs of the Kochen–Specker theorem based on 60 complex rays in four dimensions

Mordecai Waegell, P K Aravind

Published Nov 23, 2011• 32 citations
arXivHTMLPDF

Abstract

It is pointed out that the 60 complex rays in four dimensions associated with a system of two qubits yield over 10 9 critical parity proofs of the Kochen–Specker theorem. The geometrical properties of the rays are described, an overview of the parity proofs contained in them is given and examples of some of the proofs are exhibited.
DOI
10.1088/1751-8113/44/50/505303

Related papers

Generating the Kochen-Specker Sets with 36, 38 and 40 Rays in Three-qubit System: Three Simple Algorithms

S. P. Toh

We put forward three simple algorithms to generate Kochen-Specker sets used for parity proof of Kochen-Specker theorem in three-qubit system. These algorithms enables us to generate 320, 640 and 64 Kochen-Specket sets with 36, 38 and 40 rays, respectively. No any computer calculation is required, every step in the algorithms is determined by the method of picking the rays that repeat 4 times in the KS sets.

arXiv:1207.5982•Jul 25, 2012•0 citations

Parity proofs of the Kochen–Specker theorem based on the Lie algebra E8

Mordecai Waegell, P K Aravind

The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real eight-dimensional Hilbert space that contains a large number of parity proofs of the Kochen–Specker (KS) theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their KS diagram in the form of a 'basis table' showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the KS theorem found to date in eight-dimensions.

arXiv:1502.04350•Journal of Physics A Mathematical and Theoretical•May 14, 2015•21 citations

Three Simple Algorithms to Generate Kochen-Specker Sets with Thirty Six, Thirty Eight and Forty Rays in Three-qubit System

S. P. Toh

We put forward three simple algorithms to generate Kochen-Specker sets used for parity proof of Kochen-Specker theorem in three-qubit system. These algorithms enables us to generate 320, 640 and 64 Kochen-Specket sets with 36, 38 and 40 rays, respectively. No any computer calculation is required, every step in the algorithms is determined by the method of picking the rays that repeat 4 times in the KS sets.

•Jul 25, 2012•0 citations

Triacontagonal proofs of the Bell–Kochen–Specker theorem

P. K. Aravind, Justin Y. J. Burton, Guillermo Nuñez Ponasso +1 more

Abstract Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset’s polytope 4 21 in eight dimensions. We show how these projections can be modified into Kochen–Specker diagrams from which parity proofs of the Bell–Kochen–Specker theorem are easily extracted. Our construction trivially yields parity proofs of fifteen bases for all three polytopes and also allows many other proofs of the same type to be constructed for two of them. The defining feature of these proofs is that they have a fifteen-fold symmetry about the center of the Kochen–Specker diagram and thus involve both rays and bases that are multiples of fifteen. Any proof of this type can be written as a word made up of an odd number of distinct letters, each representing an orbit of fifteen bases. Knowing the word representing a proof makes it possible to infer all its characteristics without first having to recover its bases. A comparison is made with earlier approaches that have been used to obtain parity proofs in these polytopes, and some directions in which this work can be extended are discussed.

arXiv:2510.01063•Journal of Physics A Mathematical and Theoretical•Mar 23, 2026•0 citations

Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem

P. K. Aravind, Forest Lee-Elkin

It is demonstrated that the 60 rays corresponding to antipodal pairs of vertices of the 600-cell, and the 300 rays corresponding to antipodal pairs of vertices of the 120-cell, can both be used to give noncolouring proofs of the Bell-Kochen-Specker theorem in four dimensions.

Journal of Physics A Mathematical and General•Dec 11, 1998•23 citations

Proofs of the Kochen–Specker theorem based on a system of three qubits

Mordecai Waegell, P. K. Aravind

A number of new proofs of the Kochen–Specker theorem are given based on the observables of the three-qubit Pauli group. Each proof is presented in the form of a diagram from which it is obvious by inspection. Each of our observable-based proofs leads to a system of projectors and bases that generally yields a large number of ‘parity proofs’ of the Kochen–Specker theorem. Some examples of such proofs are given and some of their applications are discussed.

arXiv:1205.5015•Journal of Physics A Mathematical and Theoretical•Sep 18, 2012•44 citations
The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real eight-dimensional Hilbert space that contains a large number of parity proofs of the Kochen–Specker (KS) theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their KS diagram in the form of a 'basis table' showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the KS theorem found to date in eight-dimensions.
Abstract Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset’s polytope 4 21 in eight dimensions. We show how these projections can be modified into Kochen–Specker diagrams from which parity proofs of the Bell–Kochen–Specker theorem are easily extracted. Our construction trivially yields parity proofs of fifteen bases for all three polytopes and also allows many other proofs of the same type to be constructed for two of them. The defining feature of these proofs is that they have a fifteen-fold symmetry about the center of the Kochen–Specker diagram and thus involve both rays and bases that are multiples of fifteen. Any proof of this type can be written as a word made up of an odd number of distinct letters, each representing an orbit of fifteen bases. Knowing the word representing a proof makes it possible to infer all its characteristics without first having to recover its bases. A comparison is made with earlier approaches that have been used to obtain parity proofs in these polytopes, and some directions in which this work can be extended are discussed.