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:1207.5982

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

S. P. Toh

Published Jul 25, 2012• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1207.5982

Related papers

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

Kochen-Specker Sets with Thirty Rank-Two Projectors in Three-Qubit System

S. P. Toh

A simple three rules supplemented by five steps scheme is proposed to produce Kochen-Specker (KS) sets with 30 rank-2 projectors that occur twice each. The KS sets provide state-independent proof of KS theorem based on a system of three qubits. A small adjustment of the scheme enables us to manually generate a large number of KS sets with a mixture of rank-1 and rank-2 projectors.

arXiv:1304.6166•Apr 23, 2013•0 citations

Construction of Three-Qubit Kochen-Specker Sets

S. P. Toh

Contextuality is one of the fundamental deviations of quantum mechanics from classical physics.The Kochen-Specker theorem shows that non-contextual classical physics with hidden variables is inconsistent with the predictions of quantum mechanics.Parity proof is one of the many different approaches applied to prove Kochen-Specker theorem for quantum system with different dimensionality.This method of proof requires Kochen-Specker sets that were composed of an odd number of bases and an even number of projectors.In most of the cases, the number of Kochen-Specker sets produced is large and its production is generally aided by computer calculation.However, manual generation of the Kochen-Specker sets are also reported previously for qutrit and two-qubit systems.We put forward the first and surprisingly simple method to generate manually Kochen-Specker sets, with respect to the Mermin pentagram, that can be used for the parity proof of the Kochen-Specker theorem in a state space of eight-dimensional three-qubit system.

arXiv:1603.08282•Acta Physica Polonica A•Mar 1, 2016•0 citations

Proofs of the Kochen-Specker theorem based on the N -qubit Pauli group

Mordecai Waegell, P. K. Aravind

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the $N$-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has $O$ observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2${}^{O}$. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the $N$-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form ${2}^{N}$ for $N\ensuremath{\ge}2$. Some implications of our results are discussed.

arXiv:1302.4801•Physical Review A•Jul 1, 2013•33 citations

Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System

S. P. Toh

Kochen—Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors was proposed in 1995 [Kernaghan M and Peres A Phys. Lett. A 198 (1995) 1 ] for a three-qubit system, there have been plenty of the same type KS sets and we propose a systematic way to produce them. We also propose a probabilistic state-dependent proof of the KS theorem that mainly focuses on the values assignment for all the rank-2 projectors.

arXiv:1310.0179•Chinese Physics Letters•Oct 1, 2013•5 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
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.
Contextuality is one of the fundamental deviations of quantum mechanics from classical physics.The Kochen-Specker theorem shows that non-contextual classical physics with hidden variables is inconsistent with the predictions of quantum mechanics.Parity proof is one of the many different approaches applied to prove Kochen-Specker theorem for quantum system with different dimensionality.This method of proof requires Kochen-Specker sets that were composed of an odd number of bases and an even number of projectors.In most of the cases, the number of Kochen-Specker sets produced is large and its production is generally aided by computer calculation.However, manual generation of the Kochen-Specker sets are also reported previously for qutrit and two-qubit systems.We put forward the first and surprisingly simple method to generate manually Kochen-Specker sets, with respect to the Mermin pentagram, that can be used for the parity proof of the Kochen-Specker theorem in a state space of eight-dimensional three-qubit system.
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the NNN-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has OOO observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2O{}^{O}O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the NNN-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form 2N{2}^{N}2N for $N\ensuremath{\ge}2$. Some implications of our results are discussed.
Kochen—Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors was proposed in 1995 [Kernaghan M and Peres A Phys. Lett. A 198 (1995) 1 ] for a three-qubit system, there have been plenty of the same type KS sets and we propose a systematic way to produce them. We also propose a probabilistic state-dependent proof of the KS theorem that mainly focuses on the values assignment for all the rank-2 projectors.
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.