# Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Results

**Theorem**

**1**(MMP hypergraph reformulation of the Kochen–Specker theorem)

**.**

- (α)
- No two vertices within any of its edges are both assigned the value 1;
- (β)
- In any of its edges, not all of the vertices are assigned the value 0.

#### 2.1. Formalism

`1 2 ...9 A B ...Z a b ...z ! " #`$ % & ’ ( ) * - / : ; < = > ? @ [ ∖ ] _ ‘ { | } ~ [26]. When all of them are exhausted, one reuses them prefixed by ‘+’, then again by ‘++’, and so forth. An n-dimensional KS set with k vectors and m n-tuples is represented by an MMP hypergraph with k vertices and m edges which we denote as a k-m set. In its graphical representation, vertices are depicted as dots and edges as straight or curved lines connecting m orthogonal vertices. We handle MMP hypergraphs by means of algorithms in the programs SHORTD, MMPSTRIP, MMPSUBGRAPH, VECFIND, STATES01, and others [5,30,38,39,40,41]. In its numerical representation (used for computer processing), each MMP hypergraph is encoded in a single line in which all m edges are successively given, separated by commas, and followed by assignments of coordinatization to k vertices (see 18-9 in Section 2.2).

#### 2.2. KS Vector Lists vs. Vector Component MMP Hypergraphs

**18-9**:

`1234,4567,789A,ABCD,DEFG,GHI1,I29B,35CE,68FH.`

`{1={0,0,0,1},2={0,0,1,0},3={1,1,0,0},4={1,-1,0,0},5={0,0,1,1},6={1,1,1,-1},`

`7={1,1,-1,1}, 8={1,-1,1,1},9={1,0,0,-1},A={0,1,1,0},B={1,0,0,1},C={1,-1,1,-1},`

`D={1,1,-1,-1},E={1,-1,-1,1},F={0,1,0,1},G={1,0,1,0},H={1,0,-1,0},I={0,1,0,0}}.`

#### 2.3. Vector-Component-Generated Hypergraph Masters

`-master`, the program builds an internal list of all possible non-zero vectors containing these components. From this list, it finds all possible edges of the hypergraph, which it then generates. MMPSTRIP via its option

`-U`separates unconnected MMP subgraphs. We pipe the obtained hypergraphs through the program STATES01 to keep those that possess the KS property. We can use other programs of ours, MMPSTRIP, MMPSHUFFLE, SHORTD, STATES01, LOOP, etc., to obtain smaller KS subsets and analyze their features.

- (i)
- the input set of components for generating two-qubit KS hypergraphs (4D) should contain number pairs of opposite signs, e.g., $\pm 1$, and zero (0); we conjecture that the same holds for 3, 4, ...qubits; with 6D it does not hold literally; e.g., $\{0,1,\omega \}$ generate a KS master; however, the following combination of $\omega $’s gives the opposite sign to 1: $\omega +{\omega}^{2}=-1$;
- (ii)
- mixing real and complex components gives a denser distribution of smaller KS hypergraphs;
- (iii)
- reducing the number of components shortens the time needed to generate smaller hypergraphs and apparently does not affect their distribution.

- As for the features (ii) and (iii) above, components $\{0,\pm 1,\omega \}$ generate the master 180-203 which has the following smallest criticals 18-9, 20...22-11, 22...26-13, 24...30-15, 30...31-16, 28...35-17, 33...37-18, etc. This distribution is much denser than that of, e.g., the list-master 24-24 with real vectors which in the same span of edges consists only of 18-9, 20-11, 22-13, and 24-15 criticals or of the list-master 60-75 which starts with the 26-13 critical. In Appendix A, we give a detailed description of a 21-11 critical with a complex coordinatization and give a blueprint for its experimental implementation;
- In [19], the reader is challenged to find a master set which would contain the "seven context star" 21-7 KS critical (shown in Table 1 and Table 2). We find that $\{0,1,\omega \}$ generate the 216-153 6D master which contains just three criticals 21-7, 27-9, and 33-11, $\{0,1,\omega ,{\omega}^{2}\}$ generate 834-1609 master from which we obtained $2.5\times {10}^{7}$ criticals, and $\{0,\pm 1,\omega ,{\omega}^{2}\}$ generate 11808-314446 master from which we obtained $3\times {10}^{7}$ criticals, all of them containing the seven context star. Some of the obtained criticals are given in Appendix B;
- The 60-75 list-master contains criticals with up to 41 edges and 60 vertices, while the 2316-3052 component-master generated from the same vector components contains criticals with up to close to 200 edges and 300 vertices;
- The 60-105 list-master contains criticals with up to 40 edges and 60 vertices, while the 156-249 component-master generated from the same vector components contains criticals with up to at least 58 edges and 88 vertices;
- Components $\{0,\pm 1\}$ generate 332-1408 6D master which contains the 236-1216 list-master while originally components $\{0,\pm 1/2,\pm 1/\sqrt{3},\pm 1/\sqrt{2},1\}$ were used;
- In [37], we generated 6D criticals with up to 177 vertices and 87 edges from the list-master 236-1216, while, now, from the component-master 11808-314446, we obtain criticals with up to 201 vertices and 107 edges;

## 3. Methods

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

KS | Kochen–Specker; defined in Section 1 |

MMP | McKay-Megill-Pavičić; defined in Section 2.1 |

## Appendix A. 21-11 KS Critical with Complex States from ${\mathcal{H}}^{\mathbf{2}}\otimes {\mathcal{H}}^{\mathbf{2}}$

`8`from Figure A1. Since we are interested in the qubit states, we are going to proceed in reverse—from 4-vectors to tensor products of polarization and angular momentum states. Let us first define them:

## Appendix B. 6D Criticals from the Masters Containing the Seven Context Star.

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**Figure A2.**21-11 KS set from [19] and 27-9 are contained in three different master sets, 39-13 in two (together with 21-11 and 27-9); see the text.

**Table 1.**Vector lists from the literature; we call their masters list-masters. We shall make use of their vector components from the last column to generate master hypergraphs in Section 2.3 which we call component-masters. $\omega $ is a cubic root of unity: $\omega ={e}^{2\pi i/3}$.

dim | Master Size | Vector List | List Origin | Smallest Hypergraph | Vector Components |
---|---|---|---|---|---|

4D | 24-24 | [25,42,43] | symmetry, geometry | {0,$\pm 1$} | |

4D | 60-105 | [28,37] | Pauli operators | {0,$\pm 1,\pm i$} | |

4D | 60-75 | [27,30,37,41] | regular polytope 600-cell | $\{0,\pm (\sqrt{5}-1)/2,\pm 1,\pm (\sqrt{5}+1)/2,2\}$ | |

4D | 148-265 | [36,37] | Witting polytope | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\{0,\pm i,\pm 1,\pm \omega ,\pm {\omega}^{2},\pm i{\omega}^{1/\sqrt{3}},\pm i{\omega}^{2/\sqrt{3}}\}$ | |

6D | 21-7 | [19] | symmetry | $\{0,1,\omega ,{\omega}^{2}\}$ | |

6D | 236-1216 | Aravind & Waegell 2016, [37] | hypercube→hexaract Schäfli $\{4,{3}^{4}\}$ | $\{0,\pm 1/2,\pm 1/\sqrt{3},\pm 1/\sqrt{2},1\}$ | |

8D | 36-9 | [37] | symmetry | $\{0,\pm 1\}$ | |

8D | 120-2025 | [35,37] | Lie algebra E8 | as given in [35] | |

16D | 80-265 | [37,44,45] | Qubit states | $\{0,\pm 1\}$ | |

32D | 160-661 | [37,46] | Qubit states | $\{0,\pm 1\}$ |

**Table 2.**Component-masters we obtained. List-masters are given in Table 1. In the last two rows of all but the last column, we refer to the result [33] that there are 16D and 32D criticals with just nine edges. According to the conjectured feature (i) above, the masters generated by $\{0,\pm 1\}$ should contain those criticals; they did not come out in [37], so, we do not know how many vertices they have. The smallest ones we obtained are given in Table 1. The number of criticals given in the 4th column refer to the number of them we successfully generated although there are many more of them except in the 40-32 class.

dim | Vector Components | Component-Master Size | N^{o} of KS Criticals in Master | Smallest Hypergraph | Contains List-Masters |
---|---|---|---|---|---|

4D | {0,$\pm 1$} or {0,$\pm i$} or $\{0,\pm (\sqrt{5}-1)/2\}$ or ... | 40-32 | 6 | 24-24 | |

4D | {0,$\pm 1,\pm i$} | 156-249 | $7.7\times {10}^{6}$ | 24-24, 60-105 | |

4D | $\{0,\pm (\sqrt{5}-1)/2,\pm 1,\pm (\sqrt{5}+1)/2,2\}$ | 2316-3052 | $1.5\times {10}^{9}$ | 24-24, 60-75 | |

4D | $\{0,\pm 1,\pm i,\pm \omega ,\pm {\omega}^{2}\}$ | 400-1012 | $8\times {10}^{6}$ | 24-24, 60-105 148-265 | |

6D | $\{0,\pm 1,\omega ,{\omega}^{2}\}$ | 11808-314446 | $3\times {10}^{7}$ | 21-7, 236-1216 | |

8D | $\{0,\pm 1\}$ | 3280-1361376 | $7\times {10}^{6}$ | 36-9, 120-2025 | |

16D | $\{0,\pm 1\}$ | computationally too demanding | $4\times {10}^{6}$ | [33]. | 80-265 |

32D | $\{0,\pm 1\}$ | computationally too demanding | $2.5\times {10}^{5}$ | [33]. | 160-661 |

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Pavičić, M.; Megill, N.D.
Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces. *Entropy* **2018**, *20*, 928.
https://doi.org/10.3390/e20120928

**AMA Style**

Pavičić M, Megill ND.
Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces. *Entropy*. 2018; 20(12):928.
https://doi.org/10.3390/e20120928

**Chicago/Turabian Style**

Pavičić, Mladen, and Norman D. Megill.
2018. "Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces" *Entropy* 20, no. 12: 928.
https://doi.org/10.3390/e20120928