Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces
Abstract
:1. Introduction
2. Results
- (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
2.2. KS Vector Lists vs. Vector Component MMP Hypergraphs
2.3. Vector-Component-Generated Hypergraph Masters
- (i)
- the input set of components for generating two-qubit KS hypergraphs (4D) should contain number pairs of opposite signs, e.g., , and zero (0); we conjecture that the same holds for 3, 4, ...qubits; with 6D it does not hold literally; e.g., generate a KS master; however, the following combination of ’s gives the opposite sign to 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 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 generate the 216-153 6D master which contains just three criticals 21-7, 27-9, and 33-11, generate 834-1609 master from which we obtained criticals, and generate 11808-314446 master from which we obtained 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 generate 332-1408 6D master which contains the 236-1216 list-master while originally components 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
Appendix B. 6D Criticals from the Masters Containing the Seven Context Star.
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dim | Master Size | Vector List | List Origin | Smallest Hypergraph | Vector Components |
---|---|---|---|---|---|
4D | 24-24 | [25,42,43] | symmetry, geometry | | {0,} |
4D | 60-105 | [28,37] | Pauli operators | | {0,} |
4D | 60-75 | [27,30,37,41] | regular polytope 600-cell | | |
4D | 148-265 | [36,37] | Witting polytope | | |
6D | 21-7 | [19] | symmetry | | |
6D | 236-1216 | Aravind & Waegell 2016, [37] | hypercube→hexaract Schäfli | | |
8D | 36-9 | [37] | symmetry | | |
8D | 120-2025 | [35,37] | Lie algebra E8 | | as given in [35] |
16D | 80-265 | [37,44,45] | Qubit states | | |
32D | 160-661 | [37,46] | Qubit states | |
dim | Vector Components | Component-Master Size | No of KS Criticals in Master | Smallest Hypergraph | Contains List-Masters |
---|---|---|---|---|---|
4D | {0,} or {0,} or or ... | 40-32 | 6 | | 24-24 |
4D | {0,} | 156-249 | | 24-24, 60-105 | |
4D | 2316-3052 | | 24-24, 60-75 | ||
4D | 400-1012 | | 24-24, 60-105 148-265 | ||
6D | 11808-314446 | | 21-7, 236-1216 | ||
8D | 3280-1361376 | | 36-9, 120-2025 | ||
16D | computationally too demanding | [33]. | 80-265 | ||
32D | computationally too demanding | [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
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 StylePavičić, 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