# Taxonomy of Three-Qubit Mermin Pentagrams

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## Abstract

**:**

## 1. Introduction

## 2. Finite Symplectic Polar Spaces, Fano Plane, and Generalized Pauli Groups

## 3. Three-Qubit Observables and Positive/Negative Fano Planes

## 4. Taxonomy of Mermin Pentagrams

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Mermin, N.D. Hidden variables and the two theorems of John Bell. Rev. Modern Phys.
**1993**, 65, 803. [Google Scholar] [CrossRef] - Kochen, S.; Specker, E.P. The problem of hidden variables in quantum mechanics. J. Math. Mech.
**1967**, 17, 59. [Google Scholar] [CrossRef] - Planat, M.; Saniga, M.; Holweck, F. Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon. Quantum Inf. Process.
**2013**, 12, 2535. [Google Scholar] [CrossRef] [Green Version] - Lévay, P.; Szabó, Z. Mermin pentagrams arising from Veldkamp lines for three qubits. J. Phys. A Math. Theor.
**2017**, 50, 95201. [Google Scholar] [CrossRef] [Green Version] - Saniga, M.; Lévay, P. Mermin’s pentagram as an ovoid of PG(3, 2). EPL Europhys. Lett.
**2012**, 97, 50006. [Google Scholar] [CrossRef] - Cameron, P.J. Projective and Polar Spaces. QMW Maths Notes, 13; School of Mathematical Sciences, Queen Mary and Westfield College: London, UK, 1992. [Google Scholar]
- Havlicek, H.; Odehnal, B.; Saniga, M. Factor-group-generated polar spaces and (multi-)qudits. Symmetry Integr. Geom. Methods Appl.
**2009**, 5, 96. [Google Scholar] [CrossRef] [Green Version] - Thas, K. The geometry of generalized Pauli operators of N-qudit Hilbert space. EPL Europhys. Lett.
**2009**, 86, 60005. [Google Scholar] [CrossRef] - Planat, M.; (Institut FEMTO-ST, Besançon, France). Personal Communication, 2019.
- Borsten, L.; Duff, M.; Lévay, P. The black-hole/qubit correspondence: an up-to-date review. Class. Quantum Gravity
**2012**, 29, 224008. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**A representative of the family of negative Fano planes (top) and those of three distinct types of positive Fano planes. Negative lines are shown in bold. An observable of type A, B, and C is colored red, green, and yellow, respectively. Note that the three observables of type B lie always on a line. (We use the traditional picture of the Fano plane where one of its lines is drawn as a circle.)

**Figure 2.**A Mermin pentagram (drawn in the center) and the associated pentad of labeled Fano planes. The Fano plane at the top corresponds to the horizontal edge of the pentagram; the remaining correspondences follow readily from the rotational symmetry of the figure. Both the negative contexts of the pentagram and the negative lines of three Fano planes are boldfaced. Note that one of the negative contexts corresponds to a positive Fano plane. The Fano planes are labelled by the three-qubit observables in such a way that the closure line of the affine plane (the ‘line at infinity’) is always represented by the circle.

**Table 1.**Refined geometric classification of Mermin pentagrams. Column one (T) shows the type, column two (${C}^{-}$) the number of negative contexts in a pentagram of the given type, columns three to five (${O}_{A}$ to ${O}_{C}$) indicate the number of observables of corresponding types, column six (${F}^{-}$) the number of negative Fano planes and columns seven to nine (${F}_{a}^{+}$ to ${F}_{c}^{+}$) the distribution of types of positive Fano planes. Finally, the last column (K) indicates the number of pentagrams of a given type that lie on the ‘symmetric’ Klein quadric.

T | ${\mathit{C}}^{-}$ | ${\mathit{O}}_{\mathit{A}}$ | ${\mathit{O}}_{\mathit{B}}$ | ${\mathit{O}}_{\mathit{C}}$ | ${\mathit{F}}^{-}$ | ${\mathit{F}}_{\mathit{a}}^{+}$ | ${\mathit{F}}_{\mathit{b}}^{+}$ | ${\mathit{F}}_{\mathit{c}}^{+}$ | K |
---|---|---|---|---|---|---|---|---|---|

1 | 5 | 0 | 0 | 10 | 5 | 0 | 0 | 0 | 2 |

2 | 5 | 1 | 0 | 9 | 3 | 2 | 0 | 0 | 0 |

3 | 3 | 0 | 5 | 5 | 4 | 0 | 1 | 0 | 0 |

4 | 3 | 0 | 4 | 6 | 5 | 0 | 0 | 0 | 0 |

5 | 3 | 0 | 4 | 6 | 3 | 2 | 0 | 0 | 6 |

6 | 3 | 0 | 4 | 6 | 3 | 1 | 1 | 0 | 0 |

7 | 3 | 0 | 4 | 6 | 3 | 0 | 2 | 0 | 6 |

8 | 3 | 1 | 0 | 9 | 3 | 0 | 2 | 0 | 0 |

9 | 3 | 1 | 4 | 5 | 3 | 2 | 0 | 0 | 0 |

10 | 3 | 1 | 4 | 5 | 3 | 1 | 0 | 1 | 12 |

11 | 3 | 1 | 4 | 5 | 3 | 0 | 2 | 0 | 0 |

12 | 3 | 1 | 4 | 5 | 3 | 0 | 1 | 1 | 6 |

13 | 3 | 1 | 4 | 5 | 3 | 0 | 0 | 2 | 6 |

14 | 3 | 1 | 5 | 4 | 2 | 2 | 1 | 0 | 0 |

15 | 3 | 1 | 5 | 4 | 2 | 1 | 1 | 1 | 12 |

16 | 3 | 2 | 5 | 3 | 2 | 1 | 0 | 2 | 12 |

17 | 3 | 2 | 4 | 4 | 1 | 2 | 2 | 0 | 0 |

18 | 3 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 12 |

19 | 3 | 3 | 0 | 7 | 1 | 2 | 1 | 1 | 6 |

20 | 3 | 3 | 4 | 3 | 1 | 2 | 0 | 2 | 6 |

21 | 1 | 0 | 5 | 5 | 4 | 1 | 0 | 0 | 0 |

22 | 1 | 0 | 5 | 5 | 2 | 2 | 1 | 0 | 12 |

23 | 1 | 0 | 5 | 5 | 2 | 1 | 2 | 0 | 12 |

24 | 1 | 0 | 4 | 6 | 3 | 1 | 1 | 0 | 12 |

25 | 1 | 1 | 4 | 5 | 3 | 0 | 2 | 0 | 0 |

26 | 1 | 1 | 4 | 5 | 3 | 0 | 1 | 1 | 6 |

27 | 1 | 1 | 4 | 5 | 3 | 0 | 0 | 2 | 6 |

28 | 1 | 1 | 4 | 5 | 1 | 2 | 1 | 1 | 18 |

29 | 1 | 1 | 4 | 5 | 1 | 0 | 3 | 1 | 6 |

30 | 1 | 1 | 5 | 4 | 2 | 2 | 0 | 1 | 12 |

31 | 1 | 1 | 5 | 4 | 2 | 1 | 2 | 0 | 0 |

32 | 1 | 1 | 5 | 4 | 2 | 1 | 1 | 1 | 12 |

33 | 1 | 1 | 5 | 4 | 2 | 0 | 2 | 1 | 12 |

34 | 1 | 2 | 5 | 3 | 2 | 1 | 0 | 2 | 12 |

35 | 1 | 2 | 5 | 3 | 2 | 0 | 1 | 2 | 24 |

36 | 1 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 12 |

37 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 24 |

38 | 1 | 3 | 0 | 7 | 1 | 0 | 3 | 1 | 2 |

39 | 1 | 3 | 4 | 3 | 1 | 1 | 1 | 2 | 12 |

40 | 1 | 3 | 4 | 3 | 1 | 0 | 2 | 2 | 6 |

41 | 1 | 3 | 5 | 2 | 0 | 2 | 1 | 2 | 12 |

42 | 1 | 3 | 5 | 2 | 0 | 1 | 2 | 2 | 12 |

43 | 1 | 4 | 4 | 2 | 1 | 1 | 0 | 3 | 12 |

44 | 1 | 4 | 4 | 2 | 1 | 0 | 1 | 3 | 12 |

45 | 1 | 6 | 0 | 4 | 1 | 0 | 0 | 4 | 2 |

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**MDPI and ACS Style**

Saniga, M.; Holweck, F.; Jaffali, H.
Taxonomy of Three-Qubit Mermin Pentagrams. *Symmetry* **2020**, *12*, 534.
https://doi.org/10.3390/sym12040534

**AMA Style**

Saniga M, Holweck F, Jaffali H.
Taxonomy of Three-Qubit Mermin Pentagrams. *Symmetry*. 2020; 12(4):534.
https://doi.org/10.3390/sym12040534

**Chicago/Turabian Style**

Saniga, Metod, Frédéric Holweck, and Hamza Jaffali.
2020. "Taxonomy of Three-Qubit Mermin Pentagrams" *Symmetry* 12, no. 4: 534.
https://doi.org/10.3390/sym12040534