# Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

## Abstract

**:**

## 1. Quantum ${\mathbb{F}}_{\mathit{un}}$, and ${\mathbb{F}}_{{\mathbf{1}}^{\ell}}$

- (A)
- $q+1$ points per line,
- (B)
- any two different points are contained in precisely one line, and
- (C)
- any two different intersecting lines are contained in one axiomatic projective plane of order q.

- (1)
- property (B);
- (2)
- any two different lines intersect in precisely one point and
- (3)
- there exist four points with no three of them on the same line.

- (A${}^{\prime}$)
- each line has 2 points;
- (B${}^{\prime}$)
- any two different points are contained in one unique line;
- (C${}^{\prime}$)
- each two different intersecting lines are contained in a unique triangle.

“Can one describe a quantum theory which “sees” (fundamental aspects in) all (actual, modal, general) quantum theories?”

#### 1.1. A Virtual Deletion Machine in all Quantum Theories

#### 1.2. Overview

## 2. A Quick Review of General Quantum Theory

#### 2.1. $(\sigma ,1)$-Hermitian Forms

#### 2.2. Hermitian Forms

- for all $a,b,c,d\in V$ we have that $\nu (a+b,c+d)=\nu (a,c)+\nu (b,c)+\nu (a,d)+\nu (b,d)$;
- for all $a,b\in V$ and $\alpha ,\beta \in k$, we have that $\nu (a\alpha ,b\beta )=\sigma \left(\alpha \right)\nu (a,b)\beta $.

#### 2.3. Standard $(\sigma ,1)$-Hermitian Forms

#### 2.4. The Unitary Group $\mathit{U}(V,\phi )$

#### 2.5. GQT

## 3. The Formalism over ${\mathbb{F}}_{1}$

#### 3.1. ${\mathbb{F}}_{1}$ and ${\mathbb{F}}_{{1}^{2}}$

#### 3.2. Vector Space over ${\mathbb{F}}_{{1}^{\ell}}$, and the Affine Viewpoint: Frames

#### 3.3. The Standard Form

#### 3.4. Orthogonality

#### 3.5. Time Evolution and Hermitian Operators

## 4. The Bigger Picture

#### 4.1. The Frobenius Maps

#### 4.2. $\mathtt{Aut}\left({\mathbb{F}}_{{1}^{\ell}}\right)$

#### 4.3. Involutions of the Fields ${\mathbb{F}}_{{1}^{\ell}}$

**Lemma**

**1.**

- SUB
- m divides $r(r+2)$;
- NTRIV
- m does not divide r.

**Proof.**

**Example**

**1.**

#### 4.4. Standard Examples of Absolute Quantum Theories

**Theorem**

**1**(Unitaries and observables)

**.**

#### 4.5. Orthogonality

## 5. Dictionary

## 6. One Cannot Clone an Unknown State in Absolute Quantum Theory

## 7. Quantum Deletion in the Absolute and Actual Context

**Theorem**

**2**(Quantum deletion by almost unitary operators—actual/classical)

**.**

**Theorem**

**3**(Quantum deletion by almost unitary operators—general)

**.**

**Theorem**

**4**(Quantum deletion by almost unitary operators—absolute)

**.**

#### 7.1. Deleting Probability

#### 7.1.1. Finite Case

#### 7.1.2. Over $\mathbb{C}$, and Other Fields/Division Rings

#### 7.1.3. Interpretation

The almost unitary operator $U:{\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}\mapsto {\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}$, where ${\mathcal{H}}_{A}$ and ${\mathcal{H}}_{B}$ are copies of the same Hilbert space (over any choice of division ring or field) virtually deletes all projective wave states.

#### 7.2. Cloning for Almost Unitary Operators

## 8. Conclusions

## Funding

## Conflicts of Interest

## References

- Chang, L.N.; Lewis, Z.; Minic, D.; Takeuchi, T. Galois field quantum mechanics. Modern Phys. Lett. B
**2013**, 27, 1350064. [Google Scholar] [CrossRef] - Chang, L.N.; Lewis, Z.; Minic, D.; Takeuchi, T. Spin and rotations in Galois field quantum mechanics. J. Phys. A
**2013**, 46, 065304. [Google Scholar] [CrossRef] [Green Version] - Chang, L.N.; Lewis, Z.; Minic, D.; Takeuchi, T. Quantum 𝔽
_{un}: The q = 1 limit of Galois field quantum mechanics, projective geometry, and the field with one element. J. Phys. A**2014**, 47, 405304. [Google Scholar] [CrossRef] - Lev, F.M. Introduction to a quantum theory over a Galois field. Symmetry
**2010**, 2, 1810–1845. [Google Scholar] [CrossRef] - Schumacher, B.; Westmoreland, M.D. Modal quantum theory. Found. Phys.
**2012**, 42, 918–925. [Google Scholar] [CrossRef] - Thas, K. Absolute Arithmetic and 𝔽
_{1}-Geometry; European Mathematical Society EMS: Zürich, Switzerland, 2016. [Google Scholar] - Hirschfeld, J.W.P. Projective Geometries over Finite Fields, 2nd ed.; Oxford Mathematical Monographs; Oxford University Press: New York, NY, USA, 1998. [Google Scholar]
- Shult, E.E. Points and Lines. Characterizing the Classical Geometries; Universitext; Springer: Heidelberg/Berlin, Germany, 2011. [Google Scholar]
- Thas, K. General quantum theory. arXiv, 2018; arXiv:1712.04669. [Google Scholar]
- Veblen, O.; Young, J.W. A set of assumptions for projective geometry. Am. J. Math.
**1908**, 30, 347–380. [Google Scholar] [CrossRef] - Wootters, W.K.; Zurek, W.H. A single quantum cannot be cloned. Nature
**1982**, 299, 802–803. [Google Scholar] [CrossRef] - Dieks, D. Communication by EPR devices. Phys. Lett. A
**1982**, 92, 271–272. [Google Scholar] [CrossRef] [Green Version] - Pati, A.K.; Braunstein, S.L. Impossibility of deleting an unknown quantum state. Nature
**2000**, 404, 164–165. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aleksandrova, A.; Borish, V.; Wootters, W.K. Real-vector-space quantum theory with a universal quantum bit. Phys. Rev. A
**2013**, 87, 052106. [Google Scholar] [CrossRef] - Wootters, W.K. Entanglement sharing in real-vector-space quantum theory. Found. Phys.
**2012**, 42, 19–28. [Google Scholar] [CrossRef] - Wootters, W.K. Fundamental Theories of Physics; Springer: Dordrecht, The Netherlands, 2016; Volume 181, pp. 21–43. [Google Scholar]
- Thas, K. Projective spaces over 𝔽
_{1ℓ}. J. Combin. Des.**2019**, 27, 55–74. [Google Scholar] [CrossRef] - Thas, K. The combinatorial-motivic nature of 𝔽
_{1}-schemes. In Absolute Arithmetic and 𝔽_{1}-Geometry; European Mathematical Society EMS: Zürich, Switzerland, 2016; pp. 83–159. [Google Scholar] - Wootters, W.K.; Zurek, W.H. The no-cloning theorem. Phys. Today
**2009**, 2, 76–77. [Google Scholar] [CrossRef]

Quantum Theory | k | $\mathit{\sigma}$ | ${\mathit{k}}_{\mathit{\sigma}}$ | Standard form $\u2329\overline{\mathit{x}}|\overline{\mathit{y}}\u232a$ |
---|---|---|---|---|

Actual quantum theory | $\mathbb{C}$ | $v\mapsto \overline{v}$ | $\mathbb{R}$ | $\overline{{x}_{1}}{y}_{1}+\cdots +\overline{{x}_{m}}{y}_{m}$ |

Modal quantum theory | ${\mathbb{F}}_{{q}^{2}}$ | $v\mapsto {v}^{q}$ | ${\mathbb{F}}_{q}$ | ${x}_{1}^{q}{y}_{1}+\cdots +{x}_{m}^{q}{y}_{m}$ |

General quantum theory | division ring with involution $\sigma $ | $\sigma $ | ${k}_{\sigma}$ | ${x}_{1}^{\sigma}{y}_{1}+\cdots +{x}_{m}^{\sigma}{y}_{m}$ |

Absolute quantum theory | ${\mathbb{F}}_{{1}^{r(r+2)}}$ | $v\mapsto {v}^{r+1}$ | ${\mathbb{F}}_{{1}^{r}}$ | ${x}_{1}^{r+1}{y}_{1}+\cdots +{x}_{m}^{r+1}{y}_{m}$ |

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**MDPI and ACS Style**

Thas, K.
Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion. *Symmetry* **2019**, *11*, 174.
https://doi.org/10.3390/sym11020174

**AMA Style**

Thas K.
Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion. *Symmetry*. 2019; 11(2):174.
https://doi.org/10.3390/sym11020174

**Chicago/Turabian Style**

Thas, Koen.
2019. "Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion" *Symmetry* 11, no. 2: 174.
https://doi.org/10.3390/sym11020174