# Quantum Probability’s Algebraic Origin

## Abstract

**:**

## 1. Introduction

## 2. Quantum Logic and States

## 3. Transition Probability

**Theorem**

**1.**

- (i)
- The transition probability from p to q exists and $\mathbb{P}\left(q\right|p)=r$ iff the linear operators p and q satisfy the simple algebraic identity$$pqp=rp.$$
- (ii)
- If $p\ne 0\ne q$ holds and if both transition probabilities $\mathbb{P}\left(q\right|p)$ and $\mathbb{P}\left(p\right|q)$ exist, they are equal:$$\mathbb{P}\left(q\right|p)=\mathbb{P}(p\left|q\right).$$

**Proof.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Quantum Indeterminacy

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

- Beltrametti, E.G.; Cassinelli, G.; Rota, G.-C. The Logic of Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- Birkhoff, G.; von Neumann, J. The logic of quantum mechanics. Ann. Math.
**1936**, 37, 823–843. [Google Scholar] [CrossRef] - Varadarajan, V.S. Geometry of Quantum Theory; Van Nostrand Reinhold: New York, NY, USA, 1968; Volume 1. [Google Scholar]
- Niestegge, G. Statistische und deterministische Vorhersagbarkeit bei der quantenphysikalischen Messung. Helv. Phys. Acta
**1998**, 71, 163–183. [Google Scholar] - Niestegge, G. Non-Boolean probabilities and quantum measurement. J. Phys. Math. Gen.
**2001**, 34, 6031. [Google Scholar] [CrossRef][Green Version] - Christensen, E. Measures on projections and physical states. Commun. Math. Phys.
**1982**, 86, 529–538. [Google Scholar] [CrossRef] - Gleason, A.M. Measures on the closed subspaces of a Hilbert space. J. Math. Mech.
**1957**, 6, 885–893. [Google Scholar] [CrossRef] - Sakai, S. C*-Algebras and W*-Algebras; Springer: Berlin/Heidelberg, Germany, 1971. [Google Scholar]
- Freedman, M.; Shokrian-Zini, M.; Wang, Z. Quantum computing with octonions. Peking Math. J.
**2019**, 2, 239–273. [Google Scholar] [CrossRef][Green Version] - Bell, J.S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys.
**1966**, 38, 447–452. [Google Scholar] [CrossRef] - Cabello, A.; Estebaranz, J.; García-Alcaine, G. Bell-Kochen-Specker theorem: A proof with 18 vectors. Phys. Lett. A
**1996**, 212, 183–187. [Google Scholar] [CrossRef][Green Version] - Kochen, S.; Specker, E.P. The problem of hidden variables in quantum mechanics. J. Math. Mech.
**1967**, 17, 59–87. [Google Scholar] [CrossRef] - Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys.
**1927**, 43, 172–198. [Google Scholar] [CrossRef] - Robertson, H.P. The uncertainty principle. Phys. Rev.
**1929**, 34, 163–164. [Google Scholar] [CrossRef] - Schrödinger, E. Zum Heisenbergschen Unschärfeprinzip. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl.
**1930**, 14, 296–303. [Google Scholar] - Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics; Springer: Berlin/Heidelberg, Germany, 1979; Volume 1. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Niestegge, G. Quantum Probability’s Algebraic Origin. *Entropy* **2020**, *22*, 1196.
https://doi.org/10.3390/e22111196

**AMA Style**

Niestegge G. Quantum Probability’s Algebraic Origin. *Entropy*. 2020; 22(11):1196.
https://doi.org/10.3390/e22111196

**Chicago/Turabian Style**

Niestegge, Gerd. 2020. "Quantum Probability’s Algebraic Origin" *Entropy* 22, no. 11: 1196.
https://doi.org/10.3390/e22111196