# 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

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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