# The Born Rule is a Feature of Superposition

**Authors:** David Ellerman  
**arXiv:** [2310.04188](https://arxiv.org/abs/2310.04188)  
**Date:** 2023-10-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2310.04188v2/

## Key points

- Finite probability theory is enriched by introducing the concept of a superposition event $\Sigma S$
- Superposition events are defined based on commonalities and variations among elements in a set
- Density matrices are used to distinguish between discrete events and superposition events
- The density matrix for a discrete event $S$ is diagonal, while for a superposition event $\Sigma S$, it is obtained as an outer product of a normalized vector
- Probabilities are defined using density matrices as $\Pr(T|\rho) = \operatorname*{tr}[P_T\rho]$
- The Born Rule gives the probability of an outcome conditioned by a superposition event
- For singleton subsets, the probability is given by $\Pr(\left\{u_i\right\}|\Sigma S) = \left\langle u_i|s \right\rangle^2$
- When using complex numbers, the square becomes an absolute square: $\left| \left< u_i | s \right > \right | ^2$
- Introducing superposition events enriches finite probability theory and allows for extracting commonalities from sets while allowing variation among elements.

## Summary

Finite probability theory is enriched by introducing the mathematical notion of a superposition event $\Sigma S$, in addition to the usual discrete event $S$. The superposition event $\Sigma S$ is defined on the properties that are common to the elements of $S$, but is indefinite where the elements of $S$ differ. This concept of superposition can be seen as extracting from a set an entity that is definite on the commonalities of the elements and indefinite where they differ. To mathematically distinguish between discrete events and superposition events, density matrices are used. For a discrete event $S$, the density matrix $\rho(S)$ is diagonal, while for a superposition event $\Sigma S$, the density matrix $\rho(\Sigma S)$ is obtained as an outer product of a normalized vector $\left| s \right\rangle \in \mathbb{R}^n$. Probabilities are defined using density matrices as $\Pr(T|\rho) = \operatorname*{tr}[P_T\rho]$, where $T$ is a subset of the outcome space $U=\left(u_1,...,u_n\right)$ and $P_T$ is the diagonal projection matrix with diagonal entries $\chi_T(u_i)$. The Born Rule, which gives the probability of an outcome conditioned by a superposition event, arises naturally from this mathematical framework. Specifically, for a singleton subset $\left\{u_i\right\} \subseteq U$, the probability of obtaining outcome $u_i$ conditioned by the superposition event $\Sigma S$ is given by $\Pr(\left\{u_i\right\}|\Sigma S) = \left\langle u_i|s \right\rangle^2$. It should be noted that when using complex numbers instead of real numbers in this mathematical framework ($\mathbb{C}^n$ instead of $\mathbb{R}^n$), this square becomes an absolute square: 
    $\left| \left< u_i | s \right >  \right | ^2$. In summary, introducing superposition events to finite probability theory enriches its mathematical framework and gives rise to the Born Rule, which provides probabilities for outcomes conditioned by such events. This concept allows us to extract commonalities from sets while allowing for variation among their elements.