# Quantum mechanics as a noncommutative representation of classical conditional probabilities

## TL;DR

This paper explores how quantum mechanics can be reconstructed from classical conditional probabilities using noncommutative representations, showing that empirical data constrains these representations to align with quantum theory.

## Contribution

It demonstrates that increasing empirical data restricts noncommutative representations to uniquely match quantum mechanics, bridging classical probabilities and quantum formalism.

## Key findings

- Empirical data narrows noncommutative representations to quantum mechanics.
- Quantum representation emerges as the unique consistent model with sufficient data.
- Classical conditional probabilities can be embedded within a noncommutative framework that converges to quantum theory.

## Abstract

The aim of this paper is to analyze the reconstructability of quantum mechanics from classical conditional probabilities representing measurement outcomes conditioned on measurement choices. We will investigate how the quantum mechanical representation of classical conditional probabilities is situated within the broader frame of noncommutative representations. To this goal, we adopt some parts of the quantum formalism and ask whether empirical data can constrain the rest of the representation to conform to quantum mechanics. We will show that as the set of empirical data grows conventional elements in the representation gradually shrink and the noncommutative representations narrow down to the unique quantum mechanical representation.