# Geometric Measure of non-Commuting Simultaneous Measurement based on K-Means Clustering

## TL;DR

This paper introduces a geometric measure to quantify the non-commuting behavior of quantum measurements, linking it to initial and final states and discussing its relation to Heisenberg's uncertainty principle.

## Contribution

It proposes a novel geometric measure for non-commuting quantum measurements based on initial and final states, enhancing understanding of quantum measurement behavior.

## Key findings

- The measure is demonstrated to be rational and meaningful.
- Application examples illustrate its utility.
- Connections to Heisenberg's uncertainty principle are discussed.

## Abstract

Considering the simultaneous measurement of non-commuting observables, we define a geometric measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. The rationality of our geometric measure is demonstrated and the application of it is presented. The connection between our measure and Heisenberg's uncertainty principle is discussed as well. Our work deepens the understanding of quantum non-commuting measurement.