# Uncertainties of genuinely incompatible triple measurements based on statistical distance

## TL;DR

This paper explores the uncertainties in triple quantum measurements using statistical distance, deriving tight, state-independent inequalities that differ from pairwise measurement bounds, especially for unbiased qubit measurements.

## Contribution

It introduces new uncertainty inequalities for triple measurements based on joint measurability, with analytical bounds for unbiased qubit measurements, highlighting differences from pairwise uncertainties.

## Key findings

- Derived tight, state-independent uncertainty inequalities for triple POVMs.
- Established analytical lower bounds for unbiased qubit measurements.
- Showed measurement uncertainties differ significantly from pairwise bounds.

## Abstract

We investigate the measurement uncertainties of a triple of positive operator-valued measures (POVMs) based on statistical distance, and formulate state-independent tight uncertainty inequalities satisfied by the three measurements in terms of triple-wise joint measurability. Particularly, uncertainty inequalities for three unbiased qubit measurements are presented with analytical lower bounds which relates to the necessary and sufficient condition of the triple-wise joint measurability of the given triple. We show that the measurement uncertainties for a triple measurement are essentially different from the ones obtained by pair wise measurement uncertainties by comparing the lower bounds of different measurement uncertainties.