Testing Heisenberg's measurement uncertainty relation of three observables

TL;DR

This paper reports the first experimental verification of Heisenberg's measurement uncertainty relations for three quantum observables using a single-photon qubit, combining theoretical derivation, numerical optimization, and experimental implementation.

Contribution

It extends Heisenberg's uncertainty principle to three observables, providing a rigorous theoretical framework, numerical methods, and experimental validation.

Findings

Established MURs for three observables with a lower bound based on incompatibility

Developed a convex programming protocol for optimal measurements

Successfully tested the MURs experimentally with a single-photon qubit

Abstract

Heisenberg's measurement uncertainty relations (MUR) of two quantum observables are essential for contemporary researches in quantum foundations and quantum information science. Going beyond, here we report the first experimental test of MURs for three quantum observables. Following the proposal of Bush, Lahti, and Werner [Phys. Rev. A 89, 012129 (2014)], we first establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower-bounded by an incompatibility measure. We then develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the corresponding optimal measurements. Furthermore, we propose a novel implementation of optimal joint measurements and experimentally test our MURs using a single-photon qubit. Lastly, we discuss to analytically calculate the exact value of incompatibility measure for some symmetric triplets. We anticipate that this work may stimulate broad interests associated with the Heisenberg's uncertainty relation of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.