Single-qubit measurement of Heisenberg uncertainty lower bounds for three incompatible observables

TL;DR

This paper develops and experimentally demonstrates the first joint measurement of three incompatible quantum observables, advancing understanding of Heisenberg uncertainty bounds in quantum mechanics.

Contribution

It introduces new error trade-off relations for three incompatible observables and provides the first experimental verification using ultracold calcium ions.

Findings

Successful experimental witness of joint measurements for three incompatible observables

Development of error trade-off relations for three observables

Enhanced understanding of quantum measurement limits

Abstract

Being one of the centroidal concepts in quantum theory, the fundamental constraint imposed by Heisenberg uncertainty relations has always been a subject of immense attention and challenging in the context of joint measurements of general quantum mechanical observables. In particular, the recent extension of the original uncertainty relations has grabbed a distinct research focus and set a new ascendent target in quantum mechanics and quantum information processing. In the present work we explore the joint measurements of three incompatible observables, following the basic idea of a newly proposed error trade-off relation. In comparison to the counterpart of two incompatible observables, the joint measurements of three incompatible observables are more complex and of more primal interest in understanding quantum mechanical measurements. Attributed to the pristine idea proposed by Heisenberg in 1927, we develop the error trade-off relations for compatible observables to categorically approximate the three incompatible observables. Implementing these relations we demonstrate the first experimental witness of the joint measurements for three incompatible observables using a single ultracold $^{40}Ca^{+}$ ion in a harmonic potential. We anticipate that our inquisition would be of vital importance for quantum precision measurement and other allied quantum information technologies.