Experimental test of Heisenberg's measurement uncertainty relation based on statistical distances

TL;DR

This paper experimentally tests a refined Heisenberg measurement uncertainty relation for qubits, showing that the worst-case inaccuracy is fundamentally limited by observable incompatibility, using joint measurements on a spin system.

Contribution

It reformulates and improves the theoretical inaccuracy trade-off relation for qubits and validates it through experimental joint measurements.

Findings

The worst-case measurement inaccuracy is tightly bounded by observable incompatibility.

Experimental results confirm the improved theoretical relation.

Joint measurements of compatible but non-commutative observables verify the bounds.

Abstract

Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [Phys. Rev. Lett. 111, 160405 (2013); Phys. Rev. A 89, 012129 (2014)]. Here we reform their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible but typically non-commutative observables on one qubit are measured simultaneously.