Measurement uncertainty relation for three observables

We establish rigorously in this work a measurement uncertainty relation (MUR) for three unbiased qubit observables, which was previously shown to hold true under some presumptions. The triplet MUR states that the uncertainty, which is quantified by the total statistical distance between the target observables and the jointly implemented observables, is lower bounded by an incompatibility measure that reflects the joint measurement conditions. We derive a necessary and sufficient condition for the triplet MUR to be saturated and the corresponding optimal measurement. To facilitate experimental tests of MURs we propose a straightforward implementation of the optimal joint measurements. Lastly, by a symmetry argument, the exact values of incompatibility measure are analytically calculated for some symmetric triplets. We anticipate that our work may enrich the understanding of quantum incompatibility in terms of MURs and inspire further applications in quantum information science. This work presents a complete theory relevant to a parallel work [Y.-L. Mao et al., Testing Heisenberg-type measurement uncertainty relations of three observables, Phys. Rev. Lett. 131, 150203 (2023)] on experimental tests.

Figure 1

Target triplet ${\mathcal{M}}_{\perp }$ together with its symmetries and the optimal triplet ${\mathcal{N}}_{\perp }$ with the same symmetries.

Figure 2

Target triplet ${\mathcal{M}}_Y$ together with its symmetries and the optimal triplet ${\mathcal{N}}_Y$ with the same symmetries.

Figure 3

Exact values of incompatibility for two triplets of ideal qubit measurements with symmetries, (a) triplet ${\mathcal{M}}_{\perp }$ with $|cos2\gamma |={\vec{m}}_1·{\vec{m}}_2$ and (b) highly symmetric triplet ${\mathcal{M}}_Y$ of unbiased observables (see main text for details) according to analytical results computed with Eqs. (14) and (19), respectively. Green smooth lines: analytical; red dashed lines: numerical.