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

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.