  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.
