  The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years
ago, set a fundamental limitation on the joint measurability of complementary
observables. This limitation has long been a subject of debate, which has been
reignited recently due to new proposed forms of measurement uncertainty
relations. The present work is associated with a new error trade-off relation
for compatible observables approximating two incompatible observables, in
keeping with the spirit of Heisenberg's original ideas of 1927. We report the
first \textsl{direct} test and confirmation of the tight bounds prescribed by
such an error trade-off relation, based on an experimental realisation of
optimal joint measurements of complementary observables using a single
ultracold $^{40}Ca^{+}$ ion trapped in a harmonic potential. Our work provides
a prototypical determination of ultimate joint measurement error bounds with
potential applications in quantum information science for high-precision
measurement and information security.
