  We experimentally study the heating of trapped atomic ions during measurement
of their internal qubit states. During measurement, ions are projected into one
of two basis states and discriminated by their state-dependent fluorescence. We
observe that ions in the fluorescing state rapidly scatter photons and heat at
a rate of $\dot{\bar{n}}\sim 2\times 10^4$ quanta/s, which is $\sim 30$ times
faster than the anomalous ion heating rate. We introduce a quantum
trajectory-based framework that accurately reproduces the experimental results
and provides a unified description of ion heating for both continuous and
discrete sources.
