  In this paper we give a new way to quantify the folklore notion that quantum
measurements bring a disturbance to the system being measured. We consider two
observers who initially assign identical mixed-state density operators to a
two-state quantum system. The question we address is to what extent one
observer can, by measurement, increase the purity of his density operator
without affecting the purity of the other observer's. If there were no
restrictions on the first observer's measurements, then he could carry this out
trivially by measuring the initial density operator's eigenbasis. If, however,
the allowed measurements are those of finite strength---i.e., those
measurements strictly within the interior of the convex set of all
measurements---then the issue becomes significantly more complex. We find that
for a large class of such measurements the first observer's purity increases
the most precisely when there is some loss of purity for the second observer.
More generally the tradeoff between the two purities, when it exists, forms a
monotonic relation. This tradeoff has potential application to quantum state
control and feedback.
