  In this paper, we investigate the possibility of measuring the purity of a
quantum state (and the overlap between two quantum states) within a minimal
model where the measurement device is minimally composed. The minimality is
based on the assumptions that (i) we use a yes-no measurement on a single
system to determine the single value of the purity in order not to extract
other redundant information, and (ii) we use neither ancilla nor random
measurement. We show that the measurability of the purity within this model
critically depends on the parity of dimension of the quantum system: the purity
measurement is possible for odd dimensional quantum systems, while it is
impossible for even dimensional cases.
