Possibility of a minimal purity-measurement scheme critically depends on the parity of dimension of the quantum system

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.