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

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